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Perspectives on Molecular Thermodynamics

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Perspectives on Molecular Thermodynamics

Abstract and Figures

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 thermodynamics is linked to other topics such as chemical kinetics and fluid dynamics that are generally considered of utmost importance to chemical technology. In particular the role of entropy and more specifically of entropy production is highlighted as it guides engineers in finding process solutions of optimal efficiency.
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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 ȘW. 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