Content uploaded by Ger Koper

Author content

All content in this area was uploaded by Ger Koper on Jun 04, 2020

Content may be subject to copyright.

48

G.J.M. Koper

on Molecular

Thermodynamics

Perspectives

1

G.J.M. Koper

Perspectives

on Molecular

Thermodynamics

NUR 100

ISBN 9789464022582

Tekst en Inhoud: G.J.M. Koper

Omslagontwerp: Margo Togni

Omslagbeeld: NASA/JPL/University of Arizona

3

Preface

It was in the beginning of last year that a good colleague prof. Hans Geerlings urged me to write down

my thoughts on thermodynamics and its role in chemical technology. So far, two of my courses1 already

appeared as printed lecture notes and adding one more did not seem a tremendous job. The same

procedure could be used of first teaching the class and subsequently writing down the text. While

contemplating this idea a student came by to discuss a rather confusing article by Deepak Dhar, a

professor in Mumbai, on the “Enigma of Entropy”2. Around the same time, also some students assisting

in a molecular thermodynamics course told me they would like to have more background information

on the topic they were teaching. All in all sufficient reason to seriously consider to follow the colleague’s

suggestion.

The course “Advanced Molecular Thermodynamics” was conceived and presently given for the second

time. It will be the last time also since my retirement is near and therefore these lecture notes are now

made public. The topics dealt with are all centred around the concepts of entropy and the fluctuation

dissipation theorem. The chapters are very sketchy and should serve as a primer: in no way would it

be possible to go into more depth in the seven course lectures in one quarter of an academic year. As

background material served the course material that colleagues and myself prepared in earlier days.

There are no literature references at all. The interested reader may easily find more information using

standard searching methods on the world wide web.

In closing, let me thank my colleague Hans who read and commented on all text and the Technical

University of Delft for their support.

Ger Koper,

Leiden, April 2020.

1 An introduction to Interfacial Engineering, G.J.M. Koper, VSSD, 2007; An Introduction to Chemical Thermodynamics, G.J.M.

Koper, VSSD 2007.

2 The Enigma of Entropy, Deepak Dhar, Resonance, October 2012.

5

Contents

1. Who cares about the Second Law of Thermodynamics 7

2. Thermodynamics my way – illustrated by fuel cells 15

3. Why should a chemical engineer know about molecular thermodynamics 21

4. On the brink of chaos 25

5. The Fluctuation Dissipation Theorem: another guise of the 2nd Law? 31

6. What follows: Boltzmann’s equation from Navier-Stokes’ equations or vice versa? 37

7. Use and Abuse of the Fokker Planck and Langevin Equations 43

7

7

Who cares about the Second Law of Thermodynamics

Systems view

Like every student who entered a science or technology education I was duly exposed to the

Laws of Thermodynamics. In the beginning it was not so clear to me why people made such a

big fuzz about it. Surely, it is some notion that “heat can never pass from a colder body without

some external help” as the Clausius formulation of the Second Law – loosely – reads. But so

what? In that sense the Kelvin formulation is a bit more mysterious. It – again loosely – reads

that it is impossible to “derive a mechanical effect from a portion of matter by cooling it below

the temperature of the surroundings”. In those days, it did not seem to bear on something I

would encounter in the near future. It sounded like something mechanical as would have to do

with steam engines where – as I understood – thermodynamics was used for.

The Second Law is closely related to another obscure topic, that of entropy. I do not recall a

clear understanding of that entity from the beginning of my education. But at least there

appeared to be some quantitative statement that could be made about it. This is Clausius’

equality but that turned out to be an inequality which did not make it clearer. It reads

Q

ST

'

't

(1.1)

where S stands for entropy, measured in J/K, Q for exchanged heat, measured in J, and T for

absolute temperature, measured in K. The

'

stands for a difference, usually between the end

and the beginning of a process or process step. The equality sign holds for reversible

processes and for any natural process, being irreversible, the greater-than sign holds. The

exercises we were to do in those days all concerned reversible processes so that sense could

be made out of this equation although we did not really gather an intuitive feeling for the

magnitude of the entropy. All in all, Clausius’ inequality brought some clarity. It could be used

to grade students following the course in Thermodynamics: once exercises were done

sufficiently well, they could pass. Whether students had any understanding of the matter or not

appeared not to be relevant to me.

There are quite a number of issues related to Clausius’ inequality. The first question that comes

to mind would be “when is a process reversible” or irreversible for that matter? Then, what is

the difference between the entropy change

'

S and the heat exchange divided by temperature,

'

Q/T? And finally, what about the Entropy of the Universe constantly increasing? It implies that

negative entropy fluctuations would be forbidden, but how? In the following sections I will try

to address these issues and come to a tentative conclusion without entering into molecular

details. In the subsequent part I will deal with Boltzmann’s hypothesis on entropy which allows

a molecular view on entropy.

On reversibility

Let us first look at the kind of standard processes as are used in thermodynamics teaching,

the compression, decompression, cooling and heating of gases. This is not at all a stupid

example, provided you have the real life example in mind. My favorite is the bicycle pump. The

teaching examples require you to close off the outlet so that inside the pump you can have a

fixed portion of air. Air is sufficiently dilute so that for simple examples as the ones we shall be

discussing here it behaves according to the ideal gas laws, another obscure notion. What an

ideal gas is, will come later in this chapter. For here it suffices to say that the behavior of the

air inside the bicycle pump can be captured by the Sackur-Tetrode equation,

53ln ln

2m

n

S nR V

ªº

/

«»

¬¼

(1.2)

Remarkably, it is a formula for the absolute entropy S of the gas as a function of the number,

n, of moles, the molar volume Vm and the de Broglie length

/

of which we only need to know

8

8

here that it inversely depends on the square root of absolute temperature T. I n Figure 1.1

possible behavior of the gas has been depicted. The blue line shows the result of an isothermal

experiment which can be achieved to a good approximation if we move the piston of the bicycle

pump relatively slowly. For instance, at rest the pump contains 1 liter of air at environmental

pressure of 1 atm. We can then pus the piston down by about 20% so that the volume is

reduced to 0.8 liter. The pressure than will rise to 1.25 atm which we feel by some force exerted

on our hand that keeps the piston down. We could also pull out the piston by some 20% so

that the gas volume increases to 1.2 liter. The pressure then becomes 0.83 atm which we

notes by a pulling force on the hand that keeps the piston up. The associated entropy changes

are indicated in the graph, negative for compression and positive for decompression. These

small changes are related to heat exchanges with the environment in order to maintain the

temperature, compression provides heat to and decompression takes heat from the exterior.

We can also do the experiment in another way: very fast but controlled movements of the

piston. This would be a good approximation of an adiabatic process. When we compress the

air in the pump very fast, there is not yet heat exchange with the environment and the entropy

remains the same as it was when the bicycle pump was at rest. But very quickly we notice a

heating up of the pump and it takes a while until the thing is at room temperature again. At

constant volume then the pressure releases a bit until the pressure we had when we did the

isothermal compression experiment as described above. Likewise for the decompression

experiment.

Figure 1.1 Gas behavior during isothermal and

adiabatic volume changes.

98

8

here that it inversely depends on the square root of absolute temperature T. I n Figure 1.1

possible behavior of the gas has been depicted. The blue line shows the result of an isothermal

experiment which can be achieved to a good approximation if we move the piston of the bicycle

pump relatively slowly. For instance, at rest the pump contains 1 liter of air at environmental

pressure of 1 atm. We can then pus the piston down by about 20% so that the volume is

reduced to 0.8 liter. The pressure than will rise to 1.25 atm which we feel by some force exerted

on our hand that keeps the piston down. We could also pull out the piston by some 20% so

that the gas volume increases to 1.2 liter. The pressure then becomes 0.83 atm which we

notes by a pulling force on the hand that keeps the piston up. The associated entropy changes

are indicated in the graph, negative for compression and positive for decompression. These

small changes are related to heat exchanges with the environment in order to maintain the

temperature, compression provides heat to and decompression takes heat from the exterior.

We can also do the experiment in another way: very fast but controlled movements of the

piston. This would be a good approximation of an adiabatic process. When we compress the

air in the pump very fast, there is not yet heat exchange with the environment and the entropy

remains the same as it was when the bicycle pump was at rest. But very quickly we notice a

heating up of the pump and it takes a while until the thing is at room temperature again. At

constant volume then the pressure releases a bit until the pressure we had when we did the

isothermal compression experiment as described above. Likewise for the decompression

experiment.

Figure 1.1 Gas behavior during isothermal and

adiabatic volume changes.

9

The above story I told to many colleague scientists – typically during a dinner or at the drinking

table. But then, the important point comes. both these processes were done in an almost

reversible manner. That is to say: at each moment of time, the system passed through an

equilibrium point determined by pressure, volume and temperature. As a proof: the whole

process can be put down as lines in the pressure-volume diagram. Can you – the respected

scientist – now draw me an irreversible process path in this diagram. Typically and after some

time, a wriggly line is drawn in the diagram. At that moment I comment again, that the diagram

is the collection of points where the gas is in equilibrium at given pressure and volume so that

any line in that plane necessarily is an equilibrium process however bizarre its path. In order

to draw an irreversible path one should leave the diagram at least one time. I could imagine,

for instance, that an another axis is drawn, perpendicular to both pressure and volume, that

represents the dissipation that the system undergoes during the process. In that 3-dimensional

space I then could draw an irreversible process.

In actual fact, every process that is performed with the bicycle pump follows a path at varying

levels of dissipation depending on the rate at which it is done. But when done sufficiently

slowly, one remain so close to the plane of no dissipation – which is the equilibrium plane –

that it is as if it follows the equilibrium path. Clearly, this equilibrium path is the projection of

the irreversible path on the equilibrium plane! A reversible process does not exist in reality, it

is the limit of a series of processes that occur at less and less dissipation and hence at slower

and slower rates.

Examples of irreversible processes with the bicycle pump are abundant. For instance,

compress it first to the smallest volume one may realize and then just take off your hands. The

piston will move erratically, presumably even make oscillatory motion. What happens exactly

inside the gas is not easy to describe. Further on, we shall try to give more insight into this.

There is, however, an important aspect of thermodynamics to appreciate. If the initial point and

the final point of the process are taken the same, the entropy change involved in any process

between these points remains the same. The exchange of heat with the environment will –

after the process – be exactly the same no matter how the process is performed. So, the

dissipation will lead to an excess entropy change that must somewhere along the process path

be compensated by a heat exchange. For instance, if during one of these irreversible

processes a lot of heat is produced in the gas due to friction this heat will make the gas hotter

when it has come more or less to rest. From then on, heat is exchanged with the environment

until the system is equilibrated. The volume of the gas follows.

Maximum entropy principle

A lot of presentations on this came by before I started to appreciate the topic: how and why a

system optimizes to maximum entropy? Fluctuations that increase entropy are the only ones

that are allowed? Are there no negative entropy fluctuations then? How is this achieved.

Surely, Nature has many intricate ways of solving its “problems” but usually the causes as well

as the methods are clear in themselves albeit that it could take some time to uncover.

At last, I realized that if there is something that needs to be optimized – such as entropy –

there needs to be some handle. The fluctuations have to be in some variable. Well, that gives

a hint. Assume we have again a gas in a container, such as we dealt with above in the bicycle

pump. This time, we assume there is a divider in the container that for allows for heat

exchange, but not for volume changes or gas exchanges. The while container is closed and

isolated from the environment so that any heat that leaves the one compartment has to be

taken up by the other compartment. Of course, the gases in the two compartment are

considered homogenous at all times.

It is now not too difficult to see that the process can be followed by monitoring the temperature

difference between the two compartments; we expect it with time to vanish … on average.

However, at any moment of time the temperature difference will be slightly different from zero.

A heat flux will flow from the one compartment to the other to correct for that, etc. So,

10

10

temperature difference is a fluctuating quantity with zero average and some finite standard

deviation. What does the entropy of the whole system do now. Well, because of the heat

exchanges there will also be entropy changes for the compartments. A simple analysis teaches

us

12

12

12

12

12

11

0

dS dS dS

SS

dU dU

UU

QQ

TT

ww

ww

(1.3)

In the first line we use the fact that the two compartments are making up the total system and

hence the total entropy of the system is the sum of the entropies of the two compartments.

Likewise for changes in the entropy. The only way – in the present example – that the entropy

can change is by a change in the energy content U of the system. In the third line, we identify

the change in entropy with a change in energy as the inverse temperature. The changes in

energy of the compartments are the heat exchanges – no work can be done – and when a

heat exchange Q is positive for the one compartment it is negative for the other. So far so

good. We can now use the experimental fact that in such an experiment the temperatures of

the compartments become the same – on average - and that hence the total entropy reaches

an extremum. We can also use the Second Law formulation that the entropy maximizes and

that hence the temperature difference tends to vanish. If we wish to know what kind of

extremum the entropy will reach one should evaluate the second order derivative of entropy

versus energy. It will be negative for equal compartment temperatures and hence entropy

indeed reaches a maximum.

The above reasoning can be continued for temperature and volume changes and even

composition, volume and temperature changes. In all cases one finds maximized entropy even

if reactions are invoked to balance composition.

There is more to learn from the above experiment. We can use experimental information on

the rate of heat exchange due to a temperature difference. It is summarized as Fourier’s law

and for the purpose at hand we can say that the rate of heat exchange is proportional to the

temperature difference where the proportionality coefficient is a (positive number) that depends

on geometry and on some material properties. Assuming that the constant does not change

during the experiment we can extend the above analysis and claim that the rate of change of

the entropy, called the entropy production, is always positive or vanishes. The latter it does in

equilibrium where it fluctuates but the average will remain constant and zero. To summarize,

equilibrating processes tend to maximize entropy and to minimize entropy production.

Let us take a closer look at the entropy production.

12

11dS dQ

dt T T dt

§·

¨¸

©¹

(1.4)

It consists of two factors: a thermodynamic force, the first factor which is the difference of the

two inverse temperatures, and the heat flux, the second factor. There is always a monotonic

relationship between flux and force that close to equilibrium is usually linear. One such relation

is Fourier’s law for the heat flux used above, another example is Fick’s law between

concentration gradient and diffusive flux.

The entropy of the universe

Using the above kind of example, it is easy to demonstrate that when one considers a large

closed and isolated box filled with a gas at a temperature T, the environment, and places a

cup of hot tea in it. Obviously, the tea will cool down to the gas temperature while exchanging

heat with the gas. There is so much gas, that the temperature increase of the environment can

be neglected. From such an analysis it follows that the entropy change of the box together with

1110

10

temperature difference is a fluctuating quantity with zero average and some finite standard

deviation. What does the entropy of the whole system do now. Well, because of the heat

exchanges there will also be entropy changes for the compartments. A simple analysis teaches

us

12

12

12

12

12

11

0

dS dS dS

SS

dU dU

UU

QQ

TT

ww

ww

(1.3)

In the first line we use the fact that the two compartments are making up the total system and

hence the total entropy of the system is the sum of the entropies of the two compartments.

Likewise for changes in the entropy. The only way – in the present example – that the entropy

can change is by a change in the energy content U of the system. In the third line, we identify

the change in entropy with a change in energy as the inverse temperature. The changes in

energy of the compartments are the heat exchanges – no work can be done – and when a

heat exchange Q is positive for the one compartment it is negative for the other. So far so

good. We can now use the experimental fact that in such an experiment the temperatures of

the compartments become the same – on average - and that hence the total entropy reaches

an extremum. We can also use the Second Law formulation that the entropy maximizes and

that hence the temperature difference tends to vanish. If we wish to know what kind of

extremum the entropy will reach one should evaluate the second order derivative of entropy

versus energy. It will be negative for equal compartment temperatures and hence entropy

indeed reaches a maximum.

The above reasoning can be continued for temperature and volume changes and even

composition, volume and temperature changes. In all cases one finds maximized entropy even

if reactions are invoked to balance composition.

There is more to learn from the above experiment. We can use experimental information on

the rate of heat exchange due to a temperature difference. It is summarized as Fourier’s law

and for the purpose at hand we can say that the rate of heat exchange is proportional to the

temperature difference where the proportionality coefficient is a (positive number) that depends

on geometry and on some material properties. Assuming that the constant does not change

during the experiment we can extend the above analysis and claim that the rate of change of

the entropy, called the entropy production, is always positive or vanishes. The latter it does in

equilibrium where it fluctuates but the average will remain constant and zero. To summarize,

equilibrating processes tend to maximize entropy and to minimize entropy production.

Let us take a closer look at the entropy production.

12

11dS dQ

dt T T dt

§·

¨¸

©¹

(1.4)

It consists of two factors: a thermodynamic force, the first factor which is the difference of the

two inverse temperatures, and the heat flux, the second factor. There is always a monotonic

relationship between flux and force that close to equilibrium is usually linear. One such relation

is Fourier’s law for the heat flux used above, another example is Fick’s law between

concentration gradient and diffusive flux.

The entropy of the universe

Using the above kind of example, it is easy to demonstrate that when one considers a large

closed and isolated box filled with a gas at a temperature T, the environment, and places a

cup of hot tea in it. Obviously, the tea will cool down to the gas temperature while exchanging

heat with the gas. There is so much gas, that the temperature increase of the environment can

be neglected. From such an analysis it follows that the entropy change of the box together with

11

the cup of tea, the universe, has increased. From this simple argument one then draws the

conclusion that any process increases the entropy of the world.

Whether this is absolutely true or not is hard to argue. Our universe is not at all homogenous,

as is assumed for the box where the gas was taken to be at a fixed temperature uniformly.

Various locations of the universe are still equilibrating while at the same time reactions

continue, such as the sun producing our heat and light. The universe is not at all in equilibrium

and as such fluctuates enormously. It is not the kind of system we can easily make predictions

about!

Intermediate conclusions

In the preceding sections we discussed reversibility as a limiting condition for any process that

by nature will always be irreversible. The irreversibility can be quantified using the fluxes and

driving forces of the process and the sum of their products provides the entropy production.

Hence, the Clausius relation can be completed by adding to the exchanged heat over

temperature the time-integrated entropy production.

As demonstrated, a thermodynamic system analysis can be performed for closed, isolated

systems. Extending such results to open systems, the “Universe”, will have limited value due

to uncertainties about the nature of the environment.

Molecular view

For well-defined molecular systems it is possible to calculate the absolute entropy using the

hypothesis made by Boltzmann; it reads

ln

B

Sk : (1.5)

in which

:

is equal to the number of ways in which the energy available to the system can be

distributed over the molecules constituting the system. The proportionality constant kB has the

same dimension as the entropy. Its value in SI-units is 1.38 10-23J/K, the gas constant per

molecule. In practical systems, one prefers units per mole in which case the proportionality

constant is the molar gas constant R = 8.314 J/(Kmol).

To express Boltzmann’s hypothesis in words proves not to be that easy. My best one-liner so

far is “Entropy is a measure of the degree of dispersal of Energy”. An important aspect is that

it always deals with energy. As we shall see, other constructs that seem to use the same term

entropy in actual fact also deal with energy but this particular aspect is obscured by the

particular way such a system is discussed.

The best example on how to use Boltzmann’s hypothesis is the derivation of the so-called

micro-canonical ensemble for the ideal gas. Let us go through the basic steps in this derivation,

the math we leave aside.

First the system has to be defined as an isolated, closed box; we take the box to be cubic with

side L, volume V = L3. The box is filled with N gas molecules, for instance argon, such that the

density is low enough that collisions between molecules can be neglected. The limit of low

densities of such a gas is the ideal gas. Also collisions with the wall molecules will be ignored

under the assumption that the volume of the system is so large that surface contributions are

negligible. All gas molecules are the same, we cannot distinguish one molecule from the other,

and have molecular mass m. The energy of the gas is U and cannot change because the box

is closed and isolated. The thermodynamic variables U, V and N determine the so-called

macro-state of the system.

The second step is to determine the number of possible ways to distribute the total energy U

over the N gas molecules. The molecules were assumed not to interact (collide) and hence

they can be treated as independent entities. The energy levels of a single molecule in a box

can be calculated using quantum mechanics. The outcome is that there are three non-negative

quantum numbers, nx, ny and nz of which at least one has to be non-zero; they control the

kinetic energy in the three orthogonal directions respectively. Each of the quantum states,

(0,0,1), (0,0,2), (0,1,0), etc., has an energy proportional to the sum of the quantum numbers

12

12

squared. The distance between the energy levels in h2/8mL2, where h is Planck’s constant.

The energy levels of the total of N molecules can be depicted in a 3N-dimensional space as

dots, these are microstates that belong to a system of N molecules in a box of volume V and

having a particular energy. The dots that are in a hypersphere shell of radius

U

and a

vanishing thickness would amount to all possible ways to distribute the energy of the

molecules, i.e. the total number of microstates belonging to the macro-state defined by U, V

and N. Unfortunately, an error is involved in depicting the microstates in the 3N-dimensional

space. To do this, the molecules needed to be numbered. In order to compensate for that, the

number of microstates in the 3N-dimensional hypersphere shell needs to be divided by N!, the

number of different ways in which N molecules can be labeled.

The final step involves doing the actual mathematics where there are two limits to consider:

that of a large system, the so-called thermodynamic limit, and the vanishing hypersphere shell

thickness. One may also worry about some statistical aspects of counting the energy states

where one can prove that some small counting error is made that vanishes in the

thermodynamic limit.

The result is known as the Sackur-Tetrode equation given in Eq. (1.2) where the de Broglie

length is expressed in internal energy U rather than temperature. Boltzmann’s hypothesis

identifies the logarithm of the derived number

:

(U,V,N) as a state function, the entropy. This

implies that the derivatives versus energy, volume and particle number will provide the

temperature, pressure and chemical potential. It should not be a surprise that these yield the

familiar ideal gas relations. Some authors claim that this also allows to prove Clausius’ equality.

This is not so obvious, because this essentially is already the content of Boltzmann’s

hypothesis itself: it was assumed!

Once, the micro-canonical ensemble is defined, one may obtain the other ensembles by

considering a small subsystem in a large closed and isolated system. The canonical ensemble

represents the case where the wall between the subsystem is closed with a wall that transmits

heat. The grand-canonical ensemble arises when the wall is also permeable to the molecules.

Interactions

An important step in the above derivation was the assumption that interactions, i.e. collisions,

do not matter. As van der Waals already discussed, there are two important aspects to gas

molecule interactions, the one is the volume they exclude for each other and the second is an

at room temperature weak, short-ranged attraction. The excluded volume originates from the

idea that the electron clouds of two molecules cannot interpenetrate without violating the Pauli

exclusion principle. This gives each gas molecule a particular volume that can well be

estimated from the molecular volume of the same molecule in the liquid state. The excluded

volume is then 8 times the molecular volume. The attraction is termed van der Waals attraction

and has to do with fluctuating induced dipole – induced dipole interactions of the electron

clouds of the molecules.

Conceptually, it is not so difficult to set up an expansion in density with the ideal gas as starting

point. This led to the Kamerlingh Onnes’ virial expansion for non-ideal gases, Kirkwood-Buff

formulae for fluids and many other useful thermodynamic approaches as are used today. The

magnitude of the first correction in density to the ideal gas is the second virial coefficient divided

by the molar volume of the gas. At room temperature this ratio is proportional to the molar

volume in the liquid state divided by that in the gas state which is typically of the order one per

mille! Indeed, for more dilute gases ideal behavior may be assumed.

Configuration space

The concept of excluded volume has led to the notion of configuration space. Every microstate

belonging to the same macro-state of a system can be represented in real space by molecular

positions with around each a sphere – for simplicity gas molecules are represented as spheres

1312

12

squared. The distance between the energy levels in h2/8mL2, where h is Planck’s constant.

The energy levels of the total of N molecules can be depicted in a 3N-dimensional space as

dots, these are microstates that belong to a system of N molecules in a box of volume V and

having a particular energy. The dots that are in a hypersphere shell of radius

U

and a

vanishing thickness would amount to all possible ways to distribute the energy of the

molecules, i.e. the total number of microstates belonging to the macro-state defined by U, V

and N. Unfortunately, an error is involved in depicting the microstates in the 3N-dimensional

space. To do this, the molecules needed to be numbered. In order to compensate for that, the

number of microstates in the 3N-dimensional hypersphere shell needs to be divided by N!, the

number of different ways in which N molecules can be labeled.

The final step involves doing the actual mathematics where there are two limits to consider:

that of a large system, the so-called thermodynamic limit, and the vanishing hypersphere shell

thickness. One may also worry about some statistical aspects of counting the energy states

where one can prove that some small counting error is made that vanishes in the

thermodynamic limit.

The result is known as the Sackur-Tetrode equation given in Eq. (1.2) where the de Broglie

length is expressed in internal energy U rather than temperature. Boltzmann’s hypothesis

identifies the logarithm of the derived number

:

(U,V,N) as a state function, the entropy. This

implies that the derivatives versus energy, volume and particle number will provide the

temperature, pressure and chemical potential. It should not be a surprise that these yield the

familiar ideal gas relations. Some authors claim that this also allows to prove Clausius’ equality.

This is not so obvious, because this essentially is already the content of Boltzmann’s

hypothesis itself: it was assumed!

Once, the micro-canonical ensemble is defined, one may obtain the other ensembles by

considering a small subsystem in a large closed and isolated system. The canonical ensemble

represents the case where the wall between the subsystem is closed with a wall that transmits

heat. The grand-canonical ensemble arises when the wall is also permeable to the molecules.

Interactions

An important step in the above derivation was the assumption that interactions, i.e. collisions,

do not matter. As van der Waals already discussed, there are two important aspects to gas

molecule interactions, the one is the volume they exclude for each other and the second is an

at room temperature weak, short-ranged attraction. The excluded volume originates from the

idea that the electron clouds of two molecules cannot interpenetrate without violating the Pauli

exclusion principle. This gives each gas molecule a particular volume that can well be

estimated from the molecular volume of the same molecule in the liquid state. The excluded

volume is then 8 times the molecular volume. The attraction is termed van der Waals attraction

and has to do with fluctuating induced dipole – induced dipole interactions of the electron

clouds of the molecules.

Conceptually, it is not so difficult to set up an expansion in density with the ideal gas as starting

point. This led to the Kamerlingh Onnes’ virial expansion for non-ideal gases, Kirkwood-Buff

formulae for fluids and many other useful thermodynamic approaches as are used today. The

magnitude of the first correction in density to the ideal gas is the second virial coefficient divided

by the molar volume of the gas. At room temperature this ratio is proportional to the molar

volume in the liquid state divided by that in the gas state which is typically of the order one per

mille! Indeed, for more dilute gases ideal behavior may be assumed.

Configuration space

The concept of excluded volume has led to the notion of configuration space. Every microstate

belonging to the same macro-state of a system can be represented in real space by molecular

positions with around each a sphere – for simplicity gas molecules are represented as spheres

13

– where the sphere of another molecule cannot overlap. Each such representation is a

configuration space where the velocities of the molecules are arbitrary. As long as the total

energy of a microstate is the sum of a kinetic energy, the particle velocities, and a potential

energy, due to interactions one can consider the two energy contributions independently. The

evaluation of the kinetic part of the partition function is independent of that of the configurations.

The first is very similar if not identical to that of the ideal gas discussed above, so for non-ideal

systems the important aspects come from configuration space. In other words, one may derive

the configurational entropy independently of that for the kinetic part.

Order and disorder

For many gas microstates, the organization of spheres in configuration space will be

disordered. However, at lower temperatures and higher densities some kind of order sets in

which typically is the liquid state. The local organization of molecules in the liquid state is

experimentally found to be very similar to the closest solid state; long range order is present

only in the solid state. Solid states are often ordered in a crystal lattice and so in a liquid the

local organization around each atom is almost as in the crystal state with a well-determined

number of closest neighbors. To arrive from a gas state in a liquid state, the condensation

transition, the molecules exchange the entropy of being free moving entities in the gas state

to the enthalpy of being in close contact with neighboring molecules. This constitutes the

entropy and enthalpy of condensation.

When in the 50’s Alder and Wainwright for the first time simulated a gaseous system consisting

of “hard spheres” without an interaction, it was a surprise for them to find that the system froze

into a solid state when lowering the temperature. Since there was no attractive interaction in

their model, according to the classical view of phase transitions this could not be. It took a long

time to realize that it was a phase transition between an ordered and a disordered state: under

some conditions, the ordered phase had a higher entropy than the disordered state! This is in

contrast with the common notion that entropy would be the degree of disorder in the system.

This notion is clearly wrong! In the example at hand, it can be understood that a disordered

configuration has less entropy than an ordered configuration. It has to do with the fact, that in

an ordered state each molecule has the freedom to move within its own cell whereas at the

same density in a disordered configuration many molecules are trapped and cannot move at

all.

Conclusion

The title raised the question “Who cares about the Second Law of Thermodynamics”. Well, the

Second Law defines entropy as a physical quantity that can be measured to any desired

accuracy and computed, sometimes by analytical means but always numerically to any desired

accuracy. Its knowledge gives a handle to predict the energetics of any process, so at least

engineers should care.

15

15

Thermodynamics my way – illustrated by fuel cells

Thermodynamic efficiency

This part is very introductory but I still include it because it turns out that by far not all educations

actually deal with chemical thermodynamics anymore. Since the results of this part are very

practical, real numbers are produced, and give upper limits for open voltages and efficiencies,

this part is necessary to address the issues of the second part.

Most energy conversion technologies employed today are thermomechanical: first water is

heated, usually to steam, by coal, oil, or nuclear power, and subsequently this heat content is

used to drive a electrical generator. Despite all advances in engineering this is still the way we

do it and at a low temperature efficiency at that. By far the smarter way is to use a technology

where chemical energy is directly converted into electricity. Simply because of the lesser steps,

this must be more efficient and it is!

From Figure 2.1 it is also immediately clear that for direct conversion processes the efficiency

is much less temperature dependent than for the thermomechanical processes that are limited

by Carnot’s efficiency rule for heat engines. Let us therefore consider the polyelectrolyte

membrane fuel cell (PEMFC). As for every chemical process analysis, the stoichiometry of the

reaction is an important aspect: it tells us how many moles of reactants are needed to convert

to a given amount of product. In the present case, we are interested in the amount of electrons

that are produced and in order to determine this we need to consider the two half reactions as

take place at the two electrodes. Important to realize is that where the electrons are carried

through the external circuit, ions are carried through the internal circuit. In the case of the

PEMFC, these are the protons. The membrane is necessary to separate the ionic current from

the electron current. The stoichiometric number that is associated with the electrons will be

labeled by

Q

e; it is 2 in the present example if we fix the stoichiometric number of hydrogen to

1. Rather than proton exchange, there is also the possibility of hydroxyl exchange across a

membrane. This requires a different kind of membrane as well as a different kind of catalyst.

Although we mention this here as an alternative, we shall not go deeper into this topic here.

The celebrated Second Law of Thermodynamics is what rules the most important aspect of

any chemical reaction and hence of any fuel cell. It tells us whether a reaction will be able to

run spontaneously or not. Moreover, the difference between the initial Gibbs energy and the

final Gibbs energy is exactly equal to the amount of available work in a system. Important to

realize is that if we do not use this work, it is lost! The term “lost work” therefore is a matter of

definition of the system. In our case, we shall use this available work to make electricity. Some

of it will still be lost to heat in the process, but we can use a significant portion of it. This in

contrast to many other processes.

, , ( ) ln

j

oo o o

r j fj j o

j

p

G T p G T p S T T RT p

Q

½

§·

°°

' '

®¾

¨¸

°°

©¹

¯¿

¦ (2.1)

In order to obtain a number for the available work, we need to know the Gibbs energy at the

initial and final point of the process. The above Eq. 2.1 tells us how to obtain it from standard

tables as for instance in the “Handbook of Physics and Chemistry”, but also on many places

Figure 2.1 Comparison of thermomechanical and

electrochemical efficiencies.

16

16

on the world wide web. The values are given for standard conditions, Standard Temperature

and Pressure (STP), being 298 K and 1 Bar. Furthermore, a smart choice of reference is

chosen for all entries in the tables so that the calculation that we intend to do indeed yields the

proper result. Note that it is the absolute entropy that tells us how the Gibbs energy varies with

temperature. The above relation tells us what to do: we need to enter the stoichiometric

coefficients for the reaction (positive for products and negative for reactants), the actual

temperature and pressure. The resulting reaction Gibbs energy is per mole of fuel, the

component that we shall give stoichiometric coefficient 1; other choices are equally possible.

1

2

1 0 130.7

358.15 K, 0 60 205.2 J/mol 226 kJ/mol

1 -228,572 188.8

o

rGp

½

§ ·§ · § ·

°°

¨ ¸¨ ¸ ¨ ¸

'

®¾

¨ ¸¨ ¸ ¨ ¸

°°

¨ ¸¨ ¸ ¨ ¸

© ¹© ¹ © ¹

¯¿

(2.2)

In order to work out the equation efficiently, we use vector notation. The values from the table

are entered and the result is a typical value for a fuel as hydrogen. The electrical voltage is

easily found, since a voltage is defined as the amount of available energy per unit charge. In

this case, 2 moles of electrons are produced per mole of fuel and hence the voltage is the

reaction Gibbs energy divided by the charge of 2 moles of electrons. For the charge of 1 more

of electrons one uses the Faraday, about 100,000 C/mole. One finds 1.17 V with this example.

Now that we know the available energy, we turn to the energy content of the fuel. Although we

aim at converting all the available energy into work, some heat exchange will be required and

we need to consider this now. For that, we have the First Law of Thermodynamics. The

ultimately inefficient process will convert all the available energy into heat; it is lost then if not

further used.

^`

,

, , ()

oo o o

r j f j pj

j

HTp H T p C T T

Q

' '

¦

(2.3)

Just like the Gibbs energy, the enthalpy is tabulated and the above expression explains how

to obtain it from the tabulated data. There is, under normal fuel cell conditions, no significant

influence of pressure on the enthalpy and hence we do not take that into account. The

temperature dependence of the enthalpy is given by the heat capacity (at constant pressure).

Knowing the available work that we convert and the energy balance, we are now in a position

to compute the thermodynamic efficiency of a fuel cell, see Eq. 2.4, by considering the actual

work W performed on the system and the heat Q transferred to the system. But careful: the

result can be confusing! The naïve usage of the ratio of Gibbs energy change and enthalpy

change may yield a number larger than 1. Surely, the efficiency will not be larger than 1,

although many researchers thought so for a long time. What happens in that case is that heat

will be flowing from the exterior into the fuel cell as we required the temperature of the cell to

be constant. What happens in actual fact is that we then have a device that yields electricity

and at the same time cools our room! Unfortunately, the effect does not persist in more realistic

situations.

K

K

'

d

°'

°

®

°! d

°

¯max

0

01

G

W

QWQ H

W

QW

(2.4)

The maximum efficiency computed before is never achieved, it is a limiting value for infinitely

slow processes that deliver infinitely little current. At higher rates and higher currents there are

losses and on general grounds one may show that these are always proportional to the current

squared times a coefficient that therefore is called the resistance. The resistance value

depends partially on the specific materials used, and is partially determined by the dimensions

of the cell. Clearly, this is the field where engineering comes in and this is what we shall focus

on in the second part. To make the point, the work that is not used is lost and hence dissipated

to heat.

1716

16

on the world wide web. The values are given for standard conditions, Standard Temperature

and Pressure (STP), being 298 K and 1 Bar. Furthermore, a smart choice of reference is

chosen for all entries in the tables so that the calculation that we intend to do indeed yields the

proper result. Note that it is the absolute entropy that tells us how the Gibbs energy varies with

temperature. The above relation tells us what to do: we need to enter the stoichiometric

coefficients for the reaction (positive for products and negative for reactants), the actual

temperature and pressure. The resulting reaction Gibbs energy is per mole of fuel, the

component that we shall give stoichiometric coefficient 1; other choices are equally possible.

1

2

1 0 130.7

358.15 K, 0 60 205.2 J/mol 226 kJ/mol

1 -228,572 188.8

o

rGp

½

§ ·§ · § ·

°°

¨ ¸¨ ¸ ¨ ¸

'

®¾

¨ ¸¨ ¸ ¨ ¸

°°

¨ ¸¨ ¸ ¨ ¸

© ¹© ¹ © ¹

¯¿

(2.2)

In order to work out the equation efficiently, we use vector notation. The values from the table

are entered and the result is a typical value for a fuel as hydrogen. The electrical voltage is

easily found, since a voltage is defined as the amount of available energy per unit charge. In

this case, 2 moles of electrons are produced per mole of fuel and hence the voltage is the

reaction Gibbs energy divided by the charge of 2 moles of electrons. For the charge of 1 more

of electrons one uses the Faraday, about 100,000 C/mole. One finds 1.17 V with this example.

Now that we know the available energy, we turn to the energy content of the fuel. Although we

aim at converting all the available energy into work, some heat exchange will be required and

we need to consider this now. For that, we have the First Law of Thermodynamics. The

ultimately inefficient process will convert all the available energy into heat; it is lost then if not

further used.

^`

,

, , ()

oo o o

r j f j pj

j

HTp H T p C T T

Q

' '

¦

(2.3)

Just like the Gibbs energy, the enthalpy is tabulated and the above expression explains how

to obtain it from the tabulated data. There is, under normal fuel cell conditions, no significant

influence of pressure on the enthalpy and hence we do not take that into account. The

temperature dependence of the enthalpy is given by the heat capacity (at constant pressure).

Knowing the available work that we convert and the energy balance, we are now in a position

to compute the thermodynamic efficiency of a fuel cell, see Eq. 2.4, by considering the actual

work W performed on the system and the heat Q transferred to the system. But careful: the

result can be confusing! The naïve usage of the ratio of Gibbs energy change and enthalpy

change may yield a number larger than 1. Surely, the efficiency will not be larger than 1,

although many researchers thought so for a long time. What happens in that case is that heat

will be flowing from the exterior into the fuel cell as we required the temperature of the cell to

be constant. What happens in actual fact is that we then have a device that yields electricity

and at the same time cools our room! Unfortunately, the effect does not persist in more realistic

situations.

K

K

'

d

°'

°

®

°! d

°

¯max

0

01

G

W

QWQ H

W

QW

(2.4)

The maximum efficiency computed before is never achieved, it is a limiting value for infinitely

slow processes that deliver infinitely little current. At higher rates and higher currents there are

losses and on general grounds one may show that these are always proportional to the current

squared times a coefficient that therefore is called the resistance. The resistance value

depends partially on the specific materials used, and is partially determined by the dimensions

of the cell. Clearly, this is the field where engineering comes in and this is what we shall focus

on in the second part. To make the point, the work that is not used is lost and hence dissipated

to heat.

17

rev lost lost

rev irr irr 0

QW W

QSS S

TT T

' ' ' t (2.5)

Another term for lost work is produced entropy and Clausius’ relation, see above, tells us how

to relate the work and the entropy production. The above Eq. 2.5 is derived using the Gibbs-

Helmholtz relation that relates entropy change to the difference between Gibbs energy change

and enthalpy change over temperature. The temperature is the temperature of the environment

to where the heat is discarded.

On quite general grounds one may show, that the entropy production is always the product of

the thermodynamic force and a thermodynamic current. An example is the voltage across the

cell – remember that voltage was energy per unit charge – and the current is the electrical

current, the amount of charge per unit time. We could also have taken the reaction Gibbs

energy and the reaction rate, this gives the same result. Between the force and the current

there is a relation that is often – but not always – linear. The relation is always monotonic and

hence locally one may assume linearity. Because of this monotonicity, the entropy production

is never negative and most often positive. Only in the hypothetical equilibrium case it vanishes.

Note that in equilibrium the entropy of the system tends to a maximum whereas the entropy

production tends to a minimum.

Real systems

Fig. 2.2: Fuel cell stack by Nedstack, Arnhem NL.

We will now discuss some material aspects of fuel cells. There are many and the choice that

was made here is one of my personal interest: this is what we were working on and hence this

is where we think we know something about. Other people obviously will make other choices

and even we will make other choices a few years from now. The field is very active at the

moment and many developments are considered. Nevertheless, the choices are relevant to

the field and beyond.

Although there is still a lot that needs to be worked out, fuel cell stacks are being used as

today. Usually for stationary power applications and many as backup power systems. The

above Figure 2.2 is by NedStack, a Dutch company with which we collaborated. Each stack

consists, just like a battery pack, out of a number of cells that are separated by flow plates or

bipolar plates; we shall come back to these later. One side of the plate brings hydrogen to the

cell and the other side oxygen or air. The flow plates also carry the electricity produced in the

neighboring cells. In between the plates there are gas diffusion layers and catalyst layers

separated by a polymer membrane. The porous layers are to admit the gas to the catalyst

particles that are in electronic contact with the electrically conducting flow plates. They should

also be in contact with proton conducting material in contact with the membrane. The pores

need to be such that produced water can be carried away from the catalyst particles. Some

water is needed to hydrate the membranes and make them proton conducting.

18

18

Let us now focus on the membrane electrolyte. There are a lot of requirements imposed on

the membrane material out of which the combination of proton conductivity and hydrogen

blockage is by far the toughest: when a membrane blocks hydrogen it will surely block oxygen

or other air constituents. The canonical choice for polymer electrolyte membrane material is

Nafion, a patented product by Dupont. It is a polyelectrolyte that consists of a hydrophobic

backbone and hydrophilic side chains. The sulfonic groups on the side chains are responsible

for the proton conduction but in order to do so they need to be hydrated. A relatively unique

property of Nafion is, that it spontaneously forms a bicontinuous structure of connected

hydrophilic domains. Although the term bicontinuity suggests that also the hydrophic parts form

a connected permeating network, this is not a necessity for proper operation. It is useful to give

the membrane strength though.

In order to address the performance of membranes in fuel cells systematically*, one uses the

formalism of thermodynamic forces and fluxes (currents) as discussed before, see Figure 2.3.

The minimal set that is necessary here involves proton current, water flow and diffusion, and

hydrogen flow and diffusion. The currents are driven by in principle three thermodynamic

forces, the electrical potential gradient (electrical field) and two due to the chemical potential

gradients. Note, that transport is coupled and we shall see some important aspects of this.

The derivation of the necessary equations to assess the performance of the membrane is a bit

technical. It is presented here to show that most of it is relatively straightforward albeit that it

does require some tedious bookkeeping. The first step is to take the relevant parts from the

general expression for entropy production as can be found in many textbooks on the topic of

Non-Equilibrium Thermodynamics. It reads

^`

,,

11

w x w T h x hT x

V V

S J J j dV dV

TT

P PI

w w w {

³³

JX

(2.6)

The integrand over the total volume of the system, the membrane here, consists of the

products of fluxes and forces. The first product concerns the water flux, subscript w, the second

hydrogen, subscript h, and the last electrical charge. For that reason, we abbreviate it as the

product of two vectors, one containing the fluxes and the other the forces. In these terms, the

linear relation between fluxes and forces that we shall use here, can be written as

*Angie L. Rangel-Cárdenas and Ger Koper, Transport in Proton Exchange Membranes for Fuel Cell Applications—A Systematic

Non-Equilibrium Approach, Materials 2017, DOI: 10.3390/ma10060576.

Figure 2.3 Schematic representation of a fuel cell

membrane.

1918

18

Let us now focus on the membrane electrolyte. There are a lot of requirements imposed on

the membrane material out of which the combination of proton conductivity and hydrogen

blockage is by far the toughest: when a membrane blocks hydrogen it will surely block oxygen

or other air constituents. The canonical choice for polymer electrolyte membrane material is

Nafion, a patented product by Dupont. It is a polyelectrolyte that consists of a hydrophobic

backbone and hydrophilic side chains. The sulfonic groups on the side chains are responsible

for the proton conduction but in order to do so they need to be hydrated. A relatively unique

property of Nafion is, that it spontaneously forms a bicontinuous structure of connected

hydrophilic domains. Although the term bicontinuity suggests that also the hydrophic parts form

a connected permeating network, this is not a necessity for proper operation. It is useful to give

the membrane strength though.

In order to address the performance of membranes in fuel cells systematically*, one uses the

formalism of thermodynamic forces and fluxes (currents) as discussed before, see Figure 2.3.

The minimal set that is necessary here involves proton current, water flow and diffusion, and

hydrogen flow and diffusion. The currents are driven by in principle three thermodynamic

forces, the electrical potential gradient (electrical field) and two due to the chemical potential

gradients. Note, that transport is coupled and we shall see some important aspects of this.

The derivation of the necessary equations to assess the performance of the membrane is a bit

technical. It is presented here to show that most of it is relatively straightforward albeit that it

does require some tedious bookkeeping. The first step is to take the relevant parts from the

general expression for entropy production as can be found in many textbooks on the topic of

Non-Equilibrium Thermodynamics. It reads

^`

,,

11

w x w T h x hT x

V V

S J J j dV dV

TT

P PI

w w w {

³³

JX

(2.6)

The integrand over the total volume of the system, the membrane here, consists of the

products of fluxes and forces. The first product concerns the water flux, subscript w, the second

hydrogen, subscript h, and the last electrical charge. For that reason, we abbreviate it as the

product of two vectors, one containing the fluxes and the other the forces. In these terms, the

linear relation between fluxes and forces that we shall use here, can be written as

*Angie L. Rangel-Cárdenas and Ger Koper, Transport in Proton Exchange Membranes for Fuel Cell Applications—A Systematic

Non-Equilibrium Approach, Materials 2017, DOI: 10.3390/ma10060576.

Figure 2.3 Schematic representation of a fuel cell

membrane.

19

,

, or

w ww hw wq x w T

T

h hw hh hq x h T

qw qh qq x

J LLL

J L LL

j L LL

P

P

I

§·

w

§· § ·

¨¸

¨¸ ¨ ¸

w

¨¸

¨¸ ¨ ¸

¨¸ ¨ ¸

¨¸

w

©¹ © ¹

©¹

J LX (2.7)

The matrix

L

involves 9 different elements of which the diagonal ones are related to the water

permeability, the hydrogen permeability and the electrical conductivity. By Onsager’s

symmetry argument, we end up with 3 different off-diagonal elements. The above formulation

is not practical. Measurable quantities are liquid flow including water as well as dissolved

entities, hydrogen flow with respect to the membrane and electrical current. These three

independent variables form another base in the same vector space as where we have the

previously defined currents. Hence, one needs a base transformation to formulate the relation

in terms of experimentally accessible quantities. The entropy production is conserved with this

transformation. The result of this transformation is

0

0 0 or ' ' '

0

VV x

T

H H xH

x

JP K p

JP p

jK

V

V VI

w

§ · § ·§ ·

¨ ¸ ¨ ¸¨ ¸

w

¨ ¸ ¨ ¸¨ ¸

¨ ¸ ¨ ¸¨ ¸

w

© ¹ © ¹© ¹

J LX

(2.8)

The elements of the matrix 'L are readily interpreted. The diagonal elements are the liquid

permeability V

P, the hydrogen permeability H

P and the conductivity

V

. We only consider one

cross-coefficient, the so called electro-osmotic drag

K

V

. The other two are reported to be

negligible so we omit these.

The simplest thing to do would be to consider these four coefficients to be independent of the

location within the system, i.e. that the membrane is homogeneous. Many authors have

assumed so but that turns out not to be realistic. Next best is to assume that the membrane

consists of a center slab that is homogeneous with on the sides thin slabs that are in contact

with the electrodes. Of course, one would like to provide for a more detailed description of the

membrane but in practice it turns out to be already quite difficult to obtain coefficient values for

the above model description. Nevertheless, detailed analysis of experimental data using the

above formulation does yield satisfactory results. A comparison with literature values teaches

that the bulk coefficients prove to be very reliable, but that the contact layer values much less

so*. But then, the experiments were never aimed at obtaining values for these.

Conclusion

The above shows that using modern analysis techniques it is well possible to assess the

(thermodynamic) efficiency of devices: a first law efficiency is obtained using global

thermodynamics and the “second law efficiency” using Non-Equilibrium Thermodynamics

which allows for a much more local analysis.

21

21

Why should a chemical engineer know about molecular thermodynamics

The short answer to this question is relatively straightforward. Chemical engineers will have to

deal with new molecules and the first thing they need to know is some of their thermodynamic

properties. Of course, these could be measured but if there is a fast route to finding

approximately correct values this would be great. In addition, of new molecules by far not all

molecular details are known either. Molecular thermodynamics is exactly capable of

addressing this particular situation: it provides reasonable to very good estimates of

thermodynamic quantities using a minimum of molecular information!

Dilute gases

Doubtless the best example are the dilute gases: for monoatomic molecules one just needs

the molecular mass. For more complex molecules more details are required but many times

an idea about molar volume and boiling temperature is enough. For even more refined

information, spectroscopy in the IR and visible light suffice.

To recall how this is achieved, let us briefly review some ensemble theory, the hallmark of

molecular thermodynamics. As discussed in the first lecture, my personal favorite route starts

with the so called microcanonical ensemble. It refers to systems with a fixed volume V, number

of particles N – or their composition for mixtures – and energy U. The total number of

microstates, labeled

:

(U,V,N), that belong to the thus defined macrostate is used to obtain the

total entropy of the system using Boltzmann’s hypothesis. This leads, for the ideal gas, to the

Sackur-Tetrode equation in Eq.(3.1).

3/2

2

54

ln

23

BB

V mU

S kN kN NhN

S

ªº

§·

«»

¨¸

©¹

«»

¬¼

(3.1)

Expressing the internal energy in terms of the temperature, using

3

2B

U Nk T

, yields the more

common formulation of Eq.(3.1) given in Eq.(1.2).

Apart from 3 physical constants, i.e. Planck’s constant, Avogadro’s number (to convert to molar

quantities) and either the gas constant or Boltzmann’s constant and the 3 variables V, N and

U, there is only one molecular parameter, the molecular mass m. Nevertheless, apart from the

absolute entropy one also finds the specific heat and with that all thermodynamic data that is

available from standard tables except for the standard value for the enthalpy (at STP). The

latter sets the energy scale and depends on the standard values of other materials. One may

easily verify this for instance for Argon.

The above discussed analysis only refers to the kinetic energy of the molecule as a whole,

using its center of mass coordinates position and velocity. Apart from this, there are a few other

energy contributions that – to a high degree of accuracy – can be separated as quite distinct

energy contributions.

x Nuclear energy typically constant over quite a range of conditions. In actual fact one

needs considerable extreme conditions to change it. An exception is the class of

unstable molecules but we leave these aside.

x Electronic energy is usually constant except at high energies where collisions can

cause ionization of molecules.

x Vibrational and rotational energy do vary even at room temperature and hence need to

be taken into account.

For each n-atomic molecule, each atom has 3 kinetic energy contributions and hence such a

molecule provides 3n kinetic energy contributions. The center-of-mass motion takes 3 of those.

Rotations take up maximally nr = 3 and sometimes 2 for diatomic molecules. The remainder,

typically 3n-3-nr kinetic energy contributions define the same amount of vibrational

contributions. For both the rotations and the vibrations the partition functions are known in

terms of their frequencies. To derive these contributions, one typically uses the canonical

ensemble which considers macrostates of fixed volume, composition and temperature. It is

obtained by considering a system in thermal contact with a heat bath. For the total system one

22

22

assumes the microcanonical ensemble that should be the sum of the two microcanonical

ensembles for system and heat bath. As the energy is distributed between system and bath,

the temperature becomes the defining variable for the system. In the limit of a large heat bath

one then obtains the canonical partition function as

/( )

0

ln

B

E kT

B

Z k T A E e dE

f

:

³

(3.2)

for very many model systems, this is the ensemble of choice. As stated in Eq.(3.2), the

canonical partition function defines the Helmholtz free energy. For the rotational and vibrational

partitions, the required frequencies are experimentally obtained from IR and visible light

spectra.

For high polyatomic molecules, such as a crystal, the spectral distribution is better formulated

in terms of a frequency spectrum. Although these can be determined experimentally, some

rude but rather successful approximations have been proposed in the past. Einstein proposed

that there would be one single frequency only to consider. This did lead to some success, but

the low temperature behavior of the specific heat is not satisfactory. To amend this, Debye

proposed a spectrum in the form of the right wing of a upwards curved parabola with a

terminating frequency. This proved to be a very good approximation for many systems even

though the experimental spectrum shows much more structure. For many crystals, the Debye

frequency or corresponding temperature is listed.

Although many gases almost behave as ideal, deviations can be significant. To assess these,

the effect of molecular interactions needs to be considered. As discussed in lecture 1, this

usually does not involve the kinetic energy contributions but the positions only. In the first place,

there is the volume that molecules exclude for each other. Then there are attractive interactions

between molecules, typically due to van der Waals interactions on a relatively short range.

Evaluating their effect on thermodynamic quantities introduces the two molecular parameters

that van der Waals already introduced long time ago with his equation of state. These are the

excluded volume parameter b and the molecular attraction parameter a. The parameter b

equals 8 times the molecular volume. Its value can be quite accurately determined from the

molar volume of the same molecules in the liquid state. The parameter a can be estimated

from the enthalpy of evaporation for the gas: the latter involves the breaking of typically 6 bonds

of equal energy. Even if the enthalpy of evaporation is not known, one may estimate it using

the boiling temperature using Trouton’s rule for the entropy of evaporation, 85 J/(K mol).

The parameters find their place in the virial expansion for a gas proposed by Kamerlingh

Onnes,

2

1

mm

B

RT

pVV

§·

¨¸

©¹

"

(3.3)

and later proven theoretically along many different routes. The simplest route to obtain this

expansion involves the so called grand canonical partition function. This ensemble is obtained

much in the same way as discussed above for the canonical ensemble albeit that apart from

energy exchange there is also particle exchange. Corresponding macrostates are therefore

defined by volume, temperature and chemical potential and the corresponding thermodynamic

function is pressure. It is for that reason, that the virial expansion is so easily derived … one

just has to form an expansion in molar volume!

An assessment of the first correction to ideal gas behavior is in place. This is exactly the second

virial coefficient B2 divided by the molar volume of the gas. At room temperature, a good

estimate for the second virial coefficient is the excluded volume parameter only which equals

8 times the molar volume of the gas molecules in the liquid state. Hence, the ratio is no more

than 1%, usually less! However, near the boiling temperature serious corrections are to be

expected especially near the critical point.

Dense fluids

2322

22

assumes the microcanonical ensemble that should be the sum of the two microcanonical

ensembles for system and heat bath. As the energy is distributed between system and bath,

the temperature becomes the defining variable for the system. In the limit of a large heat bath

one then obtains the canonical partition function as

/( )

0

ln

B

E kT

B

Z k T A E e dE

f

:

³

(3.2)

for very many model systems, this is the ensemble of choice. As stated in Eq.(3.2), the

canonical partition function defines the Helmholtz free energy. For the rotational and vibrational

partitions, the required frequencies are experimentally obtained from IR and visible light

spectra.

For high polyatomic molecules, such as a crystal, the spectral distribution is better formulated

in terms of a frequency spectrum. Although these can be determined experimentally, some

rude but rather successful approximations have been proposed in the past. Einstein proposed

that there would be one single frequency only to consider. This did lead to some success, but

the low temperature behavior of the specific heat is not satisfactory. To amend this, Debye

proposed a spectrum in the form of the right wing of a upwards curved parabola with a

terminating frequency. This proved to be a very good approximation for many systems even

though the experimental spectrum shows much more structure. For many crystals, the Debye

frequency or corresponding temperature is listed.

Although many gases almost behave as ideal, deviations can be significant. To assess these,

the effect of molecular interactions needs to be considered. As discussed in lecture 1, this

usually does not involve the kinetic energy contributions but the positions only. In the first place,

there is the volume that molecules exclude for each other. Then there are attractive interactions

between molecules, typically due to van der Waals interactions on a relatively short range.

Evaluating their effect on thermodynamic quantities introduces the two molecular parameters

that van der Waals already introduced long time ago with his equation of state. These are the

excluded volume parameter b and the molecular attraction parameter a. The parameter b

equals 8 times the molecular volume. Its value can be quite accurately determined from the

molar volume of the same molecules in the liquid state. The parameter a can be estimated

from the enthalpy of evaporation for the gas: the latter involves the breaking of typically 6 bonds

of equal energy. Even if the enthalpy of evaporation is not known, one may estimate it using

the boiling temperature using Trouton’s rule for the entropy of evaporation, 85 J/(K mol).

The parameters find their place in the virial expansion for a gas proposed by Kamerlingh

Onnes,

2

1

mm

B

RT

pVV

§·

¨¸

©¹

"

(3.3)

and later proven theoretically along many different routes. The simplest route to obtain this

expansion involves the so called grand canonical partition function. This ensemble is obtained

much in the same way as discussed above for the canonical ensemble albeit that apart from

energy exchange there is also particle exchange. Corresponding macrostates are therefore

defined by volume, temperature and chemical potential and the corresponding thermodynamic

function is pressure. It is for that reason, that the virial expansion is so easily derived … one

just has to form an expansion in molar volume!

An assessment of the first correction to ideal gas behavior is in place. This is exactly the second

virial coefficient B2 divided by the molar volume of the gas. At room temperature, a good

estimate for the second virial coefficient is the excluded volume parameter only which equals

8 times the molar volume of the gas molecules in the liquid state. Hence, the ratio is no more

than 1%, usually less! However, near the boiling temperature serious corrections are to be

expected especially near the critical point.

Dense fluids

23

Near and in the liquid state, intermolecular interactions are abundant and another approach is

required. This involves assessing the local structure around the molecules. The probe of choice

is X-ray scattering as it provides the structure function, the Fourier transformed pair correlation

function g(r) for the fluid. A definition of the pair correlation is difficult to render. A proposal

would be “This is related to the probability of finding the center of a particle a given distance

from the center of another particle”. In formula form, the definition can be made as precise as

necessary but we leave that to relevant text books.

Rather we refer to Figure 3.1 where an example is given. For large separations r, the pair

correlation should tend to 1 as there should always be a particle further away and for small

values to 0 as particle centers do not overlap due to excluded volume interactions. The black

line gives the pair correlation function for a very dilute system and the light blue line for a very

dense system. As density increases, oscillations develop. The first peak signals the typical

closest approach of particles, its height is proportional to the number of neighboring molecules.

A detailed graph as in Figure 3.1 is obtained by computational techniques although present-

day X-ray scattering techniques also provide quite detailed pair correlation functions.

Computational techniques are necessary when mixtures are considered, X-ray signals do not

distinguish between molecules although varied-contrast studies do exist.

Once the structural information is obtained, Kirkwood-Buff formulae are used to obtain

thermodynamic variables. Internal energy, pressure, surface tension and line tension are

available. An example is the formula for the internal energy of the system, eq.(3.4).

2

0

32 ()()

2

B

U Nk T N u r g r r dr

SU

f

³

(3.4)

The first part of the internal energy is the ideal gas contribution. The second part is the excess

internal energy due to interactions. It involves an integral over all space where the strength of

a (derived or guessed) molecular interaction potential is gauged by the pair correlation function:

the less probable a particular separation the less the contribution to the internal energy.

The method is particularly useful to find deviations from ideality, the so called fugacity or activity

coefficients as are obtained from the excess Gibbs energy of a non-ideal system.

Conclusion

In conclusion, molecular thermodynamics provides for methods to obtain thermodynamic

information on all kinds of systems. The amount of information needed depends on the

accuracy that is required. For not too dense fluids, very accurate values can be obtained once

molecular mass, molecular volume and enthalpy of evaporation are known. More precision is

available if also rotational and vibrational spectra are available. For simple liquids, X-ray

spectra suffice whereas for more complex liquids, typically mixtures, computational techniques

are called for. The latter are currently the subject of scientific investigation.

Figure 3.1: Pair correlation fluid at varying

densities; kF sets the scale.

25

25

On the brink of chaos

In contrast to what one might think, chaos is a well-defined concept. So the first thing we need

to do is to find out about this definition. Within the realm of thermodynamics, we deal with so-

called dynamical systems. Multi-particle systems, such as a fluid, make up a dynamical system

as the motion of the individual entities is precisely described by quantum mechanics. For many

properties, the quantum aspects are not that relevant and classical mechanics is sufficient to

obtain the desired results.

As an example, let us consider a dilute gas. The positions and momenta of the particles are

governed by Newton’s equation. It is more convenient for the present purpose to resort to

another way of writing the same equations in terms of a so-called Hamiltonian. The method

resembles very much what many may recall from quantum mechanics and that alone is already

an advantage. In terms of a Hamiltonian,

H

, the equations of motion look like

w

°w

°

®w

°

°w

¯

j

j

j

j

rp

pr

H

H (4.1)

In particular when we take the Hamiltonian to be

^`

2

2

j

j

jj

pV

m

¦rH (4.2)

where the first term represents the kinetic energy and the second term the potential energy in

the system. The reader easily verifies that this is just another way to write Newton’s equations.

The advantage of this method is, that the equations of motion are given in terms of first order

differential equations.

Figure 4.1 Phase space for harmonic oscillator

To make things even more specific, let us turn to the pendulum for which the harmonic

oscillator is a good approximate model. The Hamiltonian of the harmonic oscillator is given by

2

2

1

22

j

pkx

m

¦

H

(4.3)

where mass m and spring constant k determine the oscillating frequency

/km

Z

. Given a

starting position, the trajectory of this system through phase space, the 6N-dimensional space

x

p

m

Z

26

26

of positions and momenta of all the particles, can be drawn. See Figure 4.1 for a sketch, where

only one particle position and momentum is involved. The model Hamiltonian in Eq. 4.3 is for

constant energy, so that the trajectory in phase space becomes a circle for properly scaled

axes. When damping is added, this will become a spiral going from the starting point towards

the origin.

Figure 4.2 Externally driven pendulum1: (left) initial behavior before settling into resonance, damped, (middle)

regular driven, undamped, (right) driven chaotic, undamped.

A physical pendulum is only by approximation a harmonic oscillator, because gravitation acts

through the sine of the deviation angle rather than the angle itself. For larger excursions it is

therefore non-linear. This in itself causes the pendulum already to exhibit interesting behavior

when driven. Damping may slightly reduce the effect. Examples of this behavior are shown in

Fig. 4.2.

Chaotic behavior is defined as extreme sensitivity to initial conditions for a dynamic system.

This means, that if the trajectory is started slightly removed from an earlier position the

trajectories will be completely different. It implies, that when numerical simulations are used to

study the behavior of dynamical systems one needs to focus on bundles of trajectories rather

than on individual trajectories.

Let us now analyze situations where fluid systems might become chaotic. The first example

that comes to mind is that of a fluid in a closed and isolated box. It is a standard result from

classical mechanics that N-particle systems are analytically insoluble for N larger than 2.

Hence, at any rate one has to resort to numerical techniques to analyze them. It renders the

question whether the system could be chaotic a realistic one. For an ideal gas one assumes

that collisions between the particles are negligible. In that particular case one effectively deals

with a system of independent particles and one may consider the trajectory of each particle

individually without regarding the others. This problem can be solved, even in a closed box.

The trajectories may be interesting when considering the walls and especially the corners. This

makes billiards such an interesting game! As soon as collisions come into play, the system

becomes non-linear and could become chaotic.

Moreover, the initial condition of a realistic system is hardly known. One might produce a

probability distribution of initial conditions though. As a fluid system can be modeled as a

system with a Hamiltonian, the trajectories are defined by the Hamilton equations. The

Liouville equation can now be used to describe the time evolution of the probability distribution

of initial conditions2. This implies that given an initial distribution of particle positions and

momenta, one may evaluate the distribution at any time t later. Distributions that are invariant

under the Liouville equation must be equilibrium distributions such as the canonical partition

function for a closed system kept at constant temperature.

Values for quantities that can be expressed in terms of particle coordinates – these can be

more involved than just positions and momenta and can for instance include orientation – can

then be evaluated as a function of time using the time-dependent distribution function much in

the same way as is done for the equilibrium distributions. In the special case of a stationary

distribution, that then is an equilibrium distribution, the value that one obtains for a mechanical

1 Taken from: http://demonstrations.wolfram.com/ThePendulumFromSimpleHarmonicMotionToChaos/

2 The Liouville equation is essentially is based on the conservation of the normalization of the probability distribution defined in

phase space.

2726

26

of positions and momenta of all the particles, can be drawn. See Figure 4.1 for a sketch, where

only one particle position and momentum is involved. The model Hamiltonian in Eq. 4.3 is for

constant energy, so that the trajectory in phase space becomes a circle for properly scaled

axes. When damping is added, this will become a spiral going from the starting point towards

the origin.

Figure 4.2 Externally driven pendulum1: (left) initial behavior before settling into resonance, damped, (middle)

regular driven, undamped, (right) driven chaotic, undamped.

A physical pendulum is only by approximation a harmonic oscillator, because gravitation acts

through the sine of the deviation angle rather than the angle itself. For larger excursions it is

therefore non-linear. This in itself causes the pendulum already to exhibit interesting behavior

when driven. Damping may slightly reduce the effect. Examples of this behavior are shown in

Fig. 4.2.

Chaotic behavior is defined as extreme sensitivity to initial conditions for a dynamic system.

This means, that if the trajectory is started slightly removed from an earlier position the

trajectories will be completely different. It implies, that when numerical simulations are used to

study the behavior of dynamical systems one needs to focus on bundles of trajectories rather

than on individual trajectories.

Let us now analyze situations where fluid systems might become chaotic. The first example

that comes to mind is that of a fluid in a closed and isolated box. It is a standard result from

classical mechanics that N-particle systems are analytically insoluble for N larger than 2.

Hence, at any rate one has to resort to numerical techniques to analyze them. It renders the

question whether the system could be chaotic a realistic one. For an ideal gas one assumes

that collisions between the particles are negligible. In that particular case one effectively deals

with a system of independent particles and one may consider the trajectory of each particle

individually without regarding the others. This problem can be solved, even in a closed box.

The trajectories may be interesting when considering the walls and especially the corners. This

makes billiards such an interesting game! As soon as collisions come into play, the system

becomes non-linear and could become chaotic.

Moreover, the initial condition of a realistic system is hardly known. One might produce a

probability distribution of initial conditions though. As a fluid system can be modeled as a

system with a Hamiltonian, the trajectories are defined by the Hamilton equations. The

Liouville equation can now be used to describe the time evolution of the probability distribution

of initial conditions2. This implies that given an initial distribution of particle positions and

momenta, one may evaluate the distribution at any time t later. Distributions that are invariant

under the Liouville equation must be equilibrium distributions such as the canonical partition

function for a closed system kept at constant temperature.

Values for quantities that can be expressed in terms of particle coordinates – these can be

more involved than just positions and momenta and can for instance include orientation – can

then be evaluated as a function of time using the time-dependent distribution function much in

the same way as is done for the equilibrium distributions. In the special case of a stationary

distribution, that then is an equilibrium distribution, the value that one obtains for a mechanical

1 Taken from: http://demonstrations.wolfram.com/ThePendulumFromSimpleHarmonicMotionToChaos/

2 The Liouville equation is essentially is based on the conservation of the normalization of the probability distribution defined in

phase space.

27

quantities will fluctuate around an average value. If the model is appropriately defined, this

time-averaged quantity indeed is the model representative of an experimentally observed

quantity. Even though a distribution may be stationary, all points in phase space will be visited

albeit with a frequency that is determined by the distribution: some regions more than others.

The process of averaging over all points in phase space with proper weight determined by the

equilibrium distribution is then the equivalent of the time-averaged quantities obtained from a

phase-space-average using the time-dependent distribution. This is what one usually refers to

as the ergodic hypothesis.

Stability

An associated issue is that of stability. When a system is prepared in a stationary point, it is

interesting to know whether after a small perturbation it will return to the stationary point or

whether it departs on a journey never to return. For a set of linear differential systems simply

written in the form of

y Ay

, such as dynamical systems defined by a Hamiltonian are, this

return to the stationary point will always be the case provided that the real parts of the

eigenvalues of the matrix A are non-negative. For non-linear systems one linearizes the

system near a stationary point and subsequently uses the above result. This is called Lyapunov

stability analysis. However, too large excursions may lead away from the stationary point.

Furthermore, the stationary solution may actually be a more complicated structure such as a

loop.

For the example of a fluid in a closed, isolated container, the microcanonical distribution is

absolutely stable for the ideal gas. Moreover, when particles mildly interact the system is still

Lyapunov stable because of the conserved quantities. The fixed point is then the equilibrium

distribution.

Chemical reactions

Stability is also an important aspect of chemically reacting systems and it is interesting to see

that many analogies can be drawn between the above considerations for Hamiltonian systems

and for reacting systems. The study of such systems calls for a macroscopic description of the

system.

It was Théophile de Donder (1872-1957) who quantified the concept of chemical affinity as

kk

k

A

QP

¦

(4.4)

Interestingly, this is what in other places is called the Gibbs reaction energy apart from a minus-

sign. In the sense of de Donder it is the affinity that drives a chemical reaction. The extent of a

reaction, denoted by

[

, quantifies the proceeding of the reaction. In the language of non-

equilibrium thermodynamics, a positive affinity drives the system to increase the extent of the

reaction. The corresponding entropy production is then given by their product, i.e.

10

i kk

k

A

d S dn d

TT

P[

t

¦

(4.5)

Here, stoichiometric coefficients are taken positive for products and negative for reactants. As

a final definition, the reaction rate in a system is given as

1

fb

d

r RR

V dt

[

(4.6)

and can be split in a forward rate and a backward rate. Chemical reaction rates are typically

proportional to (some power of) the concentration of the reactants. For the present purpose,

28

28

we use activities rather than concentrations. This is practical for the analysis as we shall see

but it has the happy side effect that the rate coefficients all have the same dimension as have

the reaction rates. As an example, for the reaction

AB C

DE J

U

the forward rate is given by

ABff

R ka a

DE

and the backward rate by

Cbb

R ka

J

. The powers are taken to be the stoichiometric

coefficients which only will hold for elementary reactions.

Equilibrium can now be defined in two different ways. The one situation when there is no

affinity. This leads to a condition on the chemical potentials as can be seen from eq. 4.4 which,

when the chemical potentials are expressed in terms of activities leads to the familiar

equilibrium constant. The other situation is when there is no reaction rate, i.e. when forward

and backward reaction rate balance. This leads in the standard way to identify the ratio of

forward reaction rate coefficient over the backward reaction rate coefficient with the equilibrium

constant. Also here, one has the interesting result that at equilibrium the entropy is at maximum

where the entropy production, being the product of affinity and reaction rate over temperature,

is minimal; it vanishes. However, the relation between reaction rate and affinity is non-linear

and only very near equilibrium it may be linearized.

Le Chatelier’s principle

For chemically reactive systems it is interesting to see if a stability criterion can be formulated.

Given the fact, that entropy should be maximal in equilibrium, any excursion

G[

of the extent

away from equilibrium should lower the entropy. The below calculation shows that this indeed

is the case provided that the affinity decreases in the neighborhood of equilibrium which is the

same as having a minimal Gibbs energy at equilibrium.

2

eq

00 0

eq eq

11

0

2

ii

AA A

S dS d d

TT T

G[ G[ G[

[ [ [ [ G[

[[

§· §·

ww

'

¨¸ ¨¸

ww

©¹ ©¹

³³ ³ (4.6)

In the above calculation two steps are made. First the entropy change due to a change in the

extent of the reaction is written down. The affinity is then expanded around its equilibrium

value, it vanishes there, and it is concluded that for a negative entropy change the slope of

affinity with reaction extent should be negative. This is exactly the content of Le Chatelier’s

principle which is, roughly stated “Any change in status quo prompts an opposing reaction in

the responding system” or even more strongly “The system always kicks back”.

Autocatalytic reaction in an ideal continuously stirred tank reactor (CSTR)

In order to bring a chemically reacting system in an unstable situation requires (1) a stationary

point away from equilibrium and (2) some kind of non-linearity. To study this, there is the

classical example of the ideal CSTR in which an autocatalytic reaction is done.

Figure 4.3: Performance of a catalyzed reaction in an ideal CSTR.

The performance can be gathered from Figure 4.3. For low feed rates, the reaction is almost

complete and little reactant is left in the efflux. One may increase the feed rate up to 0.25 where

2928

28

we use activities rather than concentrations. This is practical for the analysis as we shall see

but it has the happy side effect that the rate coefficients all have the same dimension as have

the reaction rates. As an example, for the reaction

AB C

DE J

U

the forward rate is given by

ABff

R ka a

DE

and the backward rate by

Cbb

R ka

J

. The powers are taken to be the stoichiometric

coefficients which only will hold for elementary reactions.

Equilibrium can now be defined in two different ways. The one situation when there is no

affinity. This leads to a condition on the chemical potentials as can be seen from eq. 4.4 which,

when the chemical potentials are expressed in terms of activities leads to the familiar

equilibrium constant. The other situation is when there is no reaction rate, i.e. when forward

and backward reaction rate balance. This leads in the standard way to identify the ratio of

forward reaction rate coefficient over the backward reaction rate coefficient with the equilibrium

constant. Also here, one has the interesting result that at equilibrium the entropy is at maximum

where the entropy production, being the product of affinity and reaction rate over temperature,

is minimal; it vanishes. However, the relation between reaction rate and affinity is non-linear

and only very near equilibrium it may be linearized.

Le Chatelier’s principle

For chemically reactive systems it is interesting to see if a stability criterion can be formulated.

Given the fact, that entropy should be maximal in equilibrium, any excursion

G[

of the extent

away from equilibrium should lower the entropy. The below calculation shows that this indeed

is the case provided that the affinity decreases in the neighborhood of equilibrium which is the

same as having a minimal Gibbs energy at equilibrium.

2

eq

00 0

eq eq

11

0

2

ii

AA A

S dS d d

TT T

G[ G[ G[

[ [ [ [ G[

[[

§· §·

ww

'

¨¸ ¨¸

ww

©¹ ©¹

³³ ³ (4.6)

In the above calculation two steps are made. First the entropy change due to a change in the

extent of the reaction is written down. The affinity is then expanded around its equilibrium

value, it vanishes there, and it is concluded that for a negative entropy change the slope of

affinity with reaction extent should be negative. This is exactly the content of Le Chatelier’s

principle which is, roughly stated “Any change in status quo prompts an opposing reaction in

the responding system” or even more strongly “The system always kicks back”.

Autocatalytic reaction in an ideal continuously stirred tank reactor (CSTR)

In order to bring a chemically reacting system in an unstable situation requires (1) a stationary

point away from equilibrium and (2) some kind of non-linearity. To study this, there is the

classical example of the ideal CSTR in which an autocatalytic reaction is done.

Figure 4.3: Performance of a catalyzed reaction in an ideal CSTR.

The performance can be gathered from Figure 4.3. For low feed rates, the reaction is almost

complete and little reactant is left in the efflux. One may increase the feed rate up to 0.25 where

29

conversion is still 50%. Beyond that rate, the performance jumps to a very low conversion

where almost only reactant is found in the efflux. Only when the feed rate is brought down

below 0.1 will the conversion recover. There is clearly hysteresis in the performance of this

system. Needless to say that this is an idealized example and in practice one will always have

to consider spatial inhomogeneities as well.

Prigogine’s minimum entropy production theorem

When processes, including chemical reactions, are operated close to equilibrium they are

commonly found to be stable. In addition, they run efficiently as demonstrated in the above

example for low feed rates. For that situation Ilya Progogine formulated in 1947 his famous

theorem “In the linear regime, the total entropy production in a system is subject to flow of

energy and matter, diS/dt, reaches a minimum value at the non-equilibrium stationary state.”.

Since then, there have been many developments in the field, but a more definite statement

could not be made so far.

In conclusion, to remain on the safe side many processes are operated close to equilibrium.

More needs to be known about systems operated far from equilibrium before a safe process

can be built around it. The present examples of chemical instability are all very interesting toy

models but fail direct contact with detailed analysis. Hence, in chemical engineering one

prefers to stay away from the brink of chaos.

31

31

The Fluctuation Dissipation Theorem: another guise of the 2nd Law?

Time scales

Physical phenomena occur on time scales that vary enormously and over many orders of

magnitude. On the scale ranging roughly from 10-16 s to 10í8 s microscopic processes occur

for which it is necessary to invoke particle mechanics using either Newton’s or Schrödinger’s

equation. On timescales from 1 s on, one considers systems to be in equilibrium. It is the

regime where classical thermodynamics applies, although many systems may still show aging

behavior. In the range from 10í8 s to 1 s one typically has the domain in which diffusion is

observed, on the low end the diffusion of (Brownian) tracer particles in a fluid and on the high

end the collective diffusion that is caused by for instance concentration gradients. This is the

regime that is of interest here and which is of utmost importance to chemical engineering.

In what follows, use will be made of a statistical mechanical description of chemical systems.

For their equilibrium properties this framework allows one to find relations between mechanical

properties and some, usually conserved, quantities such as particle numbers, volume,

temperature, and external fields. The description is statistical in the sense that averages and

magnitudes of the fluctuations around these averages are calculated. The origin of the

fluctuations lies in the fast microscopic processes. For the non-equilibrium properties a closer

examination of the fluctuations is necessary in the sense that the fast microscopic fluctuations

have to be discriminated from the slowly varying quantities that are of interest. The averages

and fluctuations around averages are calculated in such a way that the fast variations are

eliminated while the slow variations persist.

Linear response

Figure 5.1 : Demonstration of linear response, where a particular time dependent field h1 results in a time

dependent response A1 and another field h2 results in another response A2 , so that the sum field h1 + h2

results in the response A1 + A2. Dashed lines indicate the time-dependent fields and continuous lines the

responses.

It has been abundantly verified, that equilibrium systems respond linearly to small

perturbations out of the equilibrium state. This linear response is most conveniently

summarized by the expression where the time-dependence of an arbitrary observable A, which

in equilibrium has the value Aeq, is expressed in the form of the convolution integral

eq

() ( ) ( )A t A t s h s ds

I

f

f

³

(5.1)

where the stimulus h is a force field that not necessarily couples directly to the observable A.

There are three observations that need to be made with respect to this equation:

x The linearity is not only reflected by the fact that twice as strong a stimulus results in

twice as strong a response, but also that when a particular time dependent field h1

results in a time dependent response A1and another field h2results in another

response A2 , that the sum field h1 + h2 results in the response A1 + A2, see figure 5.1.

This is sometimes referred to as the superposition principle.

32

32

x It is commonly assumed, that there can be no response before a stimulus has been

applied, this is the principle of causality. Because of that, the response function

0t

I

for t < 0, so that the upper boundary of the integral in Eq.(5.1) can be taken to

be infinity even though there is no response to be expected beyond time t.

x Also, the response is assumed to be independent of the actual moment in time at which

the stimulus is applied, only the time lapse between stimulus and observation is

relevant. For equilibrium systems, this stationarity principle is obeyed but for

nonequilibrium systems, the response function may depend on time and the system is

said to age. In the latter case, one may still use a convolution integral as in Eq.(5.1)

where the age dependence of the response function, i.e.

,' 'tt t t

II

z

.

Frequency domain

In the case where the stimulus is a small oscillating field

i

() t

ht he

Z

Z

(5.2)

the response is oscillating as well with the amplitude

Ah

Z ZZ

I

(5.3)

in which the Fourier transformed response function,

i

() s

s e ds

Z

Z

II

f

f

³

(5.4)

which is, in general, complex with real part

I

c

and imaginary part

I

cc

. The Fourier transformed

response function is real valued because of causality. As a consequence the Kramers - Kronig

or dispersion relations hold, implying that once either the real or the imaginary part of the

Fourier transformed response function is fully known, the other part can be calculated. Also,

only either the real or the imaginary part is required to find the response function by inverse

Fourier transformation.

Example

Figure 5.2 : (left) Electrolyte in beaker together with two electrodes with (right) its response function.

As an example, let us consider the electrolyte in Figure 5.2 with two immersed electrodes.

Under the influence of an electric field, the ions will migrate to the electrodes; the positively

charged ions to the negative electrode and vice versa. The thus induced charge separation

induces charge displacement in the electrodes which is observed as an electrical current.

Instead of what is depicted in Figure 5.2, one typically applies an alternating electric field so

that the distance over which the ions migrate is small compared to the distance between the

electrodes. For larger amplitudes and lower oscillation frequencies, the excursions become

comparable to the distance between the electrodes and electrode polarization occurs.

For sufficiently small amplitudes neither electrode polarization nor charge transfer across the

electrolyte-electrode interface occur. In that regime the stimulus is the electric field and the

response is the electrical current density. The response function is the admittance, i.e. the

inverse of the impedance

3332

32

x It is commonly assumed, that there can be no response before a stimulus has been

applied, this is the principle of causality. Because of that, the response function

0t

I

for t < 0, so that the upper boundary of the integral in Eq.(5.1) can be taken to

be infinity even though there is no response to be expected beyond time t.

x Also, the response is assumed to be independent of the actual moment in time at which

the stimulus is applied, only the time lapse between stimulus and observation is

relevant. For equilibrium systems, this stationarity principle is obeyed but for

nonequilibrium systems, the response function may depend on time and the system is

said to age. In the latter case, one may still use a convolution integral as in Eq.(5.1)

where the age dependence of the response function, i.e.

,' 'tt t t

II

z

.

Frequency domain

In the case where the stimulus is a small oscillating field

i

() t

ht he

Z

Z

(5.2)

the response is oscillating as well with the amplitude

Ah

Z ZZ

I

(5.3)

in which the Fourier transformed response function,

i

() s

s e ds

Z

Z

II

f

f

³

(5.4)

which is, in general, complex with real part

I

c

and imaginary part

I

cc

. The Fourier transformed

response function is real valued because of causality. As a consequence the Kramers - Kronig

or dispersion relations hold, implying that once either the real or the imaginary part of the

Fourier transformed response function is fully known, the other part can be calculated. Also,

only either the real or the imaginary part is required to find the response function by inverse

Fourier transformation.

Example

Figure 5.2 : (left) Electrolyte in beaker together with two electrodes with (right) its response function.

As an example, let us consider the electrolyte in Figure 5.2 with two immersed electrodes.

Under the influence of an electric field, the ions will migrate to the electrodes; the positively

charged ions to the negative electrode and vice versa. The thus induced charge separation

induces charge displacement in the electrodes which is observed as an electrical current.

Instead of what is depicted in Figure 5.2, one typically applies an alternating electric field so

that the distance over which the ions migrate is small compared to the distance between the

electrodes. For larger amplitudes and lower oscillation frequencies, the excursions become

comparable to the distance between the electrodes and electrode polarization occurs.

For sufficiently small amplitudes neither electrode polarization nor charge transfer across the

electrolyte-electrode interface occur. In that regime the stimulus is the electric field and the

response is the electrical current density. The response function is the admittance, i.e. the

inverse of the impedance

33

iY

Z

V ZH

(5.5)

Its behavior as a function of frequency is depicted in Figure 5.2. For low frequencies, only the

ionic conductivity is observable whereas for higher frequencies the polarization due to the

charge separation dominates the admittance. The cross-over frequency only depends on

material constants, i.e. the conductivity V and the dielectric permittivity H.

Dissipation

When a field h is applied to a system, this couples to a macroscopic observable B of the

system, for instance in the case of an electric field this is the polarization due to charge

separation. The energy that is supplied to the system is given by

dW hdA

and the power,

which is the rate of energy change, is

() () ( ) ( )

t

dW dA

Pt h ht t shsds

dt dt

I

f

³

(5.6)

In the case of an oscillating field, such as in eq.(5.2), one finds that the instantaneous power

is oscillating at twice the frequency and the average over one full period, which is the actual

power absorbed, is given by

'' 2

() 2

Pt h

ZZ

ZI

(5.7)

This demonstrates the general feature that the imaginary part of the Fourier transformed

response function controls the power dissipation of the system. The real part, that is connected

to the instantaneous response of the system, represents the reversible (elastic) response of

the system whereas the imaginary part represents the irreversible (dissipative) part.

Fluctuations

In actual fact, the system’s observables are fluctuating in time and the linear response equation

(5.1) applies to a time average where the time window T is chosen such that it is large enough

to smoothen away the fast fluctuations yet small enough so that the (slower) response persists.

In equilibrium, the observable fluctuates around its average value, and the obtained value

should be independent of the time window provided that it is sufficiently large.

The equilibrium fluctuations are characterized by their correlations that are described by their

correlation function

/2

/2

1

( ) lim ( ) ( )

T

TT

Ct At s Asds

T

of

³

(5.8)

The frequency spectrum of these fluctuations is obtained by taking the Fourier transform of

eq.(5.8).

is

S C s e ds

Z

Z

f

f

³

(5.9)

In the limit of an infinite time window yields the Wiener - Khinchin Theorem that relates the

power spectrum of the equilibrium fluctuations to their correlation function.

Fluctuations about the equilibrium state decay according to the same linear laws which

describe the decay of the system from a nonequilibrium to an equilibrium state. Therefore, the

equilibrium fluctuations provide a way to probe the response of the system without actually

disturbing it. This is summarized by the Fluctuation - Dissipation Theorem that relates the

power spectrum of the equilibrium fluctuations to the imaginary part of the Fourier transformed

response system, that controls the dissipation of the system

''

2kT

S

ZZ

I

Z

(5.10)

This relation has been demonstrated to be true for many experimental systems but equally

well it has been the subject of dispute amongst theorists.

34

34

A sketch of a proof

The underlying idea of the proof is that the time-evolution of an observable of the system at

hand is solely given by the evolving microscopic coordinates of the system. Let us denote

these coordinates by a super-vector Xwhich includes positions, momenta and, for instance,

also orientations of the molecule. The value of an observable A is then obtained by taking the

average value of the corresponding “mechanical” variable

A

as it depends solely on the

coordinates

X

. An example would be the system’s center of mas for which the mechanical

variable is the sum of the coordinates of all the particles divided by their total number. As

already discussed in the previous chapter the “ensemble” average is calculated as

() ( ) ,At t d

U

³

XX XA

(5.11)

where

,t

U

X

is the probability distribution for the coordinates Xat the time t. Likewise a

correlation function can be obtained as the average over time dependent pair distribution

function.

2

(,) ( )( ') (, ;,)

AB

C tt tt d d

U

c cc c

³³

X X XX X XAB

(5.12)

In order to proceed, one needs to know the time evolution of the distribution function. This is

given by the Liouville equation also discussed in the previous chapter. Given the probability

distribution at time t’, one obtains the distribution at time t as

( , ) ( , ', ') ', ' 't G t t td

UU

³

X XX X X

(5.13)

where G is the Liouville operator. Important to note is that an equilibrium distribution eq

U

is

invariant under the operation denoted in eq.(5.13); an example would be the canonical

ensemble distribution discussed in chapter 2. In the same spirit also the pair distribution can

be obtained as

2( , ; , ) ( , ', ') ', 'tt G t t t

UU

cc

XX X X X

(5.14)

Geared up with these notions, we now turn to a hypothetical experiment in which we consider

a small stimulus h to be applied until time t = 0 at which time it is switched off. The system has

been so long under the stimilus h that it can be considered in equilibrium at the time it is

switched off. After that moment of time, the system relaxes back to an equilibrium where there

is no applied stimulus. The stimulus h couples to a mechanical variable

B

so that the

Hamiltonian of the perturbed system is enhanced by a term

h

G

XXHB

. An example

could be a magnetic field as stimulus and total magnetization as mechanical variable. The

enhancement of the Hamiltonian of the system is then the magnetic energy. Because the

stimulus is considered small, also the perturbation of the Hamiltonian is small so that the initial

distribution function at time t = 0 can be approximated as

^`

eq eq

,0 1 hB

U UE

ªº

¬¼

XXB

(5.15)

Using now the above discussed definitions and applying some – not always trivial –

mathematical operations – one finds for the response of an observable A of the system that

eq eq eq

() ()

AB

At A h C t A B

E

ªº

¬¼

(5.16)

Comparing this to eq.(5.2) for the situation where the applied field couples to the same variable

as the observable yields the so-called Green-Kubo-relation

()

()

AB

AB

dC t

tdt

IE

(5.17)

which after Fourier transformation indeed yields the required Fluctuation-Dissipation-Theorem.

3534

34

A sketch of a proof

The underlying idea of the proof is that the time-evolution of an observable of the system at

hand is solely given by the evolving microscopic coordinates of the system. Let us denote

these coordinates by a super-vector Xwhich includes positions, momenta and, for instance,

also orientations of the molecule. The value of an observable A is then obtained by taking the

average value of the corresponding “mechanical” variable

A

as it depends solely on the

coordinates

X

. An example would be the system’s center of mas for which the mechanical

variable is the sum of the coordinates of all the particles divided by their total number. As

already discussed in the previous chapter the “ensemble” average is calculated as

() ( ) ,At t d

U

³

XX XA

(5.11)

where

,t

U

X

is the probability distribution for the coordinates Xat the time t. Likewise a

correlation function can be obtained as the average over time dependent pair distribution

function.

2

(,) ( )( ') (, ;,)

AB

C tt tt d d

U

c cc c

³³

X X XX X XAB

(5.12)

In order to proceed, one needs to know the time evolution of the distribution function. This is

given by the Liouville equation also discussed in the previous chapter. Given the probability

distribution at time t’, one obtains the distribution at time t as

( , ) ( , ', ') ', ' 't G t t td

UU

³

X XX X X

(5.13)

where G is the Liouville operator. Important to note is that an equilibrium distribution eq

U

is

invariant under the operation denoted in eq.(5.13); an example would be the canonical

ensemble distribution discussed in chapter 2. In the same spirit also the pair distribution can

be obtained as

2( , ; , ) ( , ', ') ', 'tt G t t t

UU

cc

XX X X X

(5.14)

Geared up with these notions, we now turn to a hypothetical experiment in which we consider

a small stimulus h to be applied until time t = 0 at which time it is switched off. The system has

been so long under the stimilus h that it can be considered in equilibrium at the time it is

switched off. After that moment of time, the system relaxes back to an equilibrium where there

is no applied stimulus. The stimulus h couples to a mechanical variable

B

so that the

Hamiltonian of the perturbed system is enhanced by a term

h

G

XXHB

. An example

could be a magnetic field as stimulus and total magnetization as mechanical variable. The

enhancement of the Hamiltonian of the system is then the magnetic energy. Because the

stimulus is considered small, also the perturbation of the Hamiltonian is small so that the initial

distribution function at time t = 0 can be approximated as

^`

eq eq

,0 1 hB

U UE

ªº

¬¼

XXB

(5.15)

Using now the above discussed definitions and applying some – not always trivial –

mathematical operations – one finds for the response of an observable A of the system that

eq eq eq

() ()

AB

At A h C t A B

E

ªº

¬¼

(5.16)

Comparing this to eq.(5.2) for the situation where the applied field couples to the same variable

as the observable yields the so-called Green-Kubo-relation

()

()

AB

AB

dC t

tdt

IE

(5.17)

which after Fourier transformation indeed yields the required Fluctuation-Dissipation-Theorem.

35

Origin

It was Einstein who in 1905 gave the first explanation for Brownian motion in terms of random

collisions of water, or any other solvent, molecules with a particle. Over infinitely long time, the

momentum transfer due to these collisions averages out to zero but over any finite time interval

there is a net momentum transfer. Each time interval with a different magnitude and different

orientation. These make the particle move randomly. So the thermal motion of the water

molecules, the particle’s thermal motion is negligible on this scale, is responsible for the

fluctuations in the particle position and velocity. On the other hand, when a particle is driven

through water it experiences friction. The particle’s motion transfers some of the particle

momentum to the water molecules. This excess momentum is transferred to the many other

water molecules and increases their kinetic energy. The total kinetic energy of the water

increases which is noticed as an increase in temperature. The velocity of the particle is reduced

due to the drag which causes the temperature of the water to increase. As an aside, the

explanation heavily hinges on the fact that molecules and atoms exist which was barely

accepted in those days.

Later, Einstein generalized this explanation by stating that “If there is a process that dissipates

energy, turning it into heat, there is a reverse process related to thermal fluctuations.”. And

indeed, there are other processes that can be explained along the same lines. For instance

Johnson noise which relates electrical resistance of a piece of material to the voltage

fluctuations that can be measured across the same material. Then there is Kirchhoff’s law on

thermal radiation that couples the absorption coefficient to the observed black body radiation.

Finally we mention also composition fluctuations and reaction rates. Both can be observed by

fluorescence or absorption of the reacting components.

Applications

The fact that processes dissipate energy because that is turned into thermal energy has also

been generalized by Einstein. A simple example is that of a colloidal dispersion under gravity.

If the particles are very heavy, compared to the solvent, they will quickly end up at the bottom

of the vessel and likewise when they are light they will “cream” up to the top of the vessel. The

in-between situation creates a concentration gradient along the vessel, see Figure 5.3.

Figure 5.3: Barometric height distribution due to balance of diffusion and gravitation.

The colloidal particles experience a force downward, assuming they are heavier than water,

due to gravity which causes a flux as given in Fig. 5.3. The concentration gradient causes an

upward flux with a strength determined by the diffusion coefficient. In the stationary situation a

barometric height distribution develops as can be derived by equating the upward and

downward flux. The density distribution exponentially decays upward with a characteristic

length scale that is proportional to the ratio of the diffusion coefficient D and the friction

coefficient f (see Figure 5.3), yielding kBT. This ratio is the Fluctuation Dissipation Theorem

applied to this case and is more generally termed an Einstein relation. Similar reasoning yields

the Nernst equation and electronic mobility in semiconductors.

The situation described above for Brownian motion has successfully been generalized and

used in instruments to extract rheological information from the fluctuations of colloidal test

particles in a liquid. Not only does the diffusion coefficient give information about the viscosity

of the medium when the particle diameter is known. Also the power spectrum of the particle

displacements gives information on frequency dependent rheological parameters such as

36

36

storage modulus and dissipation. The technique is – for obvious reasons – called

microrheology. The Brownian motion of the test particles can be observed by microscopy

enhanced by image analysis techniques for relatively low frequencies and by light scattering

for the higher frequencies.

Aging

In the beginning of this chapter we mentioned stationarity as a requirement for linear response

to hold. In actual fact this is never true and indeed, linear response in itself is a limiting law in

the sense discussed in earlier chapters. Realistic systems decay, since their “birth”, towards

equilibrium. This can actually be monitored by the fluctuation dissipation relation where only in

the long time limit one will observe a constant value. At present, soft matter systems are in

vogue and these often are glassy in nature. This means that they are locked into a metastable

situation not to reach equilibrium within finite time … if ever. For these systems the FDT holds

up to a constant as has been shown by many examples nowadays.

Conclusion

The name of the chapter is actually a question: “The Fluctuation Dissipation Theorem: another

guise of the 2nd Law?”. In order to answer this question, we follow Einstein’s argument.

Systems out of equilibrium experience fluxes that are driven by energy gradients. The

proportionality factor indicates the dissipation strength. These fluxes create entropy gradients

that in themselves drive opposing fluxes. The proportionality factor between entropy gradient

and associated flux is one that dictates the magnitude of fluctuations. In equilibrium these

fluxes balance and this leads to a relation between the proportionality factors, both involving

rate information. More precisely, their ratio is in equilibrium a constant times thermal energy.

It is the Second Law that predicts that these fluxes will ultimately balance which leads to the

Fluctuation Dissipation Theorem. Conversely, as a consequence of the FDT the fluxes will

ultimately balance which is the content of the Second Law.

3736

36

storage modulus and dissipation. The technique is – for obvious reasons – called

microrheology. The Brownian motion of the test particles can be observed by microscopy

enhanced by image analysis techniques for relatively low frequencies and by light scattering

for the higher frequencies.

Aging

In the beginning of this chapter we mentioned stationarity as a requirement for linear response

to hold. In actual fact this is never true and indeed, linear response in itself is a limiting law in

the sense discussed in earlier chapters. Realistic systems decay, since their “birth”, towards

equilibrium. This can actually be monitored by the fluctuation dissipation relation where only in

the long time limit one will observe a constant value. At present, soft matter systems are in

vogue and these often are glassy in nature. This means that they are locked into a metastable

situation not to reach equilibrium within finite time … if ever. For these systems the FDT holds

up to a constant as has been shown by many examples nowadays.

Conclusion

The name of the chapter is actually a question: “The Fluctuation Dissipation Theorem: another

guise of the 2nd Law?”. In order to answer this question, we follow Einstein’s argument.

Systems out of equilibrium experience fluxes that are driven by energy gradients. The

proportionality factor indicates the dissipation strength. These fluxes create entropy gradients

that in themselves drive opposing fluxes. The proportionality factor between entropy gradient

and associated flux is one that dictates the magnitude of fluctuations. In equilibrium these

fluxes balance and this leads to a relation between the proportionality factors, both involving

rate information. More precisely, their ratio is in equilibrium a constant times thermal energy.

It is the Second Law that predicts that these fluxes will ultimately balance which leads to the

Fluctuation Dissipation Theorem. Conversely, as a consequence of the FDT the fluxes will

ultimately balance which is the content of the Second Law.

37

What follows: Boltzmann’s equation from Navier-Stokes’ equations or vice versa?

Fluid element

Hydrodynamics is a form of continuum mechanics which, at the time when the Navier-Stokes

equations were devised, was just fine. When they would need the concept of a fluid element –

for instance to describe the effect of velocity gradients around a point in space – they could

take any length scale they could think of. Since we now know that fluids are actually composed

of molecules some justification is necessary. The common saying is that a fluid element needs

to be small enough so that velocity and pressure gradients are negligible while being large

enough so that physical observables such as pressure, temperature and density are well-

defined. That statement leaves a lot to wish for. For instance, does such an element actually

exist and if so, what size would it have? To answer this question, we consider the situation of

a pocket of fluid with varying length scale d, see Figure 6.1.

Figure 6.1: (left) Sketch of a fluid element of length scale d and

(right) graph of density versus length scale.

For very large length scales, there are very many molecules in the pocket and the density,

being the number of molecules over the volume of the pocket, is well defined with negligible

fluctuations. For very small length scales, say of the order of the size of the molecules, the

density will fluctuate between zero – no molecule in the pocket – or some finite number

corresponding to the number of molecules that maximally fit in the pocket. The Law of Large

Numbers from statistics predicts that the relative magnitude of the fluctuations in particle

number and hence in the density will scale with the inverse square root of the particle number.

The same result can be obtained from Molecular Thermodynamics. If we decide that we find a

relative uncertainty in physical observables should be one per mille (i.e. 10-3) the number of

molecules should be at least one million. For water molecules this would be a pocket of 30 nm

diameter. Sufficiently small for the purpose of describing the flow of water as a continuum.

Mass continuity

Mass and energy conservation are essential notions in science. But since hydrodynamics is

dealing with velocity fields – it is a continuum mechanics – conservation of fluid needs to be

added to be able to come to actual predictions, that is to say solutions of the flow equations.

The general statement on mass conservation reads “influx + generation – efflux = storage”.

For a fluid pocket of any size, the influx and efflux are described by

U

v

, the product of the local

density and velocity.

,

VV

dt d dS

dt

UU

w

³³

r r vn

(6.1)

One usually does not consider chemical reactions in hydrodynamics but if one would, there

could be (positive or negative) generation of matter. The storage is thus described by the

change in mass in the fluid pocket of volume V under consideration, see the left side of eq.

(6.1). On the right side is the summation of the flux through the surface over the total surface

area

Vw

of the fluid pocket. Using Gauss’ divergence theorem this can be cast in a more useful

local form as

38

38

, ,,t tt

t

UU

wªº

¬¼

wr r vr

(6.2)

although in order to explain it, one needs to resort to finite fluid pockets again …

Even if chemical reactions are not considered, there is yet another reason why the density can

locally increase or decrease – usually under the action of compression or expansion. Where

gases are certainly exhibiting this phenomenon it is rare for liquids such as water. Therefore,

one usually makes the assumption that the fluid is incompressible, i.e. that the divergence of

the velocity field vanishes

0 v

. As a consequence, the total derivative of the density – in

the field called the hydrodynamic derivative vanishes, see eq. (6.3).

,, 0

tt

Dt t

UU U

{w r rv

(6.3)

Local flow field

Figure 6.2: Local flow modes (left) rotation

(middle) expansion and (right) elongation.

Before we discuss momentum conservation, it is useful to first have a closer look at the velocity

field in an arbitrary point in the fluid. For that we consider the derivative of the velocity which is

written as

T

x x x xx xy xz

y y y x y z yx yy yz

z z z zx z y zz

v vvv

v vvv v v v

v vvv

§·

w w www

§·§· §· ¨¸

¨¸¨¸ ¨¸

w w w w w

¨¸

¨¸¨¸ ¨¸

¨¸¨¸ ¨¸ ¨¸

w w www

©¹©¹ ©¹ ©¹

v

(6.4)

As is shown in eq. (6.4) this is a tensor and as such it has three Eigenmodes, that is to say

three Eigenvalues and three Eigenvectors. One Eigenmode is solid body rotation, belonging

to a complex eigenvalue, see Figure 6.2 left. The second Eigenmode is expansion, see Figure

6.2 middle, with a real Eigenvalue and the third is elongation also with a real eigenvalue, see

Figure 6.2 right. The commonly known shear flow is a combination of solid body rotation and

elongation. In engineering, elongation is the important flow mode as it is responsible for the

breaking of jets into droplets as occurs for instance with emulsification.

Momentum conservation

Although the term momentum conservation sounds fancy, it is nothing more that Newton’s

second law that couples acceleration to force per unit mass. But in continuum mechanics it

already gives a daunting equation. The complication here is the friction that a fluid experiences

while flowing. Euler ignored friction in which case one arrives at

p

t

UU U

w

wv vv g

(6.5)

The left side of eq. (6.5) is the change in momentum of a fluid element and the first term on

the right side is the momentum flux out of the fluid element. The other two terms are forces

where the pressure gradient gives a force on the surface of the fluid element and an external

force such as gravity the body force acting on the fluid element as a whole. Despite the fact

that Euler’s equation (6.5) is a simplification of reality, it does lead to some important

consequences. For instance, it leads to Bernouilli’s equation predicting liquid flow velocity at

the opening of a vessel as a function of the liquid height inside the vessel. It is a consequence

3938

38

, ,,t tt

t

UU

wªº

¬¼

wr r vr

(6.2)

although in order to explain it, one needs to resort to finite fluid pockets again …

Even if chemical reactions are not considered, there is yet another reason why the density can

locally increase or decrease – usually under the action of compression or expansion. Where

gases are certainly exhibiting this phenomenon it is rare for liquids such as water. Therefore,

one usually makes the assumption that the fluid is incompressible, i.e. that the divergence of

the velocity field vanishes

0 v

. As a consequence, the total derivative of the density – in

the field called the hydrodynamic derivative vanishes, see eq. (6.3).

,, 0

tt

Dt t

UU U

{w r rv

(6.3)

Local flow field

Figure 6.2: Local flow modes (left) rotation

(middle) expansion and (right) elongation.

Before we discuss momentum conservation, it is useful to first have a closer look at the velocity

field in an arbitrary point in the fluid. For that we consider the derivative of the velocity which is

written as

T

x x x xx xy xz

y y y x y z yx yy yz

z z z zx z y zz

v vvv

v vvv v v v

v vvv

§·

w w www

§·§· §· ¨¸

¨¸¨¸ ¨¸

w w w w w

¨¸

¨¸¨¸ ¨¸

¨¸¨¸ ¨¸ ¨¸

w w www

©¹©¹ ©¹ ©¹

v

(6.4)

As is shown in eq. (6.4) this is a tensor and as such it has three Eigenmodes, that is to say

three Eigenvalues and three Eigenvectors. One Eigenmode is solid body rotation, belonging

to a complex eigenvalue, see Figure 6.2 left. The second Eigenmode is expansion, see Figure

6.2 middle, with a real Eigenvalue and the third is elongation also with a real eigenvalue, see

Figure 6.2 right. The commonly known shear flow is a combination of solid body rotation and

elongation. In engineering, elongation is the important flow mode as it is responsible for the

breaking of jets into droplets as occurs for instance with emulsification.

Momentum conservation

Although the term momentum conservation sounds fancy, it is nothing more that Newton’s

second law that couples acceleration to force per unit mass. But in continuum mechanics it

already gives a daunting equation. The complication here is the friction that a fluid experiences

while flowing. Euler ignored friction in which case one arrives at

p

t

UU U

w

wv vv g

(6.5)

The left side of eq. (6.5) is the change in momentum of a fluid element and the first term on

the right side is the momentum flux out of the fluid element. The other two terms are forces

where the pressure gradient gives a force on the surface of the fluid element and an external

force such as gravity the body force acting on the fluid element as a whole. Despite the fact

that Euler’s equation (6.5) is a simplification of reality, it does lead to some important

consequences. For instance, it leads to Bernouilli’s equation predicting liquid flow velocity at

the opening of a vessel as a function of the liquid height inside the vessel. It is a consequence

39

of energy conservation as the flow inside the vessel is rather low so that there is hardly any

friction.

The full Navier-Stokes’ equation adds a viscous pressure tensor to the pressure in Euler’s

equation (6.5). The viscous pressure tensor is given in eq. (6.6) and admittedly, it looks

daunting!

sym

1

Tr()2() Tr()

3

9K

½

®¾

¯¿

Ȇ Y Y Y

(6.6)

Let us therefore have a closer look. The first term is the unit tensor with a coefficient

9

times a

factor that is representative of expansion, see Figure 6.2 middle. Hence the coefficient

9

is

called the bulk viscosity. The second term consists of twice the “normal” shear viscosity K times

a factor that is representative of elongational flow, see Figure 6.2 right. Solid body rotation is

missing as it does not cause friction. The expression (6.6) for the viscous pressure tensor is a

simplification of a more general case albeit that with these two coefficients, many times only

the shear viscosity, one may describe all kinds of interesting flow behavior. Introducing more

parameters will certainly make the predictions one can make more reliable but without a good

way of estimating those this does not make good sense. It is important to realize, that both

viscosities appear in the entropy production of fluid flow where the viscous pressure tensor,

eq. (6.6), is the force where the local flow field, eq. (6.4), is the flux term.

In conclusion so far, we now have seen that Navier-Stokes’ equation is nothing more than

Newton’s Second Law albeit with a smart guess on the nature of the friction pressure tensor.

At low Reynolds number, slow flow fields, the non-linearity introduced by the term with velocity

field “squared”, i.e.

vv

, can be neglected. The thus obtained equation, valid only for

incompressible flow, is called the Stokes equation. It is in particular for this regime where most

of the practical solutions are available. In order to solve the Navier-Stokes equation, mass

continuity is required. The behavior of energy and entropy can also be obtained from these

equations albeit not a day to day practice.

Kinetic theory

The elementary kinetic theory of gases is a classical theory that describes a gas as a large

number of submicroscopic particles – nowadays called atoms or molecules – which are in

constant, rapid, motion with random velocity and direction. The randomness arises from the

many collisions that the particles have with each other and with the walls of the container. The

theory accounts for macroscopic properties of gases, such as pressure and temperature, by

considering their molecular composition and motion. Gas pressure results from the momentum

transfer due to particle collisions with the walls of a container. Temperature is associated with

the average kinetic energy of the particles as is further detailed by Maxwell’s molecular velocity

distribution. Kinetic theory furthermore accounts for diffusion coefficient, viscosity, thermal

conductivity and many more properties of gases. Collisions between molecules are in actual

fact not described. The only necessary parameter in the theory is the mean collision free path,

the mean distance between two successive collisions. Furthermore, it is assumed that

collisions randomize the velocities. In order to improve on this and describe fluids at higher

densities, it is clear that a more detailed description of what occurs during collisions is needed.

Boltzmann’s equation

Rather than what is done in Liouville’s equation which takes the ensemble of particles in the

system, the distribution function that Boltzmann’s equation considers is about the position and

velocity (momentum) of a single particle. For simplicity we assume identical particles here

although that is not really necessary. However, more than one particle type makes the notation

even more cumbersome than it already is. Hence, the basic variable is the probability density

(, ,)ftrv

that a particle at time t is at position rwith velocity

v

. The time evolution of this

distribution can then be formulated as

40

40

>@

(, ,)

d fff

f t Cf

dt t

www

www

rv v g

rv

(6.7)

where the time-derivative of position is the velocity and the time derivative of the velocity is the

acceleration or force per unit mass

g

. The right hand side of the equation,

>@

Cf

, stands for

the effect of collisions on the distribution function.

Figure 6.3 Sketch of a molecular collision.

In order to proceed with the collision operator in eq. (6.7) one needs to describe molecular

collisions. In Figure 6.3. a situation is sketched where two particles with respective velocities

v

and

c

v

collide at a position

r

and at one instant of time t. The collision results in two particles

leaving the site with velocities

1

v

and

1

c

v

. The collision is assumed to take place infinitely fast

which of course in real physical situations will not really be the case. But because the time

lapse of the collision itself is small compared to the other time scales in this problem this will

not lead to undesired side effects. Furthermore, three important aspects of the molecular

collisions will be taken into account

1. Momentum conservation, for identical particles this implies

11

cc

vv v v

.

2. Energy conservation.

3. Molecular reversibility, i.e. the particle trajectories can be reversed.

With these ingredients, the collision operator is written as

^`

1 1 1111 11

,, , , , ,C f t d d f f ff

c c c c c cc

ªº

: :

¬¼

³³

rv v v vv v v v v vv

(6.8)

where momentum conservation is already taken into account, i.e. the velocity c

vcan be

expressed in terms of the three other velocities. We have not done so to make the equation

more readable. The first term describes the increase in probability density due to collisions that

yield the velocity

v

and the second term the loss in probability density. The factor : contains

what physicists call the collision cross section and describes the success rate of the collision

with given velocities. The simplified notation for the probability densities speaks for itself where

the product of two probability densities accounts for the fact that the two particles with given

velocities have to be present at the same position and time. Interestingly, apart from the above

mentioned three properties of the collisions no more information is needed on the precise

nature of the factor : except that it is an existing entity.

However, the Boltzmann equation as given above is not precise as it assumes just like kinetic

theory that after a collision the velocities are completely randomized. However, when two

particles collide in a dense fluid, there is a significant probability that a series of collisions

develops such that the particles that initially collided get an additional collision. These

correlations are negligible for low densities but become serious at higher densities. Theories

have been developed to overcome this issue by taking multiple collisions into account such as

the Chapman-Enskog-equation are therefore more frequently cited when quantitative results

are needed.

Boltzmann’s H-theorem

4140

40

>@

(, ,)

d fff

f t Cf

dt t

www

www

rv v g

rv

(6.7)

where the time-derivative of position is the velocity and the time derivative of the velocity is the

acceleration or force per unit mass

g

. The right hand side of the equation,

>@

Cf

, stands for

the effect of collisions on the distribution function.

Figure 6.3 Sketch of a molecular collision.

In order to proceed with the collision operator in eq. (6.7) one needs to describe molecular

collisions. In Figure 6.3. a situation is sketched where two particles with respective velocities

v

and

c

v

collide at a position

r

and at one instant of time t. The collision results in two particles

leaving the site with velocities

1

v

and

1

c

v

. The collision is assumed to take place infinitely fast

which of course in real physical situations will not really be the case. But because the time

lapse of the collision itself is small compared to the other time scales in this problem this will

not lead to undesired side effects. Furthermore, three important aspects of the molecular

collisions will be taken into account

1. Momentum conservation, for identical particles this implies

11

cc

vv v v

.

2. Energy conservation.

3. Molecular reversibility, i.e. the particle trajectories can be reversed.

With these ingredients, the collision operator is written as

^`

1 1 1111 11

,, , , , ,C f t d d f f ff

c c c c c cc

ªº

: :

¬¼

³³

rv v v vv v v v v vv

(6.8)

where momentum conservation is already taken into account, i.e. the velocity c

vcan be

expressed in terms of the three other velocities. We have not done so to make the equation

more readable. The first term describes the increase in probability density due to collisions that

yield the velocity

v

and the second term the loss in probability density. The factor : contains

what physicists call the collision cross section and describes the success rate of the collision

with given velocities. The simplified notation for the probability densities speaks for itself where

the product of two probability densities accounts for the fact that the two particles with given

velocities have to be present at the same position and time. Interestingly, apart from the above

mentioned three properties of the collisions no more information is needed on the precise

nature of the factor : except that it is an existing entity.

However, the Boltzmann equation as given above is not precise as it assumes just like kinetic

theory that after a collision the velocities are completely randomized. However, when two

particles collide in a dense fluid, there is a significant probability that a series of collisions

develops such that the particles that initially collided get an additional collision. These

correlations are negligible for low densities but become serious at higher densities. Theories

have been developed to overcome this issue by taking multiple collisions into account such as

the Chapman-Enskog-equation are therefore more frequently cited when quantitative results

are needed.

Boltzmann’s H-theorem

41

Continuity equations for mass, momentum and energy can be directly obtained from

Boltzmann’s equation, but the more interesting quantity is what we nowadays would call the

entropy. Consider

>@

, , (, ,)ln (, ,) 1tstkdftft

U

³

r r v rv rv

(6.9)

which represents the entropy density as we would define it naively. Boltzmann used this

definition of entropy density integrated over the volume of the system. By using the three

properties of the collision operator as mentioned above, i.e. microscopic reversibility,

momentum and energy conservation he could demonstrate that

U

t

³

, ,0

dd

St d ts t

dt dt rr r

(6.10)

It demonstrates that microscopic reversibility leads to macroscopic irreversibility and that was

a great breakthrough at the time. Improvements of the Boltzmann theory to make it applicable

for higher density fluids do not essentially change this finding.

Conclusion

Boltzmann’s equation, or elaborations such as by Chapman and Enskog, can be used to derive

the Navier-Stokes equation and associated continuity equation. With some effort, Boltzmann’s

equation can be derived from Navier-Stokes equation. However, the important contribution of

the Boltzmann equation is that it resolves an intriguing issue, namely that microscopic

reversibility leads to macroscopic irreversibility. Needless to say, that also the entropy as

calculated using the Navier-Stokes equation leads to a statement of macroscopic irreversibility.

However, there it is not clear that microscopic reversibility is implied.

43

43

Use and Abuse of the Fokker Planck and Langevin Equations

Liouville equation

Let us review the action of the Liouville equation. It acts on probability density functions that

are a function of phase space coordinates represented by the vector X that contains positions,

momenta, orientations, etc. of the molecules in a system. Given an initial distribution

U

0

,tX

it describes the time evolution of the initial density as

UU

³

0

0

( , ) ( , ', ) ', 't G t t tdX XX X X

(7.1)

Application of the Liouville equation provides the time evolved probability density of finding a

particular configuration, i.e. positions, momenta, etc., at time t. The Green function G

represents the action of the Liouville equation. The expectation value of any observable on the

system can then be obtained using this time dependent distribution as

U

³

() ( ) ,At t dXX XA

(7.2)

The methodology that the Liouville formalism provides is identical to the situation for

equilibrium statistical mechanics albeit that here the probability density is a function of time as

well.

Fluctuations

Figure 7.1 Sketch of a fluctuating quantity with indication of the time-averaged profile

and the magnitude of the noise.

In Figure 7.1 we have sketched evolution of a time-dependent quantity A of the system. The

average value decays to a time-independent level Aeq and around this the values fluctuate with

an amplitude that is proportional to the inverse square root of system size. One typically is

interested in the slowly varying average value and not so much in the fast varying fluctuations.

The information provided by the Liouville equation is hence too detailed, much less detail would

suffice.

Master equation

Figure 7.2 Sketch of the action of a “projection operator”.

44

44

This idea, of separating slow and fast variations, can best be illustrated for the case of an

observable that only takes discrete values, such as the total spin of a cluster of atoms. One

might imagine that one could identify pockets in phase space in each of which the observable

attains one discrete value. Stepping from one value to the other is then jumping from one

pocket to another pocket in phase space. The fast fluctuations then represent evolution within

the pockets. In such a way, phase space is partitioned into a complete, possibly countably

infinite, set of pockets. Because of continuity, the pockets will be internally completely

connected and the pockets are – by definition – mutually exclusive.

To define the probability (density) that a certain observable A attains a particular value at time

t one uses a projection operator, in actual fact a filter that sieves out that part of phase space

as provides the value, i.e.

GU

³

(,) ( ) ,Pat a t dX XXA

(7.3)

Note, that the observable might also be a vector, such as a direction, of a tensor, such as the

complete polarizability of a crystal.

Master equation

The time evolution of the probability density defined above, see eq.(7.3), can now be

formulated with a so-called Master equation that reads

wccc

w

³

,,Pat Maa Patda

t

(7.4)

The important quantity is the probability M(a|a’) that the system jumps from one pocket with

value a’ to another pocket with value a. Surely it can be obtained from the Liouville equation

as described above but we will leave the details for the reader to work out (if so desired). There

are actually two contributions to the transition probability M, a reversible part that originates

from fluxes and forces acting on the system. It is asymmetrical in equilibrium, i.e.

wccc

w³

,,

R

a Pat M aa Patda

a

(7.5a)

and an irreversible part, that is symmetrical in equilibrium,

cc c

eq eqII

MaaP a MaaP a

(7.5b)

The latter is the detailed balance condition that is quite similar to the condition for reaction

rates in chemical equilibrium.

The more familiar form of the master equation is given below,

^`

ww cc c c

ww

³

,, , ,Pat a Pat Taa Pat T aaPat da

ta

(7.6)

where the first term represents the reversible part, the second term the probability gain and

the last term the probability loss. An example of its use, maybe without being aware, is the

application of Fermi’s Golden Rule to obtain the transition probability between radiation states.

There is a strong tendency in the literature to link the time dependent probability (density) P(a,t)

to the entropy using Boltzmann’s prescription. As, formally spoken, this expression is provided

only for the microcanonical ensemble, care has to be taken. However, for the canonical and

grand-canonical ensemble one may convince oneself that Boltzmann’s expression for entropy

is also valid. This, however, in itself does not guarantee the same for any time-dependent

probability even if obtained by a projection from probability distribution used in the Liouville

formalism.

Random walk

To illustrate the use of the Master equation, we consider the random walk on a line, see

Figure 7.3. The probability to make one step to the right per time step is p and to the left q. If

the sum of the two probabilities is 1 there is a finite probability to remain on the initial site. The

Master Equation now reads

4544

44

This idea, of separating slow and fast variations, can best be illustrated for the case of an

observable that only takes discrete values, such as the total spin of a cluster of atoms. One

might imagine that one could identify pockets in phase space in each of which the observable

attains one discrete value. Stepping from one value to the other is then jumping from one

pocket to another pocket in phase space. The fast fluctuations then represent evolution within

the pockets. In such a way, phase space is partitioned into a complete, possibly countably

infinite, set of pockets. Because of continuity, the pockets will be internally completely

connected and the pockets are – by definition – mutually exclusive.

To define the probability (density) that a certain observable A attains a particular value at time

t one uses a projection operator, in actual fact a filter that sieves out that part of phase space

as provides the value, i.e.

GU

³

(,) ( ) ,Pat a t dX XXA

(7.3)

Note, that the observable might also be a vector, such as a direction, of a tensor, such as the

complete polarizability of a crystal.

Master equation

The time evolution of the probability density defined above, see eq.(7.3), can now be

formulated with a so-called Master equation that reads

wccc

w

³

,,Pat Maa Patda

t

(7.4)

The important quantity is the probability M(a|a’) that the system jumps from one pocket with

value a’ to another pocket with value a. Surely it can be obtained from the Liouville equation

as described above but we will leave the details for the reader to work out (if so desired). There

are actually two contributions to the transition probability M, a reversible part that originates

from fluxes and forces acting on the system. It is asymmetrical in equilibrium, i.e.

wccc

w³

,,

R

a Pat M aa Patda

a

(7.5a)

and an irreversible part, that is symmetrical in equilibrium,

cc c

eq eqII

MaaP a MaaP a

(7.5b)

The latter is the detailed balance condition that is quite similar to the condition for reaction

rates in chemical equilibrium.

The more familiar form of the master equation is given below,

^`

ww cc c c

ww

³

,, , ,Pat a Pat Taa Pat T aaPat da

ta

(7.6)

where the first term represents the reversible part, the second term the probability gain and

the last term the probability loss. An example of its use, maybe without being aware, is the

application of Fermi’s Golden Rule to obtain the transition probability between radiation states.

There is a strong tendency in the literature to link the time dependent probability (density) P(a,t)

to the entropy using Boltzmann’s prescription. As, formally spoken, this expression is provided

only for the microcanonical ensemble, care has to be taken. However, for the canonical and

grand-canonical ensemble one may convince oneself that Boltzmann’s expression for entropy

is also valid. This, however, in itself does not guarantee the same for any time-dependent

probability even if obtained by a projection from probability distribution used in the Liouville

formalism.

Random walk

To illustrate the use of the Master equation, we consider the random walk on a line, see

Figure 7.3. The probability to make one step to the right per time step is p and to the left q. If

the sum of the two probabilities is 1 there is a finite probability to remain on the initial site. The

Master Equation now reads

45

, 1 1, 1,PmN pPm N qPm N

(7.6)

Figure 7.3 (left) random walk on a line and (right) evolution

of the site occupation probability when started from the origin.

Assuming the random walker to be initially in the origin, the time evolution of the probability to

find the random walker at a particular site after time t0 is sketched in Figure 7.3 (left): it stays

centered at the origin, but the excursions of the random walker grow with the square root of

time. Time and again, the random walker always visits the origin, i.e. it does not disappear.

Other examples involve the so called Chemical Master Equations, which are particularly useful

if very complex networks of chemical reactions, such as occurring in biochemistry, need to be

analyzed. There is an enormous activity in this field, mainly focused on implementing

simplifications because the original chemical master equations are near untractable at realistic

time scales.

Fokker-Planck equation

Even though the Master Equation is already a significant reduction in complexity compared to

the full-fledged Liouville equation, there is still quite some information required to define all the

transition probabilities. Studying the behavior of the transition probabilities for random

processes one noticed that the transition probability density has a rather uniform shape of

which the mean value and the coefficient of variation vary slowly as a function of its variables.

This calls for a further simplification to the form

DD

ww w

w

w ww w

2

12

2

,, ,

1

,2

P at a P at a P at

P at a

t aa a

(7.7)

in which act two moments of the distribution, the mean Į1 and the coefficient of variation Į2.

Important to notice here is that, since these are two moments of the same distribution where

one represents the average value and the other the “noise”, there is – according to the

Fluctuating Dissipation Theorem – a relation that couples the two.

Langevin equation

Completely independent of the stochastic equations described up till here, Langevin proposed

a different kind of equation that considers stochastic variables itself. The Langevin equation

for a Brownian particle reads

JK

dv

m vt

dt

(7.8)

With damping Ȗ and noise ȘW. There is a mathematical difficulty with this equation in the sense

that it is not immediately clear how the derivative of a stochastic variable is to be calculated.

Of course, one may invoke the associated difference equation but that is exactly where the

problem arises. In other words, does one take time instants t and t + h for time step h, or t – h

and t, the symmetrical difference t – h/2 and t + h/2 or yet another choice. The choice

determines the outcome, so care is to be taken.

On the other hand, one only considers quantities such as averages, e.g. the average velocity

or the velocity correlation function. Also the parameters of the Langevin equation cannot be

46

46

chosen independently. There has to be a relation between the damping and the noise

according to the Fluctuation Dissipation Theorem. Despite its mathematical difficulties, the

Langevin equation is very popular probably because of its apparent simplicity.

Conclusion

The purpose of all equations described here is to describe the time evolution of a dynamical

system. The Liouville equation provides all detail as one may wish for but that implies that in

many occasions it is too detailed and, in addition, by far not all parameters needed for its full

evaluation are available. Besides that, it is – except in simple cases – intractable if alone

because of the large number of coordinates. Yet it is very important as it serves as a reference

for all other descriptions of the same system, much like the function of the microcanonical

ensemble. Also the microcanonical ensemble is hardly used in practice but it does provide for

a solid background.

The master equation is very useful in practice, certainly now that sufficient computational

power is readily available. Moreover, it is often used to model systems such that one makes

educated guesses on the nature of the transition probabilities. What should not be forgotten is

that detailed balance should hold even if the system never reaches equilibrium on reasonable

time scales.

The Fokker Planck and Langevin equations are over-simplifications that are – most of the time

– easily solvable if only numerically. They provide for a simple way to study the essential

ingredients for a particular phenomenon at hand. Their use is justified by the transparency of

the equations and the nature of their solutions. The abuse comes from parameter assumptions

that do not satisfy the Fluctuation Dissipation Theorem. This needs to be checked in both

cases. For the Langevin equation, one then also has to worry about the precise definition of

the derivative of the random variables.

4746

46

chosen independently. There has to be a relation between the damping and the noise

according to the Fluctuation Dissipation Theorem. Despite its mathematical difficulties, the

Langevin equation is very popular probably because of its apparent simplicity.

Conclusion

The purpose of all equations described here is to describe the time evolution of a dynamical

system. The Liouville equation provides all detail as one may wish for but that implies that in

many occasions it is too detailed and, in addition, by far not all parameters needed for its full

evaluation are available. Besides that, it is – except in simple cases – intractable if alone

because of the large number of coordinates. Yet it is very important as it serves as a reference

for all other descriptions of the same system, much like the function of the microcanonical

ensemble. Also the microcanonical ensemble is hardly used in practice but it does provide for

a solid background.

The master equation is very useful in practice, certainly now that sufficient computational

power is readily available. Moreover, it is often used to model systems such that one makes

educated guesses on the nature of the transition probabilities. What should not be forgotten is

that detailed balance should hold even if the system never reaches equilibrium on reasonable

time scales.

The Fokker Planck and Langevin equations are over-simplifications that are – most of the time

– easily solvable if only numerically. They provide for a simple way to study the essential

ingredients for a particular phenomenon at hand. Their use is justified by the transparency of

the equations and the nature of their solutions. The abuse comes from parameter assumptions

that do not satisfy the Fluctuation Dissipation Theorem. This needs to be checked in both

cases. For the Langevin equation, one then also has to worry about the precise definition of

the derivative of the random variables.

These are the lectures on Molecular Thermodynamics that

I gave the last two years before my retirement from the

Delft University of Technology. They reﬂect my view on how