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The Thermal Ratchets Model

for Transport of Diffusive Particles

Masters Thesis

Abhranil Das

Department of Physics

Indian Institute of Science Education and Research

Kolkata

under the supervision of

Dr. Soumitro Banerjee

6 May 2013

Certificate

This is to certify that the thesis entitled `The Thermal Ratchets Model for Trans-

portation of Diusive Particles', being submitted to the Indian Institute of Science

Education and Research, Kolkata, in partial fulllment of the requirements for the

award of the Integrated MS degree, embodies the research work done by Abhranil

Das under our supervision at the aforementioned institute. The work presented here

is original and has not been submitted so far, in part or full, for any degree or diploma

of any other university/institute.

Dr. Soumitro Banerjee

Contents

1 Introduction 4

1.1 Background ......................................... 5

1.2 The Thermal Ratchets Model ............................... 6

1.3 Literature Survey ...................................... 8

1.4 Scope of the Present Work ................................. 14

1.5 Methodology ........................................ 15

1.5.1 Some Tools and Ideas ............................... 15

1.5.1.1 Single Ratchet Unit with Periodic Boundaries ............ 15

1.5.1.2 Averaging Transport over a Spatial Period .............. 17

1.5.2 Solution Methods .................................. 17

1.5.2.1 Monte Carlo Simulation ......................... 17

1.5.2.2 Numerical Solution of the Drift-Diusion Equation .......... 17

2 Theoretical Analysis 19

2.1 Diusion ........................................... 19

2.2 Dissipation and Drift .................................... 19

2.3 Fick's Laws and the Drift-Diusion Equation ...................... 20

2.4 Evolution Equation for Flux ................................ 21

2.5 Equivalence of Innite Lattice and Single Ratchet Period ............... 22

2.6 Transport Averaged over a Spatial Period is the Macroscopic Transport ....... 23

2.7 The Einstein-Smoluchowski Relation ........................... 25

2.8 Temperature-Dependence of the Drift-Diusion Equation ............... 25

2.9 Temperature-Dependence of

D

and

γ

........................... 26

2.10 Spread from a Point Source ................................ 27

2.11 Settling Time ........................................ 31

2.12 Equilibrium Concentration Distribution ......................... 33

2.12.1 Boltzmann Equilibrium Distribution ....................... 33

2.12.2 Equilibrium Distributions in the `O' and `On' States ............. 34

2.13 Energetics in the Thermal Ratchets Model ........................ 38

2.13.1 Source of the Energy ................................ 38

2.13.2 Consumption of the Energy ............................ 40

2.13.2.1 Energy Dissipation due to Collisions .................. 45

2.14 Derivatives at Switching: Undened yet Indispensable ................. 48

2.15 Conclusions ......................................... 53

2

CONTENTS

3

3 Monte Carlo Simulation 55

3.1 Constructing Hopping Probabilities from Potential Gradient .............. 55

3.2 Preliminary Results and Visualizations .......................... 56

3.3 Fitting the Flux with Analytical Results ......................... 60

3.4 Flux vs Degree of Asymmetry ............................... 63

3.5 Transport Rate vs Switching Period ........................... 65

3.6 Drawbacks of the Monte Carlo Approach ........................ 66

3.7 Conclusions ......................................... 67

4 Numerical Solution 68

4.1 The Counting Problem ................................... 68

4.2 Equilibrium Concentration Distribution ......................... 68

4.3 Asymptotically Periodic Dynamics for Fast Switching ................. 70

4.4 Asymptotic Transport is Uniform over Space ...................... 74

4.4.1 Smoothing Asymptotic Transport by Mesh Renement ............. 75

4.5 Transport vs

τ

o

...................................... 76

4.6 Transport Rate vs

τ

on

and

τ

o

.............................. 77

4.7 Transport Rate vs Diusion Constant .......................... 79

4.8 Transport Rate and Transient Lifetime vs Temperature ................ 80

4.9 Conclusions ......................................... 82

5 Conclusions 83

5.1 Summary of Important Results .............................. 83

5.2 Scope for Future Work ................................... 85

Bibliography 86

Chapter 1

Introduction

Average displacement of an unbiased diusing particle from its starting point is zero at all times.

Hence, a population of such particles may only be transported in a desired direction by application

of an external potential which induces a drift in the particles. However, a potential that exerts a

force whose space-average is zero, such as the following, is not expected to result in a transport (i.e.

a ow whose space-average is non-zero) of diusive particles along a one-dimensional lattice.

Figure 1.1: The ratchet potential.

Such a potential is called a `ratchet' potential, because its asymmetric jagged shape resembles the

mechanical rectifying device known as a ratchet that is ubiquitous in most mechanical applications,

being responsible, for example, for why pedalling necessarily moves a bicycle forwards but a forward

motion of the cycle does not necessarily cause the pedals to rotate, allowing a cyclist to rest the

pedals as the cycle cruises.

Now, a non-intuitive fact is that if such a potential is turned on and o periodically, it results

in a transport of the particles in the direction of the gentler slope. This model of transport is

known as the Thermal Ratchets model. This model explains several transport phenomena observed

in nature, including the transport of vital materials by molecular walkers along the cytoskeleton

inside the living cell.

4

CHAPTER 1. INTRODUCTION

5

Several papers in the 1990s studied this model and the resultant ux via analytical, computa-

tional and experimental approaches. The simulation approach (Monte Carlo) is rather computa-

tionally intensive and does not yield smooth dynamics. The analytical approach is only applicable

with an assumption of adiabatic adjustment, manifested in this case in the requirement that the

potential stays in its `on' condition long enough for the particle distribution to settle at a steady

state.

The approach taken in this thesis is instead a numerical solution of the drift-diusion equation

that governs the evolution of the particle density. This is computationally economical, yields smooth

dynamics, and does not demand any restrictive assumptions on the system. With this approach

we have been able to probe the regime of high alternating rate where adiabatic adjustment is not

allowed. A surprising result is that even in such a case the system settles into periodic behaviour

asymptotically, without requiring to reach a steady state in the `on' phases.

1.1 Background

It is easy to understand that diusing particles do not, on the average, have a tendency to move in a

particular direction. In absence of external forces, such particles may be thought of as exhibiting an

unbiased random walk, whose mean displacement after any specied time from the starting position

is zero. However, in many situations across diverse scenarios and elds, it is often necessary to drive

such particles in a particular direction. The problem of transportation of diusing particles is thus

a pertinent and important one across many elds and applications.

As an example, one may consider the problem of transportation of vital material and organelles

from one part to another of the living cell by motor proteins that `walk' along cellular laments

that stretch across the length of the cell. Examples are myosin-I, II and V, which walk along actin

laments, and kinesin and dynein, which walk along microtubules.[1]

A complete understanding of the mechanism of such molecular walkers has still not been

achieved. However, for some of the motor proteins such as the myosin group, it appears that

directed motion may be the underlying mechanism [2], which happens to be a very energy-ecient

mode of transport. In that case, one can no longer think of the motor proteins as random-walking

particles. However, a less sophisticated mechanism based on random walk with bias may have been

the mechanism at work in earlier stages of evolution, and may furthermore be the mechanism for

some kinds of motor proteins even today. Also, such a mechanism being simpler to understand

merits some discussion and analysis, as it may serve to illuminate greatly the understanding of

more complicated models.

This is just one scenario in which an analysis of the transport of diusing particles is relevant. In

more general terms, this has been a problem much thought about in the realms of thermodynamics

and general physics, as it has deep connections with the second law and the idea of entropy. If

it is possible to transport diusing particles exploiting nothing but their thermal energy, it is also

possible to generate useful work out of that transport, and hence violate the second law. An

analogous problem was rst raised by Gabriel Lippmann in 1900, via a ratchet and pawl model.

Consider the following gure:

CHAPTER 1. INTRODUCTION

6

Figure 1.2: Ratchet and Pawl

This is the much-discussed ratchet and pawl model. The left chamber contains a gas heated to

a certain temperature, with a vane introduced into the box. The other end of the shaft through this

vane connects to a ratchet wheel with an asymmetric jagged surface. A pawl rests on this surface,

which serves to prevent backward rotation and thus recties the motion. As gas molecules strike

the vane in the left chamber from random directions, every so often it manages to turn the ratchet

wheel in the forward direction, serving to lift a load tied to the shaft. If this mechanism works, the

nal result should be a chamber of cooled atoms and a lifted load. Thus, thermal energy contained

in the gas molecules would have been converted to useful work.

A qualitative explanation for why the device fails to do the envisaged task followed in 1912

from Polish physicist Marian Smoluchowski.[3] In 1962, Richard Feynman, as part of his celebrated

lectures, was the rst to demonstrate a quantitative analysis of why this `brownian ratchet' model

fails [4, Chap. 46]. In 1996 though, Juan Parrondo and Pep Español [5] used a variation of this

model to point out a aw in Feynman's argument, although the conclusion about the ineectiveness

of the device remained intact. There has since been much discussion about this model. [6,7]

Transporting diusive particles using their thermal uctuations as energy source is an analogous

problem. However, it has been established that this is not possible for the same reasons responsible

for the failure of the above device. However it is often useful and necessary to consider a transport

of diusive particle seven at the expense of energy. Various models aim to achieve this, one of which

is the thermal ratchets model, the current subject of study. This mechanism achieves transport of

random-walking particles using a ratchet potential, and even against an externally imposed force.

A full description of the model shall be elaborated in later chapters and an explanation for why it

does not violate the second law of thermodynamics shall be presented in detail in sec.2.13.

1.2 The Thermal Ratchets Model

Before we take a look at the existing study of the thermal ratchets model it is necessary to obtain

an outline of the model itself. The following is a simple gure to illustrate the model:

CHAPTER 1. INTRODUCTION

7

Figure 1.3: The thermal ratchets model

Diusive particles are executing random walk along an extended channel. A potential that

causes a drift of these particles in addition to the diusion is switched on and o alternately.

This ratchet potential is spatially asymmetric. For purposes of simplication, we shall consider a

sawtooth prole formed by straight line segments.

It has been established that such a spatially periodic and asymmetric, and temporally alternating

potential causes diusive particles to be driven in the direction of the gentler slope of the sawtooth.

Here is an illustration to help explain how the mechanism achieves this transport:

Figure 1.4: The thermal ratchets mechanism, courtesy

Mechanics of the Cell

[1, p. 432]

CHAPTER 1. INTRODUCTION

8

As may be understood from the gure, the asymmetric location of the neighbouring potential

peaks on either side of the valley from which the particle density originates following the potential

release causes dierent fractions of it to tumble to the left and right valleys. Thus, there is a net ux

to the right (in this case). Since this same phenomenon occurs in each repeating spatial unit, there

is no depletion or accumulation of particles in each region. The particles that have been lost from

a valley (to the right, mostly, in this case) are exactly compensated by those owing in from the

left. This compensation is made possible by the fact that these particles do not interact with each

other, and hence their dynamics are completely independent of the distribution of other particles.

The net eect, therefore, of one cycle is a transport of particles to the right throughout the innite

lattice array.

If the switching time for the potential is too short, though, the particles will not have enough

time to gather at the potential valleys in the `on' state, so that at the moment the potential is

turned o, they shall not diuse from a sharp peak as in this illustration, and the explanation for

the transport mechanism is not as simple. However, the overall eect of the potential is still to

drive transport towards one direction.

For an understanding of how such a system may arise in a biological context of molecular walkers

on the cytoskeleton, refer to the following illustration:

Figure 1.5: Thermal ratchets in biolaments and molecular walkers, courtesy

Mechanics of the Cell

[1, p. 432]

1.3 Literature Survey

In 1992 Charles R. Doering and Jonathan C. Gadoua [8] considered the problem of overdamped

diusing particles crossing a uctuating potential barrier using their thermal energy. The potential

barrier they worked with was a piecewise linear one that switched between a high and a low state as a

CHAPTER 1. INTRODUCTION

9

Markov process. It was, however, a symmetric tent function, as opposed to the asymmetric sawtooth

of the Thermal Ratchets potential. Because of the Markovian process, the probabilities of the barrier

staying in either the high or the low state drops exponentially with time with an exponent of

γ

,

called the ipping rate. The authors solved this system exactly with a master equation approach

and also via a Monte Carlo simulation. They reported a resonance-like phenomenon, manifested in

a local minimum of the mean time of rst passage of the barrier (MFPT) against the ipping rate,

which implies a local maximum of the transport rate of the particles:

Figure 1.6: Mean rst passage time vs barrier uctuation rate in a Markov ratchet potential,

Doering and Gadoua [8]

In 1993 M. Magnasco [9] considered transport of diusive particles under the asymmetric ther-

mal ratchets potential. Assuming the alternation rate to allow for the adiabatic adjustment time

(implying that the system exhibits periodic behaviour in time), the author arrived at an exact

expression for the particle ux via a Fokker-Planck equation approach [10,11]. This was plotted

against the amplitude of the external force for various temperatures and against the temperature

of the system for a xed amplitude. In both cases the ux showed a local maximum:

CHAPTER 1. INTRODUCTION

10

Figure 1.7: Flux for a square-wave ratchet vs excitation amplitude, Magnasco [9]

Figure 1.8: Flux vs temperature, Magnasco [9]

R. Dean Astumian and Martin Bier in 1994 [12] theoretically discussed the directional motion

of diusing particles under the inuence of a thermal ratchets potential exerting a zero space-

averaged force, and noted agreement with experimental results for the motion of kinesin molecules

CHAPTER 1. INTRODUCTION

11

along biopolymers [13]. However, theirs had been a simplied approach in which as the ratchet

potential was turned o, the particle concentration was assumed to start each time from delta-

function peaks at the potential valleys. This is again physically possible only when the potential is

left in the `on' state longer than the adiabatic adjustment time for the system.

Prost et al in the same year [14] further studied the average ux of the transported particles

against various parameters of the system. For this they employed both analytical calculations and

an averaged Monte Carlo simulation of the dynamics of the random-walking particles. They found

a similar resonance phenomenon against the ipping rate of the potential barrier as Doering et al

[8]:

Figure 1.9: Transport rate vs switching period, Prost et al [14]

The horizontal axis here is the period of the switching, and the vertical axis is the average ux

rate or velocity of the particle transport.

This paper also reported a resonance phenomenon against the excitation amplitude in agreement

with Magnasco [9]. Their graph is presented below. The horizontal axis this time is the excitation

amplitude.

CHAPTER 1. INTRODUCTION

12

Figure 1.10: Transport rate vs excitation amplitude, Prost et al [14]

Rousselet et al in the same year [15] studied the thermal ratchets mechanism in an experimen-

tal context with colloidal particles suspended in solution and subjected to a switching dielectric

potential, validating the results of the former papers and discussing, in addition, the possibility of

separation techniques utilizing the mechanism. They produced plots of the transport rate against

τ

o

, the duration of the `o' state of the potential for several particle sizes (particle size aects the

diusion and damping coecients). They found a rising asymptotic curve for this and compared

with theoretical results assuming a gaussian spread from the potential valleys (which is, again, a

periodic phenomenon only for ipping times longer than the adiabatic adjustment time):

CHAPTER 1. INTRODUCTION

13

Figure 1.11: Cumulative transport vs

τ

o

, Rousselet et al [15]

In 1997 R D Astumian [16] studied uctuation-driven transport in connection with the Thermal

Ratchets model, and discussed applications in particle separation techniques and the design of

molecular motors and pumps. In this paper resonance phenomena for the ux were shown against

both the forcing amplitude and the temperature, in agreement with previous literature:

CHAPTER 1. INTRODUCTION

14

Figure 1.12: Flux vs excitation amplitude and temperature, Astumian [16]

In 1998 Kamegawa et al [17] analysed the energetics of the thermal ratchets model constructed

by Magnasco [9] and challenged the idea that a maximum ux at a nite temperature implies

that the eciency of the machine converting chemical energy to work also reaches a maximum

at a nite temperature. They showed that upon calculating the eciency of the system, it is, as

usual, maximum in the absence of any thermal uctuation whatsoever, in which state there is no

transport.

1.4 Scope of the Present Work

As has been discussed in the literature survey, all analysis on the thermal ratchets model has so

far been conned to the allowance of adiabatic adjustment time. This requirement is manifested

in the case of this system by allowing the ratchet potential to stay on long enough for the particle

density to settle at the potential valleys before turning it o again. This ensures that the dynamics

of the density is repeated in every cycle of ipping.

Also, the approaches so far explored to study this system were Fokker-Planck calculations for

exact analytical results (which work only for ipping times longer than the adiabatic adjustment

CHAPTER 1. INTRODUCTION

15

time), or a Monte Carlo approach which in theory is applicable to any situation, but requires

statistical smoothing before its results may be used. In this work, the approach has been through

a numerical solution of the Smoluchowski drift-diusion equation (2.7) applied with the thermal

ratchets potential. Since this equation describes the evolution of the particle density itself, it

yields dynamics which is automatically smooth without requiring statistical averaging. In this way

it is computationally much faster than a Monte Carlo approach. However, it has an advantage

over analytical calculations too (which also yield smooth dynamics) in that it is a general solution

technique, applicable to any potential and any ipping rate. In this work though, we shall consider

the usual asymmetric sawtooth potential that alternates discretely in time.

We shall primarily study the quantities that have been explored in the previous works, only in

this extended, more generalized approach. This yields some behaviour which only reveals itself in

the high ipping rate domain, such as transient dynamics and asymptotic periodicity. We shall

study these in detail and also focus on the length of the transient lifetime and its dependence on

temperature.

1.5 Methodology

The system was analysed in two ways: rst, via a Monte Carlo simulation, and second, a numerical

solution of the Smoluchowski drift-diusion equation (2.7). For both the stochastic model and the

numerical integration, and for various plots, python scripts with various modules have been used.

For brevity of presentation, in this and all following sections, dots (

˙

) and primes (

0

) have been

meant to imply partial derivatives with respect to time and space respectively.

1.5.1 Some Tools and Ideas

Before we give an overview of the two solution approaches, we discuss two ideas that are applicable

in the implementation of either of these approaches.

1.5.1.1 Single Ratchet Unit with Periodic Boundaries

The ratchet potential ideally extends innitely in space. Initially, this was approximated by con-

structing only several repeating units of the spatially periodic potential, as illustrated below:

CHAPTER 1. INTRODUCTION

16

Figure 1.13: Thermal ratchets simulation model

However, for simulation purposes it is computationally rather intensive to construct the potential

and observe concentration dynamics for too long a lattice. In order to solve this problem, only the

smallest repeating unit of the spatially periodic potential was constructed, and periodic boundary

conditions were built in so that particles exiting from one edge re-appear through the other. This

is illustrated below:

Figure 1.14: Single ratchet unit with periodic boundary conditions

CHAPTER 1. INTRODUCTION

17

The logical justication to claim that these two represent the same system and will produce

identical dynamics is explained in sec. 2.5.

1.5.1.2 Averaging Transport over a Spatial Period

Most of the results that shall be discussed relate to a measure of the transport of particles (naturally,

for transportation

is

the purpose of the model). Thus, it is important to establish how we measure

transport.

As was done for many of the results in the following chapters, one may choose to measure the

transport through a point, that is the number of particles that pass through the point. But is this

an accurate measure of the transport that the model achieves from one point to another over a

large scale?

It may be shown by logical arguments that it is not, and in fact, only an average of the transport

over a spatial period of the periodic lattice corresponds to the macroscopic transport free of internal

ows (discussed and derived in sec. 2.6). This shall be implemented later in the analysis of our

results (g. 4.4)

1.5.2 Solution Methods

We now give a brief overview of the two solution methods that have been covered here, the Monte

Carlo simulation and the numerical solution. In addition, there is the analytic solution approach

through a Fokker-Planck method which has not been covered here.

1.5.2.1 Monte Carlo Simulation

We rst construct a Monte Carlo simulation of the Thermal Ratchets model that shows us a

visual of the potential, particle concentration dynamics and yields plots for various quantities.

This is implemented over a one-dimensional lattice with iterated time steps. The diusion was

implemented via a random walk of the particles, in which at every iteration each particle has

predened probabilities of hopping to either of its neighbouring sites or staying in the same site.

The relative tendency of staying in the current site versus moving out is an indirect measure of the

temperature of the system. In addition, the relative probabilities of a left and right hop may not

be the same, which is the basis for a stochastic drift. For this, the drift velocity at a point was

calculated via nite dierencing the ratchet potential at that point, and the drift was implemented

by tilting the probabilities of a left versus a right hop on the basis of this slope (details in sec. 3.1).

1.5.2.2 Numerical Solution of the Drift-Diusion Equation

A Monte Carlo simulation needs to be averaged over many stochastic elements before we may

observe any smooth dynamics. This requires a lot of computational power. Instead, for the rest of

the study we shift to a numerical solution of the Smoluchowski drift-diusion equation (2.7) that

is computationally much less intensive. Naturally, the numerical integration occurs over a discrete

lattice and discrete time-steps. However, the resolution of either may be modied in the model to

get the desired accuracy.

For the numerical integration no special renement technique (such as Runge-Kutta) was em-

ployed as it is a partial dierential equation. Instead, the usual Euler method [18, p. 45], [19, p. 35]

was employed with suciently small time steps to achieve the required accuracy. In order to further

CHAPTER 1. INTRODUCTION

18

improve the scheme, spatial symmetry was built into the integration process by considering a three-

point central dierence for the rst derivative

(vc)0

in the equation 2.7.

v(x, t)

is further obtained

from the rst derivative

U0(x, t)

, which was again evaluated using a symmetric three-point method.

The second derivative

c00

is automatically symmetric. Thus, the nal simulation is symmetric in

space, implying that mirroring the potential mirrors the dynamics, as it should.

Chapter 2

Theoretical Analysis

2.1 Diusion

The average displacement of a population of particles diusing from a point in a medium is zero,

at any time from the start:

¯x= 0,∀t.

However, the spread of the particles from the point of injection keeps increasing. This does not

increase at a linear rate with time though. Instead, the spread, measured in the standard deviation

of the particle positions, increases with the root of the time:

σx=√2Dt.

(2.1)

D

is known as the diusion coecient.

This result may be arrived at by considering a lattice of positions on which the individual

particles can hop at iterated time steps, i.e. a random walk. For a derivation, see Berg [20].

2.2 Dissipation and Drift

In a dissipative environment the drag force on a particle is given by:

F

drag

=−γv,

(2.2)

where

γ

is the drag coecient. In our system, the diusing particles are in a highly damped

environment. In addition to this drag force, however, they experience an external force due to the

thermal ratchets potential, and their net dynamics is governed by:

m˙v=F

ext

−γv

.

Now consider a particle starting from rest under this dynamical equation. The right side is then

positive, and its speed

v

climbs from 0. But as it does, the magnitude of the acceleration drops.

Thus, under a slowing acceleration, the particle reaches a terminal velocity

v

term

=F

ext

γ=−1

γ

∂U

ext

∂x ,

(2.3)

19

CHAPTER 2. THEORETICAL ANALYSIS

20

at which point

F

drag

=−F

ext

. This, however, only happens asymptotically with an exponent

γ

.

For a particle at a higher speed than

F

ext

/γ

, a retardation brings it down to this speed. In highly

dissipative media, the exponent

γ

is large, and the terminal velocity is reached quickly, so that

under a force even small distances may be considered to be travelled with this uniform speed, called

the

drift speed

.

2.3 Fick's Laws and the Drift-Diusion Equation

Fick's Laws [21,22] describe diusion that arises out of independent random walks of a population

of particles and provide a way to measure the diusion constant. The dynamics of a concentration

of particles which do not inuence each other's trajectories to a great extent are well approximated

by these laws, and may be derived from a continuum extension of a random walk model on a lattice

[20]. Since we consider densities of independent diusing particles for our model, they are also

governed by these laws.

The rst law states that the ux of the particles due to diusion is proportional to the concen-

tration gradient:

~

J

di

=−D∇c.

(2.4)

The second law describes the time evolution of the particle density prole:

˙c=−∇.~

J

di

=∇.(D∇c).

(2.5)

If there is no additional drift arising out of external forces, this equation is sucient to describe

the evolution of the particle concentration. However, in our case there is a drift due to the ratchet

potential and the second law will not suce. For this one must consider the advective ux due to

the drift speed:

~

J

adv

=~vc

which gives:

˙c=−∇.(~

J

di

+~

J

adv

) = ∇.(D∇c−~vc).

(2.6)

This is known as the Smoluchowski drift-diusion equation [23].

In one dimension, and considering that the diusion and damping coecients are space-independent,

this reduces to:

˙c=Dc00 −(vc)0=Dc00 +1

γ(U0

ext

c)0.

(2.7)

This is the governing equation for the particle concentration in our model. Taking the appro-

priate drift speeds

v(x, t)

that result from the ratchet potential and solving for

c(x, t)

gives us the

dynamics of the particle concentration for our system. Note here that since the ratchet potential is

piecewise dened in space, the drift speed

v

obtained from the potential via 2.3 is space-dependent

and cannot be taken out of the space derivative on the right (see details in sec. 2.14).

CHAPTER 2. THEORETICAL ANALYSIS

21

2.4 Evolution Equation for Flux

The drift-diusion equation describes the evolution of the particle concentration in space and time.

From this we can nd an equation that describes the evolution of uxes at dierent points in space

against time. For this, we start from the expression for ux at a point:

J(x, t) = v(x, t)c(x, t)−Dc0(x, t)

(2.8)

Dierentiating partially with respect to time, we have:

˙

J=˙

(vc)−D˙c0

The diusion coecient

D

can always be taken out of time- and space-derivatives as it is

independent of both. If we further assume that the drift speed is independent of time, we may take

it out of the time derivative:

˙

J=v˙c−D˙c0

Now we may use 2.6, which can be simplied as:

˙c=−J0

(2.9)

This can be used to rewrite the equation in terms of ux alone:

˙

J=DJ00 −v˙

J0

(2.10)

Here the notation

˙0

implies

∂tx(= ∂xt )

. Re-written in this standard notation it is:

∂tJ+v∂txJ−D∂xxJ= 0.

This is the evolution equation for the ux. The drift speed

v

is now outside the space derivative,

as opposed to the case for the concentration equation (2.7). Thus, there are no more issues with

the kink in drift speed due to the piecewise dened ratchet potential.

However, a new problem is introduced. Whereas eq. 2.7 gave me a way to compute

˙c

in terms

of the

c

at any point of time, this equation requires me to know

˙

J0

also at a point of time in order

to compute what

˙

J

would be. At rst glance this appears impossible, because one would need to

know how

J

evolves rst in order to calculate how its space-derivative evolves. So because of this

bootstrapping dilemma, this equation brings new complications with it. This problem may mean

simply that without knowledge of the concentration, it is not possible to compute the evolution of

the ux alone in terms of itself.

There is an additional point that we have overlooked so far. We have made an assumption of a

zero

time-derivative

of the drift speeds to obtain this equation. If instead the drift speeds change

in time (which they will at switching), we must account for the extra term in the equation:

˙

J=DJ00 −v˙

J0+ ˙vc

(2.11)

Unfortunately, the drift speed derivative comes coupled with the concentration again, which we

were trying to get rid of in the rst place. If we must write the evolution equation in terms of ux

alone, we must integrate 2.9 indenitely with respect to time, to give:

CHAPTER 2. THEORETICAL ANALYSIS

22

c=−ˆJ0dt

Introducing this, the evolution equation of ux is changed to a more complicated integro-

dierential equation, but in

J

alone:

˙

J=DJ00 −v˙

J0−˙vˆJ0dt

2.5 Equivalence of Innite Lattice and Single Ratchet Period

In section 1.5.1.1 we argued that replacing an innite lattice with just one segment the length of

a spatial period of the ratchet potential, and putting periodic boundaries around it are the same

thing. The logical basis for this is explained in the following argument.

The total number of particles on the lattice is xed. Now if in an innite lattice we initiate the

system with a concentration that is uniform, or more generally periodic with the potential period,

then an equal number of particles shall be in each unit not only at the start, but at any point of

time (because the dynamics is identical in each unit). Thus, the number of particles in each ratchet

unit must also be unchanging in time. Therefore, the ux through one edge of a ratchet unit at

any point must exactly equal that through the other edge, but cancel each other in sign. This can

be shown backwards mathematically by invoking the continuity equation (2.5) which yields Fick's

second law (discussed later in sec. 2.3):

˙c(x, t) = −J0(x, t)

⇒ˆb

a

˙c(x, t)dx =J(a, t)−J(b, t)

Now, if there is a spatial periodicity of

p

in the dynamics, we have:

J(x, t) = J(x+p, t)∀x.

(2.12)

Thus, choosing such a period gives us:

ˆa+p

a

˙c(x, t)dx =J(a, t)−J(a+p, t)=0

Now, since a partial dierential w.r.t time can be exchanged with an integral w.r.t. space, the

quantity on the left is:

∂tˆa+p

a

c(x, t)dx =∂tMa+p

a

where

Ma+p

a

is the mass of particles in the region from

a

to

a+p

. Thus, this is an expression

for the rate of change of this mass. Since the right side is zero, it means there is no change in

CHAPTER 2. THEORETICAL ANALYSIS

23

mass in any region of length equal to the period of the dynamics. The number of particles entering

through one edge of such a region is the same number of particles exiting through the other edge

(

J(a, t) = J(a+p, t)

). And the particles exiting from the right boundary of such a region shall

enter through the left boundary the next. And since this same behaviour is exhibited by all units,

the same number of particles shall be entering from the left boundary of the former, fed by its left

neighbour.

Thus, physically an innite lattice is the same as a single unit with periodic boundaries. And

since this unit has periodic boundaries, it does not matter exactly how we choose this unit from the

lattice (where we choose its start point). Even if we shift this unit around, it represents the same

innite lattice because of its periodic boundaries, and hence the physical dynamics that we observe

shall not change.

2.6 Transport Averaged over a Spatial Period is the Macro-

scopic Transport

In section 1.5.1.2 we argued that the measure of transport that corresponds to the macroscopic

transport from one place to another in the large scale is the space-averaged transport over one

spatial period of the lattice. The explanation behind this is laid out in the following argument.

Consider the following scenario: we start with a peak of accumulated of particles at the center

of each ratchet unit, which over some time settles due to diusion into a at concentration prole

over the length of this unit. Assume that the edges of each ratchet unit are impervious to the

passage of particles (say by a high potential peak there), meaning that in this time no particles

travel from one ratchet unit to another. Thus, over the large scale of space, no transportation has

been achieved in any direction. An innite series of peaks regularly spaced at the center of spatial

units have diused into at heights in their respective units.

But now consider that we were measuring transport through a point in the ratchet unit that

was to one side of the initial particle peak at the center. As this peak spreads out through diusion,

we shall register transport through that point in one direction. If we keep integrating this ux over

time, i.e. measure the cumulative transport achieved until the at concentration prole has been

reached, this shall be a non-zero quantity, indicating spuriously that transport has occurred.

This is quite true, because transport has indeed occurred through that point. But at a point

symmetrically opposite to this with respect to the center, exactly the same amount of transport

has occurred in the opposite direction, which was not measured.

Thus, for certain situations, the transport measured at a single point on the ratchet unit may

not faithfully reect the transport that is occurring in the large scale. Some amount of the transport

registered at one point may be due to internal ows which do not contribute towards the large-scale

transport that the model was designed to generate.

The solution to this is averaging the transport over all points along a ratchet unit. Internal ows

shall be cancelled out by the summation, and the quantity that shall persist is the space-averaged

ow. The interesting property of the thermal ratchets model is that with a force whose space-

average is zero, it achieves a ow whose space-average is non-zero, as stated rst at the beginning

of this chapter (1).

Now one may object that although the ratchet potential has a dened periodicity, choosing the

start (and hence, end) point of what we shall call a repeating spatial unit is entirely subjective.

This may be a peak-to-peak segment, or a valley-to-valley, or any other segment of the same length.

CHAPTER 2. THEORETICAL ANALYSIS

24

But this does not introduce any ambiguity in the idea of the space-averaged ow, since no matter

which segment we choose to take as a unit of spatial periodicity, the space-averaged ow over it

will come to the same number. This is because the integral of a periodic function is the same,

irrespective of our choice of the beginning and end points of the period. This may be illustrated by

some mathematics.

If we take

J

av

(a)

as the averaged ow over a spatial period starting at

a

, i.e.:

J

av

(a) = 1

pˆa+p

a

J(x)dx,

then we need to show that this integral is the same as it would be if we had chosen a dierent

start point

b

, i.e.

J

av

(a) = J

av

(b),∀a, b.

(2.13)

Let

k=b−a

. Then we have:

J

av

(b) = 1

pˆb+p

b

J(x)dx =1

pˆa+k+p

a+k

J(x)dx =1

pˆa+k+p

a+k

J(x−p)dx

(because of periodicity: 2.12)

Now we eect a change of variable

x0=x−l

, which changes the integral to:

1

pˆa+k

a+k−p

J(x0)dx0.

This may be re-written by breaking down the range of integration into three parts:

1

p[ˆa

a+k−pl

J(x0)dx0+ˆa+p

a

J(x0)dx0+ˆa+k

a+p

J(x0)dx0]

The middle of these terms is

J

av

(a)

. Now we show that the rst and third in fact cancel each

other.

Eecting a variable change

x00 =x0+p

on the rst we have the integral:

ˆa+p

a+k

J(x00 −p)dx00 =ˆa+p

a+k

J(x00)dx00

(because of periodicity: 2.12)

=−ˆa+k

a+p

J(x00)dx00 .

This is the negative of the third term.

Thus, we have established 2.13, and hence that space-averaged ux over a spatial period starting

at any location is the same, and equal to that over the entire lattice. Therefore, this quantity serves

as a measure of the macroscopic transport rate free of internal ows.

CHAPTER 2. THEORETICAL ANALYSIS

25

2.7 The Einstein-Smoluchowski Relation

The Einstein-Smoluchowski equation relates the diusion constant and the damping coecient of

particles in dynamic equilibrium in a diusive and dissipative medium with the temperature of the

medium, and was derived independently by Einstein [24] and Smoluchowski [25]. The relation is

the following:

γD =k

B

T.

(2.14)

Here

γ=F

ext

v

term

is the drag coecient (see 2.3),

D

is the diusion coecient,

T

is the temperature

of the system and

k

B

is the Boltzmann constant. If we thus know the temperature-dependence of

either

D

or

γ

for a system, we can use 2.14 to get the temperature-dependence of the other, and

re-mould eq. 2.7 purely in terms of temperature.

Additionally, in the limit of low Reynold's number, the drag coecient for an object in the uid

is proportional to the viscosity coecient of the uid, given by the Stokes law:

γ= 6πηr.

Putting this into 2.14 gives the Stokes-Einstein relation:

D=k

B

T

6πηr.

2.8 Temperature-Dependence of the Drift-Diusion Equation

We have commented before that although the thermal energy of the heat bath in which the particles

are suspended is not what drives the transport, the thermal energy is indeed vital for the transport

to work. Therefore, we should expect some dependence of the dynamics on temperature. At rst

glance, however, the governing dierential equation (2.7) does not appear to have any dependence

on temperature:

˙c=Dc00 −(vc)0

But let us now rewrite the drift speed

v

in terms of the external potential and the drag coecient,

using eq. 2.3:

˙c=Dc00 +1

γ(U0c)0

Now, in general the diusion and drag coecients vary with temperature:

˙c=D(T)c00 +1

γ(T)(U0c)0

(2.15)

This is where the temperature-dependence of dynamics comes in.

At rst glance it appears that in order to investigate the variation in dynamics with respect to

the change of temperature, we would have to know both the temperature-dependences

γ(T)

and

D(T)

. However, since we know the Einstein-Smoluchowski relation (2.14), we may re-write 2.15

in terms of only one of these dependences, because knowing one of them for a particular case of a

liquid and suspended particles gives us the other.

CHAPTER 2. THEORETICAL ANALYSIS

26

In terms of

γ(T)

alone eq. 2.15 is:

˙c=1

γ(T)(k

B

T c00 + (U0c)0),

while in terms of

D(T)

it is:

˙c=D(T)(c00 +1

k

B

T(U0c)0).

2.9 Temperature-Dependence of

D

and

γ

Temperature-dependence of

γ

and

D

is found to be widely varied depending on the kind of material

in which the diusion and drag are occurring, the phase of the materials etc. One of the wider used

models for

D(T)

follows from the Arrhenius equation (see Laidler [26]):

D(T) = D0e−Ea/RT

(2.16)

where

Ea

is the activation energy and

D0

is the diusion constant at innite temperature, to

which this function asymptotes. Fluids for which the Arrhenius equation may be applied are called

Arrhenius uids.

The graph for this function looks like the following:

Figure 2.1: Diusion coecient as a function of temperature for an Arrhenius uid

Qualitatively, this curve is not hard to understand. With increasing temperature, the bom-

bardment of the suspended diusing particles by uid particles increases, and so does its rate of

diusion, and hence the diusion coecient, until it reaches an asymptotic value.

From this function, we may get

γ(T)

using the Einstein-Smoluchowski relation (2.14):

γ(t) = k

B

T

D(T)=k

B

D0

T eEa/RT .

(2.17)

The graph for this function is the following:

CHAPTER 2. THEORETICAL ANALYSIS

27

Figure 2.2: Drag coecient as a function of temperature for an Arrhenius uid

Thus, there is a temperature at which the drag coecient is minimum.

2.10 Spread from a Point Source

In our model, we shall consider the particles to be both diusing and drifting in response to our

ratchet potential. Thus, if we inject such a population of particles at a point in a medium, their

density will start to drift with the drift speed calculated in 2.3 and also start to spread with an

increasing standard deviation as calculated in 2.1. The solution to the diusion equation 2.5 for a

point source in the form of a Dirac delta function as the initial condition is the following spreading

gaussian:

P(x, t) = N(F

ext

γt, √2Dt) = e−(x−

F

ext

γt)2

4Dt

√4πDt .

The notation

P

suggests that this may also be considered as the probability distribution of a

single particle injected in such a medium.

Now consider the thermal ratchets model. If we leave the potential on for long enough, the

particle concentration accumulates at the valleys. At the steady state, they may be approximated

by a Dirac delta or a thin, tall gaussian form. Thus, when the potential is turned o, the spreading

of the concentration from these sites follows the equation above. Refer to the following gure, where

a

and

b

are the widths of the two slopes of the ratchet:

CHAPTER 2. THEORETICAL ANALYSIS

28

Figure 2.3: Spread from neighbouring valleys

Consider the spread of the concentration from only the left valley as the potential is turned o.

In absence of the potential there is no drift, and the density at a certain distance from the point of

spread, say at the potential peak, rst rises as the spreading gaussian reaches it, and then falls as

it spreads further out. This is given by:

ρ(a, t) = M e−a2

4Dt

√4πDt .

(2.18)

Here

M

is the total mass of the particles at each potential valley when the potential is turned

o. This function has been plotted against time for two positions, one (brown) twice as distant as

the other (red) from the point of spreading:

CHAPTER 2. THEORETICAL ANALYSIS

29

Figure 2.4: Probability density height vs time for two distances from the source of spread

Using Fick's 1st Law (2.4) we may also obtain the instantaneous ux at that point. This is

given by:

J(a, t) = M ae−a2

4Dt

4t√πDt .

(2.19)

Now we come to a more important quantity, the cumulative ux through a point. This is the

total ux of particles that has travelled through a point near the diusion source upto a time

t

. In

the gure of the ratchets potential at the beginning of this section, there are two diusion sources

that spread as a gaussian from either side of the potential peak as the potential is turned o. The

one on the left contributes to positive cumulative ux, while the other reduces it. Using 2.18, we

may write the net ux through the position of the potential peak as:

F(t) = M

√4πDt (ˆ∞

a

e−x2/4Dtdx −ˆ−b

−∞

e−x2/4Dtdx).

With some rearrangement this becomes:

F(t) = M

√4πDt ˆb

a

e−x2/4Dtdx =M{

erf

(b

√2Dt )−

erf

(a

√2Dt )}.

(2.20)

Here

erf

is the error function, the integral of the gaussian.

The following is a plot for this function with mass

M= 1

,

D= 1/2

,

a= 5

and

b= 15

. Later

we shall compare this with a plot obtained from a simulation using the same parameter values.

CHAPTER 2. THEORETICAL ANALYSIS

30

Figure 2.5: Analytical expression for cumulative transport through a point vs time

Note that we may also have arrived at an expression for this cumulative ux through a point

by integrating the instantaneous ux at that point over time:

F(t) = ˆt

0

J(a, t0)dt0.

For a single point source at a distance

a

from the point in consideration, we may use 2.19, and

comparing the two approaches, this implies that:

F(t) = M

√4πDt ˆ∞

a

e−x2/4Dtdx =M a

4√πD ˆt

0

e−a2/4Dt

t03/2dt.

In the following gure, the three quantities discussed so far, the concentration, instantaneous

ux and the cumulative ux, are compared at the position of the potential peak as a function of

time elapsed after the potential is turned o. They are scaled to the same height:

CHAPTER 2. THEORETICAL ANALYSIS

31

Figure 2.6: Scaled comparison of three quantities evaluated at a point vs time

The three are similar curves, and they each reach a maximum, but at dierent points of time.

Since the thermal ratchets model is a mechanism for transport, we