characteristics of work fluctuations in chaotic system projects repository 2015... ·...

National University of Singapore

Department of Physics

Characteristics of Work

Fluctuations in Chaotic Systems

Author:

Alvis Mazon Tan

Supervisor:

Professor Gong Jiangbin

Mentor:

Jiawen Deng

A thesis submitted in partial fulfilment of the requirements

for the degree of Bachelor of Science

(HONOURS)

4 April 2016

ABSTRACT

Miniaturisation has become a trademark of our modern society. Technological de-

vices such as electronic chips are decreasing in size throughout the years. This has

forced us to re-look into the problems facing nanoscale thermodynamics which is

the characteristics of a few body problem. For system with finite degrees of free-

dom, fluctuation is comparable to the ensemble mean [1]. In order to increase the

efficiency and performance of these small systems it is critical that we minimise

these work fluctuations. Much work has been done by Gong and his team on the

effects of adiabatic protocol on classical and quantum system in the suspression of

work fluctuations [2].

In this paper, we will explore the characteristics of work fluctuations in chaotic

systems, where the Sinai billiard will be our candidate. We will sample our trajec-

tories from the microcanonical and canonical ensemble and expose it to adiabatic

protocols in an attempt to study the behaviour of work fluctuations. Simultane-

ously, it will also be relevant for us to compare work fluctuations in chaotic and

non chaotic models. Lastly, we will review some of the thermodynamics concepts

pertinent to small systems.

1

ACKNOWLEDGEMENTS

‘we can only see a short distance ahead, but we can see plenty that needs to be

done’. -Alan Turing

The above quote succinctly summarised my FYP journey in this one year. Indeed

the feeling is none other then surreal. Just when I thought I had it all, life struck

me hard and presented me with yet another set of challenges. This project will

not have been possible if not for the determination and grit of the team.

I would like to express my heartfelt gratitude to my supervisor Professor Gong

Jiangbin. I am extremely honoured and privileged to have the opportunity to

work under a great team. As much as inheriting valuable knowledge from this

project, this journey enable me to better understand myself in times of stress and

pressure. Being pushed out of my comfort zone is definitely the best way to grow.

I started out this journey with minimal knowledge in computing, nevertheless it

was all worth the effort. There was this intangible sense of achievement when you

finally get the program up and running.

Much fun and laughter have permeated the GS room throughout this one year,

moments like these are hard to come by and it is a timely reminder that we are

all humans and that taking a break might not be a bad idea after all. Special

thanks to a very important person, my mentor Jiawen . I would like to offer my

heartfelt gratitude to him for his unconditional help. Of course for introducing

2

Contents

me to C++, which came as a shock because I thought that Matlab was already

torturous enough. Without his help and encouragement, I will not be able to see

myself through. I am truly appreciative of the knowledge that he imparted me and

I thank him for putting up with all my incessant questions albeit some of them

being trivial.

Special mention has to be made to exceptional individuals. Joel Wong and Ng Yien

of whom I had meaningful and constructive discussions with on Matlab. Without

them, this journey would be fraught with perils and uncertainties. Not forgetting,

Yong Sheng who provided me with valuable insights and tips for creating this doc-

ument.

I would like to thank my dear family for being so understanding in times like this.

Enabling me to work in peace and of course, tolerating my frequent mood swings.

Last but not least to my understanding partner, Juan ,this journey is made pos-

sible by your understanding and patience.

The purpose of university is to unlearn what we have learn. These 4 years, know-

ing what you do not know is more important than knowing what you have already

known.

I dedicate this thesis to all whom have made me who I am today.

3

CONTENTS

Abstract 1

Acknowledgements 2

1 Introduction 11

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Hamiltonian Mechanics 13

2.1 The concept of phase space . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Hamilton’s equations and Liouville’s theorem . . . . . . . . . . . . . 14

2.2.1 Invariant measure in Liouville’s dynamics: Phase space volume 15

3 Ergodicity and Chaos 17

3.1 On Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The Ergodic Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 On Microcanonical ensemble (MCE) . . . . . . . . . . . . . . . . . 18

3.4 Ergodic Adiabatic Invariant . . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Physical interpretation of the ergodic adiabatic invariant . . 24

3.5 Chaos theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.1 Visualizing Chaos : The Poincare Surface of Section (P.O.S) 28

4

Contents CONTENTS

4 Statistical mechanics in small system 31

4.1 Meaning of temperature in statistical mechanics . . . . . . . . . . . 31

4.1.1 Relationship between the surface and volume entropy . . . . 32

4.1.2 The Henon Heiles Oscillator: An application . . . . . . . . . 34

5 Fluctuation theorems 38

5.1 Crook’s fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . 38

5.1.1 Crook’s relation for MCE . . . . . . . . . . . . . . . . . . . 39

5.2 Jarzynski Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 Jarzynski Equality in classical system . . . . . . . . . . . . . 40

5.3 Work fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 The Sinai billiard 44

6.1 Adiabatic invariant of Sinai billiard . . . . . . . . . . . . . . . . . . 46

7 Methodology and objectives 48

7.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.2.1 Generating the ensembles . . . . . . . . . . . . . . . . . . . 51

7.2.2 Adiabatic variation of the Hamiltonian . . . . . . . . . . . . 53

7.2.3 Adiabatic expansion of wall . . . . . . . . . . . . . . . . . . 54

8 Numerical simulations and results 56

8.1 Work fluctuations in MCE . . . . . . . . . . . . . . . . . . . . . . . 56

8.2 Work fluctuations in Canonical ensemble . . . . . . . . . . . . . . . 58

8.2.1 Derivation of 〈e−βW 〉 and 〈e−2βW 〉 for Sinai and modified

Sinai billiards. . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.2.2 Derivation of 〈e−2βW 〉square for Square . . . . . . . . . . . . 61

8.2.3 Expansion protocol . . . . . . . . . . . . . . . . . . . . . . . 63

8.2.4 Determination of δL for expansion protocol: φ =5

4. . . . . 64

8.2.5 Comparison of work fluctuations for each model at different

β: Expansion protocol . . . . . . . . . . . . . . . . . . . . . 65

8.2.6 Contraction protocol . . . . . . . . . . . . . . . . . . . . . . 67

5

Contents CONTENTS

8.2.7 Determination of δL for contraction protocol : φ =4

5. . . . 68

8.2.8 Comparison of work fluctuations for each model at different

β: Contraction protocol . . . . . . . . . . . . . . . . . . . . 69

8.2.9 Analysis of results: The work fluctuation of each model at

different β. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.2.10 Analysis of results: On the smoothness of convergence for

the expansion and contraction protocol. . . . . . . . . . . . . 72

8.3 Work fluctuations in chaotic and non chaotic model: MCE . . . . . 76

8.4 Work fluctuations in chaotic and non chaotic models: Canonical

ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.4.1 Comparison of work fluctuation across different models: Ex-

pansion protocol . . . . . . . . . . . . . . . . . . . . . . . . 78

8.4.2 Comparison of work fluctuations across different models:

Contraction protocol . . . . . . . . . . . . . . . . . . . . . . 80

8.4.3 Analysis: Work fluctuation for chaotic and non chaotic model

by canonical sampling . . . . . . . . . . . . . . . . . . . . . 82

9 Conclusion 83

10 The step forward 85

Appendices 87

A Derivation of the adiabatic invariant for 2D Sinai system 88

B Matlab codes 89

B.1 Sinai billiard: MCE . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.2 Modified Sinai billiard: MCE . . . . . . . . . . . . . . . . . . . . . 95

B.3 Poincare surface of section for Henon Heiles oscillators . . . . . . . 102

C C++ Codes 105

C.1 Sinai billiard: Canonical . . . . . . . . . . . . . . . . . . . . . . . . 105

C.2 Modified Sinai billiard: Canonical . . . . . . . . . . . . . . . . . . . 112

6

List of Figures CONTENTS

Bibliography 122

7

LIST OF FIGURES

3.1 Microcanonical representation in phase space . . . . . . . . . . . . . 20

3.2 Evolution of energy surface for 1D integrable system . . . . . . . . 22

3.3 Evolution of energy surface for MCE . . . . . . . . . . . . . . . . . 25

3.4 Sample trajectory for Poincare surface of section . . . . . . . . . . . 28

3.5 P.O.S (Low energy) . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6 P.O.S (Medium energy) . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 P.O.S (High energy) . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1 P.O.S of Henon Heiles oscillator at E = 110

. . . . . . . . . . . . . . 36


. . . . . . . . . . . . . . . 36


. . . . . . . . . . . . . . . 36

6.1 The Sinai billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2 Matlab simulation for Sinai billiard. . . . . . . . . . . . . . . . . . . 45

6.3 Matlab simulation for modified Sinai billiard. . . . . . . . . . . . . . 46

7.1 Sinai billiard: Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 Modified Sinai billiard . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.3 Square billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.4 Expansion protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.1 Work fluctuations in MCE . . . . . . . . . . . . . . . . . . . . . . . 56

8.2 Relative work fluctuations in MCE . . . . . . . . . . . . . . . . . . 57

8

List of Tables LIST OF FIGURES

8.3 Comparison of work fluctuation for square: Expansion . . . . . . . 65

8.4 Comparison of work fluctuation for modified Sinai billiard: Expansion 66

8.5 Comparison of work fluctuation for Sinai billiard: Expansion . . . . 66

8.6 Contraction protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.7 Comparison of work fluctuation for square: Contraction . . . . . . . 69

8.8 Comparison of work fluctuation for semi-circle model: Contraction . 70

8.9 Comparison of work fluctuation for Sinai model: Contraction . . . . 70

8.10 Chaotic vs non-chaotic models in MCE . . . . . . . . . . . . . . . . 76

8.11 Comparison of work fluctuation for expansion at β=0.1. . . . . . . . 78

8.12 Comparison of work fluctuation for expansion at β=0.01. . . . . . . 79

8.13 Comparison of work fluctuation for expansion at β=0.001 . . . . . . 79

8.14 Comparison of work fluctuation for contraction at β=0.1 . . . . . . 80

8.15 Comparison of work fluctuation for contraction at β=0.01 . . . . . 81

8.16 Comparison of work fluctuation for contraction at β=0.001 . . . . . 81

9

LIST OF TABLES

8.1 Determination of δL for expansion protocol. . . . . . . . . . . . . . 64

8.2 Determination of δL for contraction protocol. . . . . . . . . . . . . 68

8.3 Table of test for δ 〈e−2βW 〉 for ergodic systems: circle and semi-circle 75

8.4 Table of test for δ 〈e−2βW 〉 for the non- ergodic system: Square. . . 75

10

Chapter 1INTRODUCTION

1.1 Motivation

The advent of modern technology has forced us to re-look into the problem facing

nano-scale thermodynamics, where quantum mechanical effects have to be taken

into account. Technonlogical devices have shrunk in size over the years and the

thermodynamics of few bodies systems is in the limelight. Much research has been

done on nano scale thermodynamical system such as the single ion heat engine [3]

and the single molecule opto-mechanical system [4].

In a small system with few degrees of freedom, thermal and work fluctuations can-

not be neglected [2]. It is critical to minimise these work fluctuation to improve

the work output these nano scale heat engine.

Another problem that is pertinent to our discussion is the ability for us to define

meaningful thermodynamical quantities for few body system. Gibb’s theory of sta-

tistical ensemble allows us to make statistical interpretation of system with infinite

degrees of freedom based on the laws of large numbers. Equlilbrium conditions

are necessary for us to make meaningful interpretation from statistical mechanics.

So that we are able to describe macroscopic observable like temperature and pres-

sure [5]. This very property is often not found in small system where fluctuations

due to work and heat are dominant [6].

11

Introduction 12

To our surprise, Berdichevsky and team proposed that even for small system, if

the system is chaotic enough and exhibit ergodicity then we are still able to draw

meaningful conclusions of their thermodynamics [7].

The scope of this paper will focus on exploring the characteristics of work fluctu-

ations in chaotic system, which in our case we have chosen the Sinai Billiard for

the purpose of this study.

1.2 Thesis overview

This paper will be divided into 3 parts. The first part of the paper, Chapter 2-5

will be dedicated to reviewing some of the key concepts that is necessary for us to

better understand this project. Chapter 2, will be a brief overview on Hamiltonian

mechanics and the concept of phase phase. While Chapter 3 and Chapter 4 will be

on the discussion of Chaos and Ergodicity and their roles in statistical mechanics.

Next, Chapter 5 will be more involved as we will delve into statistical mechanics

in non-equilibrium regime mainly through the use of fluctuation theorems.

For the 2nd part, Chapter 6 and 7, we will discuss on the properties of the Sinai

billiard and review the methodology for this project.

Lastly, we will end off with some discussion on the results that we have obtained

from our computational simulations in Chapter 8 .

12

Chapter 2HAMILTONIAN MECHANICS

2.1 The concept of phase space

Phase space forms an integral part of the studies of dynamical system. Generally

speaking, phase space is described into position q and momentum p, namely the

generalized coordinate and momenta of the system. Classically we are able to

identify a state of a system by defining q and p of the system at a given time t.

A point in phase space will then represent the state of the system. The formalism

of phase space is critical for us to analyse Hamiltonian systems.

A system’s state in phase space can be represented byq ,p

The phase points

will evolve under the Hamiltonian equation of motions. For a time independent

Hamiltonian, Hamiltonian dynamics will then demand that no two trajectories can

ever cross in phase space because any points in phase space will be governed by

the Hamilton’s equations of motion which are linear and deterministic.

13

Hamiltonian Mechanics 14

2.2 Hamilton’s equations and Liouville’s theo-

rem

Various form of formalism have been developed in the field of mechanics and

dynamics; of which the Hamiltonian formalism has the most direct correlation to

quantum mechanics and statistical mechanics. Under this formalism, we seek to

solve the equation of motion by using first order equations, which is known as the

Hamilton’s equations of motion.

p = −∂H∂q

(2.1)

q =∂H

∂p(2.2)

dH

dt=∂H

∂t(2.3)

with generalized momentum p = (p1...pN), coordinate q = (q1...qN) and H is the

Hamiltonian of the system and in our case it is just the sum of its kinetic and

potential energy.

Along with Hamiltonian dynamics is the classical Liouville dynamics. More popu-

larly known Liouville’s theorem, first formulated by the German physicist Joseph

Liouville in 1838. The theorem gives an invariant measure to our Hamiltonian sys-

tem which is the phase space volume. Liouville’s theorem states that the density

ρ(q,p, t) of representative points in phase space corresponding to the motion of

the system remains constant during the motion [8]. This is due to the incompress-

ibility of flows in the Hamiltonian systems. For the subsequent derivations I will

omit the (q,p, t) dependence in ρ for neatness but one should always be aware of

these dependence.

14


Liouville’s theorem states that

dρ

dt= 0 (2.4)

By application of the chain rule

dρ

dt=∂ρ

∂t+

n∑i=1

(∂(ρqi)

∂qi+∂(ρpi)

∂pi

)=∂ρ

∂t+

n∑i=1

(qi∂ρ

∂qi+ pi

∂ρ

∂pi

)=∂ρ

∂t+

n∑i=1

(∂H

∂pi

∂ρ

∂qi− ∂H

∂qi

∂ρ

∂pi

)=∂ρ

∂t+ ρ,H

(2.5)

Where i is the number of independent equations born out of the constraints sus-

tained by the dynamical system and the poisson bracket ρ,H =∑n

i=1

(∂H∂pi

∂ρ∂qi− ∂H

∂qi

∂ρ∂pi

).

Thus the evolution of the phase space density in Hamiltonian mechanics is given

by the compact form

∂ρ

∂t= −ρ,H (2.6)

2.2.1 Invariant measure in Liouville’s dynamics: Phase

space volume

The Hamiltonian evolution of the system can be regarded as a series of canonical

transformations in phase space.

For canonical transformations, there exists a symplectic structure given by

MJMT = J (2.7)

15


where M is the Jacobian matrix and J is defined as 0 1

−1 0

The volume element will undergo a canonical transformation from

(dη) = dq1dq2...dqndp1dp2...dpn (2.8)

to a new volume element

(dζ) = dQ1dQ2...dQndP1dP2...dPn (2.9)

This transformation relation is governed by the Jacobian determinant

dζ = ||M||dη (2.10)

To find M we take the determinant of both sides for the symplectic condition in

Eq. (2.7) to arrive at

||M||2||J|| = ||J|| (2.11)

It is clear that Eq. (2.11) gives a value of M= ±1. Referring to Eq. (2.10) we can

see that Hamiltonian dynamics of a statistical ensemble preserves the phase space

volume. This idea will be an anchor point from which we will explore the concept

of adiabatic invariant.

16

Chapter 3ERGODICITY AND CHAOS

3.1 On Ergodicity

The study of ergodicity is abstract and usually restricted to that of pure mathe-

matics. A multidisciplinary approach has to be taken as the concept of ergodicity

involves ideas from probability theory, number theory and vector fields on mani-

fold etc.

Simply put, ergodic theory is the mathematical theory of dynamical system pro-

vided with an invariant measure. For the usual Hamiltonian system that we will

be studying, this invariant form will be that of the phase space volume (Ω) in Eq.

(3.10). The concept of ergodicity is however, relevant to physics as it forms the

cornerstone for our interpretation of statistical mechanics. If a dynamical system

is ergodic then the particles trajectories will fill the available phase space over time

subjected to its initial constraints.

That being said, it remains impossible for a particular trajectory to cross path

with every point in the available phase space. For a high dimensional phase space

despite long time the 1D trajectory may be ‘lost’ in phase space. It can only come

arbitrary close to the neighbourhood of every point in the available phase space.

17

Ergodicity and Chaos 18

3.2 The Ergodic Hypothesis

The Ergodic Theorem is a central concept in the study of ergodicity. Mathemati-

cians and physicists have made efforts directed to obtain a proof of the validity

of the ergodic hypothesis in particular mechanical systems, although the efforts

did not lead to a solution of the original problem, many more interesting results

emerged from this field of research ranging from number theory to information

theory. Hence we can only give a somewhat vague definition of what the theorem

really encompasses [9].

Despite its mathematical rigour the Ergodic Hypothesis has a rather straightfor-

ward interpretation, at least to physicists. It implies that the time average of a

particle’s trajectory is equivalent to its ensemble average over phase space.

〈A (q,p)〉 =

∫dNq dNpρ(q,p)A(q,p) (3.1)

〈A(q(t),p(t)

)〉t = lim

T→∞

1

T

∫ T

0

A(q(t),p(t)) dt (3.2)

Hence

〈A(q,p)〉 = 〈A(q(t),p(t)

)〉t (3.3)

The expression in Eq. (3.3) forms the basis of statistical mechanics and introduces

us to the idea of Gibb’s ensembles in the interpretation of statistical mechanics. Let

us then take a closer look at the most fundamental ensemble: The microcanonical

ensemble, in order to better understand the role that Ergodic Hypothesis plays in

the establishment of statistical mechanics.

3.3 On Microcanonical ensemble (MCE)

The microcaonical ensemble forms the backbone of statistical mechanics in which

all other ensembles could be derived from. Thus it is appropriate for us to devote

18


some time to understand the MCE in this part.

Being the most fundamental statistical ensemble, the thermodynamics of the MCE

is governed by energy conservation which at equilibrium, forbids heat or matter

exchange with the surrounding.

We will now introduce the term Ξ(E, V,N, α) which is known as the statistical

weight, α is an additional parameter that has to be defined if the system is in the

non equilibrium state. To each value of α we will have the corresponding statisti-

cal weight of Ξ(E, V,N, α); the number of microstates comprising that macrostate.

In the language of phase space, a microstate is represented by ε = (q, p) and ε, the

macroscopic equilibrium, is ergodic with respect to the Hamiltonian dynamics [10].

The phase space, Ξ , can be divided into a finite number of K disjoint cells, each

cell will then be a microstate of the system.

For the microcanonical ensemble the postulate of equal a priori probabability which

states that; for an isolated system, all microstate which are compatible with the

constraint, (E,V,N) in this case, will have an equal probability of occurring.

pi =1

Ξ(3.4)

subjected to the constraint ofk∑i=1

pi = 1

Thus in the MCE the role of ergodicity is distinct. It guarantees that a particle

will visit every single microstate in its available phase space, constrained by E,

over time. The probability of the system being in any of the available microstate

is equal. In the equilibrium case, we can derive a particular property in thermo-

dynamics, the Entropy.

19


The entropy of the system is given by

S(pi) = −k∑i=1

pi ln pi = k ln Ξ (3.5)

In other words, the value of the entropy will be at its maximum in an equilibrium

state.

For the MCE , the whole ensemble will occupy a volume of a thin layer of shell,

black region, bounded by E + ε and E.

E

E + ε

Phase space volume

Figure 3.1: The volume occupied by the micocanonical ensembles will just be athin layer of shell bounded by the energy surface.

The volume bounded by the energy shells is the phase space volume. This concept

will be important when we visit the subsequent section on the ergodic adiabatic

invariants.

Let us now explore the Boltzmann entropy, which shall be assigned a symbol (S)

with regards to Fig (3.1).

S = k ln(εω) (3.6)

where ε is a small energy constant required to make the argument of the logarithm

dimensionless and ω is the density of states (d.o.s) in phase space of the system.

Hence the product εω = Ξ will give the total number of microstates for the given

constraint.

20


It is relevant to point out that there is another form of entropy of the so called

volume entropy (S) given by

S = k ln Ω (3.7)

We shall discuss more on these two types of entropies in the next chapter, where

we will cover statistical mechanics of small systems.

The importance of entropy should should never be downplayed as it is a funda-

mental quantity in which all other thermodynamical variables, such as temperature

and pressure, can be derived.

3.4 Ergodic Adiabatic Invariant

Adiabatic invariants are well studied in the field of physics and it manifest itself

in various definitions. In the field of thermodynamics an adiabatic process is one

which forbids heat exchange with its surrounding. For this the entropy is the in-

variant if the process is reversible. In quantum mechanics adiabaticity implies that

the change in Hamiltonian is slow compared to the time scale set by the energy

difference of the eigenstates of H0 [?]. This ensures that no transition takes place

during the adiabatic process and the quantum number is invariant.

Let us begin with the analysis of a classical 1D-integrable system in classical me-

chanics.

All 1D systems are integrable and hence solvable. An adiabatic evolution can be

described as an evolution of its energy surface from t = 0 at E = E(0) to a new

energy surface at t = τ with value E(τ). see Fig 3.2.

21


𝑡 = 0 𝑡 = τ

𝑝

𝑞 𝑞

𝑝

𝐻 = 𝐸(0) 𝐻 = 𝐸(τ)

Figure 3.2: During an adiabatic evolution, the energy surface evolve from E(0) toE(τ). The area bounded by the energy surface is known as the ‘action’.

Fig 3.2 represents a physical picture for a 1 dimensional integrable system, it can

be easily extended into system with multiple degrees of freedom thus forming a

multi-dimensional phase space.

The area covered by the loop, which is the surface of constant energy will remain

invariant if an adiabatic protocol is enforced onto it. This area is known as the

‘action’ in classical mechanics and is usually denoted by the letter I. The classical

adiabatic theorem states that if an external parameter is changing slowly as com-

pared with the time scale of a classical integrable system, then the action variable

I will be an invariant. I can be calculated from a circulation integral in phase

space.

I =1

2π

∮pdq (3.8)

For an ergodic dynamical syetem under an adiabatic protocol, there exists an er-

godic adiabatic invariant Ω [11].

22


For a conservative dynamical system characterised by a time dependent Hamilto-

nian, we have

H = H(p,q, λ(t)

)(3.9)

where p, q are N vectors and N represents the degrees of freedom in the system.

The explicit slow time dependence of H is encapsulated in λ. By adiabatic, we

meant that the rate of change of the Hamiltonian is much slower than the natural

frequency of the system.

The ergodic adiabatic invariant is defined as

Ω(E, λ(t)

)=

∫∫V

U[E −H(p,q, λ(t)

)] dNp dNq (3.10)

where U is the unit step function.

The expression in Eq. (3.10) is a full integral over phase space. The step function

U reminds us that we are only considering microstates whose whose Hamilton is

less than the prescribed energy E. Thus Ω(E, λ(t)

)is measuring the phase space

volume that is bounded by the surface of constant energy E, which is an invariant

property [12].

We know that Ω(E, λ(t)

)is strictly a function of energy, E and λ(t). Hence we

can always express E in (q,p) representation, without loss of mathematical rigour,

to arrive at a more general expression for Ω.

Ω(q, p, λ(t)

)= Ω

(E(q, p, λ(t)

), λ(t)

)(3.11)

I shall name Ω(E(q, p, λ(t)

), λ(t)

)as Ω

(E, λ(t)

)so that we are neater with the

expression.

From Eq. (3.11), we can then show that E and Ω(E, λ(t)

)are bijection of each

23


other. Meaning to say that for a value of E there will only be one unique Ω

corresponding to it.

3.4.1 Physical interpretation of the ergodic adiabatic in-

variant

From subsection (2.2.1), we know that for any Hamiltonian system the phase space

volume is an invariant. This is also true for ergodic system that obeys Hamilton’s

equation of motion. Now there is additional property about ergodic systems that

makes it stands out amongst the non ergodic one upon exposure to an adiabatic

protocol.

Suppose that we sample some initial points, (qn,pn), where n is the labelling for

the particle’s number, we have n = 1, 2 and 3 for this example. This is done at

time t = 0, from a surface of constant energy E0 i,e from a MCE. We will then

expose this ensemble under an adiabatic protocol, that is to say we will consider the

variation of λ(t) so that the Hamiltonian of the system is changing adiabatically.

For the overall protocol, we have from t = 0 to t = τ

∆λ = λ(τ)− λ(0) (3.12)

24


(q3, p3)

(q2 , p2 )

(q1, p1)

Adiabatic protocol

(q’1,p’

1 )

(q’2,p’

2 )

(q’3,p’

3 )

Δλ

E0 Eτ

Figure 3.3: During an adiabatic evolution, the energy surface evolve from E0 toEτ . The phase space volume remains a constant during the protocol hence playingthe role as an adiabatic invariant.

The ergodic adiabatic invariant is then the phase space volume which is bounded

by E0 and Eτ . The adiabatic evolution will change the energy of the ensemble but

it will preserve the phase space volume. This invariance will be important in the

following section when we discuss more on statistical mechanics of small systems.

From Fig (3.3) , the sampled points will evolve to a new energy surface Eτ , in the

primed representation. What is interesting to note here is that these points will

have the same energy. They lie on a equi-energy surface. This is, in general, not

true for non-ergodic systems, as the final trajectories will have different energies.

This is a valuable insight. Now we know that if we sample our ergodic system from

a microcanonical ensemble and we vary its Hamiltonian adiabatically, then the fi-

nal state of the ensemble will also have similar energy, which is a microcanonical

state as well.

This has serious implications for statistical mechanics in small systems. If the final

state is a MCE then we can adopt Gibb’s interpretations of statistical mechanics

25


to get statistical information on the small system after the protocol. Hence we can

re-apply the usual laws of statistical mechanics.

On a side note, we are also able to gain additional insights, from Hamilton’s

equation of motion in Eq. (2.3):

dH

dt=∂H

∂λλ (3.13)

If the system is ergodic, for an infinitesimal change in λ, λ(t + dt) = λ(t) + λdt.

The trajectory will have cover the whole of its available phase space in that time.

Hence, it will be apt at this to investigate the expectation value of the expectation

value of Eq. (3.13).

dH

dt= 〈∂H

∂λ〉λλ (3.14)

Taking the appropriate time derivative, we have

Efinal − Einitial = −∫ λ(τ)

λ(0)

Fλdλ (3.15)

Where Fλ = 〈∂H∂λ〉λ

is the microcanonical average.

From Eq. (3.15), if the initial sampled state was microcanonical then the final en-

ergy of the ensemble is independent of the initial conditions. It is only dependent

on the λ parameter which defines the way we implement the protocol.

It is important to point out that there is a distinction between the invariance of the

phase space volume mentioned in Liouville’s theorem in Eq. (2.10) and the ergodic

adiabatic invariant from the above-mentioned. The invariant measure, the phase

26


space volume, is due to Liouviile’s dynamics because the flow is incompressible in

phase space, it is preserved in a Hamiltonian system. Whereas in the context of

the ergodic adiabatic invariant, the phase space volume is also preserved. However

this volume is bounded by a surface of constant energy if the initial sampling was

done at the microcanonical state. An adiabatic evolution of the system will change

this energy surface but preserves the volume that it initially bounds.

3.5 Chaos theory

Determinism has been a trademark of physics ever since the 19th century. Given

the initial position and momentum of a classical particle, we are able to predict its

final state. To our surprise, nature is simply not that trivial. Most of the natural

phenonmenon such as weather pattern and planetary motion are often chaotic [8].

Chaotic dynamical systems are extremely sensitive to initial conditions. A small

change in initial condition will result in a totally different outcome thus rendering

long term prediction impossible [13]. That is to say chaos occurs when a system

depends in a sensitive way on its previous state.

This sensitivity is characterised by the local instability of the phase space orbits.

Much to our surprise and popular belief, a chaotic system is deterministic, i.e a

given set of initial conditions we are still able to predict the final outcome of the

trajectories. Hence the term deterministic chaos. Deterministic chaos is a trade-

mark for non linear system and non-linearity is a necessary condition for chaos but

not a sufficient one [8].

In general, Chaotic motions are those that lies between regular deterministics

trajectories, that were derived from solutions of integrable equations, to that of

unpredictable stochastic behaviour characterized by complete randomness [13].

Chaotic dynamics cannot be solve analytically and have to be analyse numerically

and dealt with in its full complexity.

27


3.5.1 Visualizing Chaos : The Poincare Surface of Section

(P.O.S)

Chaotic behaviour manifest itself in irregular trajectories in phase space. A more

useful approach to visualize chaos will be to use the Poincare surface of section

representation.

The Poincare surface of section is to provide an analysis using a 2D slice through

a 3D energy surface given by H(px, py, qx, qy) = H0. One always have a choice as

to decide on which parameters to fix, for instance if we decide to fix qy then we

will be studying motions in the (qx, px) plane. If the system is bounded then after

a certain time interval the trajectory will return and intersect the 2D plane again.

That is to say the trajectories are bound to intersect with that same section of

state space chosen after some time, this is in fact a necessary property for us to

adopt the surface of section approach.

Figure 3.4: The Poincare surface of section for a quasi-periodic orbit, notice thatthe trajectory will still intersection that same section of state space after somefinite time

Some characteristics from for the P.O.S map are:

1. Characteristics of Poincare map

A unique point or multiple points: System is periodic

A closed curved: System is quasi-periodic

28


A cloud of points: System is chaotic

The poincare map will thus prove a pictorial representation on the interpretation

of the dynamics of the system [14].

The P.O.S is useful for studying the behaviour of the system if we vary its energy

parameter. For example, a conservative system with 2 degrees of freedom, we will

have a 3D energy surface with a surface of section in 2D [15].

Figure 3.5: ThePoincare map forlow energy.

Figure 3.6: ThePoincare map formedium energy.

Figure 3.7: ThePoincare map forhigh energy.

As observed, as the energy of the system is increased there are fewer periodic

orbits and more random points on the Poincare map. These shows that chaotic

behaviour is dominant in this particular system with increasing energy. A chaotic

system will almost occupy the whole of the available space in the P.O.S, this has

yet another implication to the concept of ergodicity and these two concepts are

deeply intertwined.

The difference between chaos and ergodicity is subtle. In fact it remains almost

impossible, or at least mathematically abstract to draw a link between these two.

Having said that there are still some relations that we can observe between these 2

concepts. A more chaotic system will accelerate the process of achieving ergodicity.

A system with chaotic dynamics have the tendency to span its motion across

29


the whole configuration space. This is indeed the ingredients needed to establish

ergodicity. Hence, in general a completely chaotic system will be ergodic as well.

Thus for the rest of the discussion we will inter-switched these two terms.

30

Chapter 4STATISTICAL MECHANICS IN SMALL SYSTEM

4.1 Meaning of temperature in statistical me-

chanics

Statistical mechanics is an asymptotic theory valid in the limit of an infinite de-

grees of freedom. Hence there is a need for us to re-modify some of the concepts

that we know in classical statistical mechanics to fit into our current concept. In

this chapter we will investigate the notion of entropy in system with finite degrees

of freedom and its implications to thermodynamics.

There are currently two widely accepted views of entropy; they are the surface

entropy (S) and the volume entropy (S) associated with Boltzmann and Gibbs

respectively. Our concern will be the role that entropy plays in the MCE and the

associated definition of Temperature.

Firstly we will define the surface entropy as such

S = k ln(εω) (4.1)

where ε is a small energy constant required to make the argument of the logarithm

dimensionless and ω is the density of states (d.o.s) in phase space of the system.

31

Statistical Mechanics in Small Systems 32

On the other hand the volume entropy is of the form,

S = k ln Ω (4.2)

Ω is this case is the phase space volume that is enclosed by the surface of constant

energy in a MCE.

For a system with finite degrees of freedom, the most common form of entropy

adopted is the surface entropy as given in Eq. (4.1). In the thermodynamical

limit, where N →∞ the surface and volume entropy are equivalent.

With two different definitions for the entropies, we can define two types of temper-

atures for the MCE which we shall call it the Gibb’s (TG) and Boltzmann’s (TB)

temperature respectively.

TG =Ω(E)

ω(E)(4.3)

TB =ω(E)

ν(E)(4.4)

where ν(E) =∂ω(E)

∂E.

As mentioned, TG and TB will be equivalent in the thermodynamic limit. Hence

for system with finite degrees of freedom the temperature of the system will not

be similar to that of a classical ensemble.

4.1.1 Relationship between the surface and volume en-

tropy

We will now explore the relationship between the surface and volume entropy.

The aim of this section is to find a relation connecting the surface and volume

entropy. These definitions of surface and volume entropy challenged our normal

32


understanding of the meaning of temeprature. In fact, in an MCE there is no

unique definition of temperature [16].

The d.o.s of a system can be interpreted as

ω(E) =∂Ω(E)

∂E(4.5)

The derivation of the relationship between the two entropies is direct. We cast the

entropies in the following exponential form for the rest of the derivation we will

set Boltzmann’s constant k = 1.

We multiply ε to Eq. (4.5) for convenience and we want to express ω as a logarithnic

function so that it will be more convenient later when we express it in its entropy

form.

For any given protocol we will have a change in d.o.s. We have the expression:

ln(εωf)− ln(εωi) = ln(ε∂Ωf

∂E)− ln(ε

∂Ωi

∂E)

e(ln(εωf)−ln(εωi)) = e

(ln∂Ωf∂E

/∂Ωi∂E

)

=∂Ωf

∂E/∂Ωi

∂E

(4.6)

Let’s revert our attention to the Gibbs temperature, TG. From Eq. (4.3),

1

TG=∂ ln Ω

∂E

=1

Ω

∂Ω

∂E

(4.7)

With reference to Eq. (4.6) and Eq. (4.7) we then have the expression relating

surface to volume entropy through the relation

33


∂Ωf

∂E/∂Ωi

∂E=TiTf

Ωf

Ωi

(4.8)

Therefore with reference to Eq. (4.6) and Eq. (4.8) we have the following relation:

e(ln(εωf)−ln(εωi)) =Ti

Tf

Ωf

Ωi

(4.9)

Now we have the conversion formula which relates the surface to volume entropy

by a temperature factor of TiTf

. To simplify matters, we can write Eq. (4.9) as

e(Sf−Si) =TiTfe(Sf−Si) (4.10)

Hence we will have a clean relation connecting the surface and the volume entropy

as described.

4.1.2 The Henon Heiles Oscillator: An application

One classical example of non-linear dynamics will be that of celestial mechanics.

The Henon Heiles model,developed by Michel Henon and Carl Heiles while work-

ing on the problem of non-linear motion of a star around a galactic center where

the motion is restricted to a plane [17].

This section will briefly introduce a classic example of a chaotic system with low

degrees of freedom and it will provide a better understanding on the role of chaos

for system with low degrees of freedom. The Henon Heiles oscillator has been well

studied. For the high energy regime, the oscillator exhibits chaotic motion and its

statistical mechanics resembles that of a small system [7].

H =1

2(p2x + p2

y) +1

2(x2 + y2) + λ(x2y − y3

3) (4.11)

The non linear term in λ give rise to chaotic motions.

34


The Hamiltonian equation has the following form

p = −∂H(q,p, λ)

∂q(4.12)

q =∂H(q,p, λ)

∂p(4.13)

If λ is fixed, trajectories of the system will sample the surface of constant energy

E which will bound a phase space volume Ω(E, λ).

It is interesting to note that at high energy vibration typically for energy of E ≈ 16

the motion can be approximately ergodic. This can be seen from the Poincare’s

surface of section at different energy levels for the Henon Heiles oscillator. The

following simulation has been done for this system. The Poincare’s plane is fixed

at q1 = 0 hence we will be studying the dynamics in the q2 and p2 plane.

35


q2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

p2

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Poincare surface of section at E=1/10

Figure 4.1: The Poincare map for HenonHeiles oscillator at E = 1

10

q2

-0.4 -0.2 0 0.2 0.4 0.6

p2

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5Poincare surface of section at E=1/8


8

q2

-0.4 -0.2 0 0.2 0.4 0.6 0.8

p2

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Poincare surface of section at E=1/6


6

It can be seen that as the energy of the Henon Heiles oscillators is increased there

will be a gradual breakdown of the invariant tori apparent in Fig 4.2. At the

threshold energy of E = 16

the motion of the oscillator is chaotic as represented by

the clouds of points in the Poincare’s map [18].

An adiabatic variation of λ introduces an adiabatic protocol to the Henon Heiles

system, which in turn generates an ergodic adiabatic invariant. This adiabatic

invariant is just Ω(E, λ). We will then make use of this ergodic adiabatic invariant

36


to perform thermodynamical calculations, one such example will be to derive the

entropy of the ergodic Henon Heiles Hamiltonian:

S(E, λ) = ln Ω(E, λ) + Constant (4.14)

Following which, thermodynamical quantities like temperature (T) can also be

defined∂S

∂E=

1

T(4.15)

in its usual form.

Thus the presence of an ergodic adiabatic invariant is the key starting point for us

to make any sense of statistical mechanics in small systems.

37

Chapter 5FLUCTUATION THEOREMS

5.1 Crook’s fluctuation theorem

In equilibrium regime , microscopic time reversibility implies that any process and

its time reverse will occur equally frequently.

For non equilibrium processes, Crooks relation is exceptionally useful to help us

understand fluctuations in a non equilibrium regime. The relation is derived from

the canonical ensemble and is a form of the entropy production formula given as

PF(W )

PR(−W )= e

∆Sk (5.1)

Here ∆S is the entropy production of the driven system over some time interval,

PF(W ) is the probability distribution of the forward protocol and PR(W ) is the

probability distribution of the entropy production when the system is driven in a

time reversed manner [19].

As expected, in a system that is equilibrated we will find Eq. (5.1) to have a value

of 1. That is to say, during equilibrium there will be no net heat exchange and

hence no production of entropy. This is expected because the entropy of a system

will attain a maximum value at equilibrium [20]. Crook’s relation is extremely use-

ful for us to investigate the behaviour of system far away from equilibrium. Thus

fluctuation theorem will be useful for us as we are studying system with finite

38

Fluctuation theorems 39

degrees of freedom where non-equilibrium statistical mechanics plays a dominant

role.

5.1.1 Crook’s relation for MCE

It is useful for us to understand Crook’s fluctuation theorem from a fundamental

point of view. We will derive a version of the relation from a microcanonical point

of view. If we prepare our initial state at a MCE then we are sampling our states

from an energy shell of Hi = E, the work obtained is W = Hf (xf )−Hi(xi). Where

xf and xi is the final and initial position respectively. Work(W) being a random

variable is given by [21]

PE(W ) =

∫dxiδ(Hi(xi)− E)δ(W −Hf (xf ) +Hi(xi))

Ωi(E)(5.2)

Since the system is microscopically reversible we could also write the reverse prob-

ability distribution as

PE+W (−W ) =

∫dxfδ(Hi(xf )− E −W )δ(Hf (xf )−W −Hi(xi))

Ωf (E +W )(5.3)

The Jacobian for the transformation from dxi to dxf is 1. Therefore we can equate

Eqs. (5.2) to (5.3) and we have

PE(W )

PE+W (−W )=

Ωf (E +W )

Ωi(E)= e

Sf (E+W )−Si(E)

kB (5.4)

To further extract information from Eq. (5.4) we can adopt the 1st and 2nd law

of thermodynamics.

dF = dU − TdS (5.5)

For an isolated system dU = W thus the change in entropy of a system for a non

adiabatic process is

39


∆S = W−∆FT

(5.6)

In the thermodynamic limit where E → ∞ and the work distribution converges

to P(W ) and P(−W ) one recovers the canonical form of the Crook’s relation, as

explored in Eq.( 5.1) [19].

P(W )

P(−W )= e

∆SkB = eβ(W−∆F ) (5.7)

By integration we can retrieve the Jarzynski equality which we are about to discuss

in the next section.

5.2 Jarzynski Equality

Jarzynski equality is a benchmark for us to study the effect of non equilibrium

statical mechaics and thermodynamics [22]. The equation is 〈e−βW 〉 = e−∆F , the

expected exponential of work applied to a system during a force protocol is equiv-

alent to the exponential of Helmholtz free energy difference F between the two

thermally equilibrated states. This powerful insight allows us to relate the non-

equilibrium quantity W with the equilibrium quantity ∆F .

This chapter will be a review of the Jarzynski equality in classical system and its

derivation will be of due importance as well.

5.2.1 Jarzynski Equality in classical system

The Jarzynski equality relates work statistics with the Helmholtz free energy dif-

ference. The first thing to make clear is the definition of work in the classical

system that we are considering. Here we follow the approach of inclusive work,

whereby the work is given by the energy difference between the initial and final

state of the system. Consider a system described by the Hamiltonian H(λ(t), z(t))

evolving from t=0 to t= τ , where

40


z(t) = [p(t), q(t)] (5.8)

is the evolution trajectory of the system and λ(t) is a time dependent parameter

of the Hamiltonian.The inclusive work done is given by

Wτ = H(λ(τ), z(τ))−H(λ(0), z(0)). (5.9)

Beginning with an initial sample prepared at a Gibbs distribution, With (λ(0), z(0)

being the initial condition, the probability distribution at t=0 will be

ρ(λ(0), z(0)) =e−βH(λ(0),z(0))

Z0

, (5.10)

where

Zt =

∫Ω

e−βH(λ(t),z(t))dz(t), (5.11)

is the partition function of the system at time t. The expected exponential of work

done to the system during the protocol is then given by

〈e−βW 〉 =

∫Ω

ρ(λ(0), z(0))e−βWτdz(0)

=

∫Ω

e−βH(λ(0),z(0))

Z0

e−β[H(λ(τ),z(z(0),τ))−H(λ(0),z(0))]dz(0)

=ZτZ0

The Helmholtz free energy expressed by partition function is F = − 1β

lnZ. To-

gether with the expression in Eq. (5.12), we obtained the Jarzynski equality in

classical system:

〈e−βW 〉 =e−βFτ

e−βF0= e−β∆F . (5.12)

The expression takes the centre stage in small system thermodynamics. No matter

how fast we apply apply a force protocol to a system we are still able to retrieve

useful information,free energy changes ∆F,from it as long as the final and initial

Hamiltonian of the system is known. Thus the Jarzynski equality allows us to

41


harvest information on equilibrium state i.e ∆F from non-equilibrium properties

like work done on the system.

Jarzynski equality can also be used to verify the 2nd law of thermodynamics by

the use of the so called Jensen’s inequality where f(x) is a convex function.

〈f(x)〉 > f(〈x〉) (5.13)

The relation will follow naturally from Eq. (5.12),

〈W 〉 > 〈F 〉 (5.14)

For an adiabatic process we will obtain an equality sign for (5.14) thus all the work

incurred will be transferred to the free energy of the system.

5.3 Work fluctuations

The study of work fluctuations is the primary goal of our research. Small systems

may not reach their optimal performance as they are operating in non-equilibrium

conditions [6]. In such syatem, the work fluctuations is substantial. Work fluctu-

ations in quantum and classical system was well studied [2]. Under an adiabatic

protocol the work fluctuation of a system will indeed be minimised. In our case

we wished to study the characteristics of work fluctuations of chaotic systems with

finite degrees of freedom.

For a microcanonical ensemble the work fluctuation is

δ2(W ) =1

N

N∑i=1

[Wi − 〈W 〉]2 (5.15)

For a canonical ensemble the work fluctuation is expressed as

δ2(e−βW ) = 〈e−2βW 〉 − 〈e−βW 〉2 (5.16)

42


By minimising the variance as expressed in Eqs. (5.15) and (5.16), we will then

minimise the work fluctuations required to improve the efficiency of our small

system. It is good to keep these definitions on hand as we will be using them quite

frequently in later parts of the discussion.

43

Chapter 6THE SINAI BILLIARD

We have chosen the Sinai billiard to be our ergodic system of study. The Sinai bil-

liard is a well studied ergodic model characterised by motion which is highly non-

linear. [23] It is fully ergodic in its phase space, the model can also be extended

to the so called Lorentz gas model where it is particularly useful for the study of

kinetic theory of gases [24].

The particle is bounded by 4 walls and a circular domain, all these boundaries

are of infinite potential. On traversing the region Φ, the particle is experiencing

zero potential and hence performing free motion. A particle will experience spec-

ular reflection at the walls and the circular surface, obeying the ”Law of reflection.

The dispersing nature of the circular domain as depicted in Fig 6.1 is the feature

that give rise to the chaotic motion of the billiard system This divergence is what

give rises to a chaotic orbits, making the Sinai system highly ergodic.

44

Sinai billiard 45

Φ

Figure 6.1: Sinai billiard has been provento be highly ergodic due to the dispers-ing nature of the circular domain. In thebounded region Φ the potential is zero[25].

V =

0 if within Φ,

∞ at boundary.

A Matlab simulation reveals that the configuration space is indeed ergodic for the

Sinai billiard set up refer to Fig 6.2.

Figure 6.2: Simulation of particles trajectories of Sinai billiard using n=5 and timescale of 80

45

Sinai billiard 46

For the purpose of this paper we will also explore another model of the billiard

system known as the modified Sinai billiard, which has a semi-circular domain

compared to the Sinai’s circular one.

Figure 6.3: Simulation of particles trajectories for modified Sinai billiard usingn=5 and time scale of 80

The modified Sinai billiard shown in Fig 6.3 is less chaotic compared to the cir-

cular configuration. This can be observed from the distribution of the trajectories

covering the configuration space. The prescence of the flat surface in the modified

version will result in the trajectories to be less divergent and more regular as com-

pared to its circular counterpart. This explains the more sparse distribution of its

trajectories across the configuration space.

6.1 Adiabatic invariant of Sinai billiard

The Sinai billiard is a 2D system and hence possess 4 degrees of freedom. From Eq.

(3.10) we can obtain a more concrete expression, for a more detailed derivation

(refer to (A.1)). In the context of the Sinai system the adiabatic invariant reads

as

Ω(E, λ(t)

)=

∫∫V

U(E −H)(p,q, t

)dNp dNq (6.1)

46

Sinai billiard 47

By direct integration we have:

Ω(E, λ(t)

)= 2πmEA (6.2)

Where E and A is the energy and area of the system respectively.

Neglecting all the relevant constant and setting m=1. We approach the condition

necessary for an adiabatic change.

Ω0 = Ωt (6.3)

Correspondingly, we will have:

E(0)A(0) = E(t)A(t) (6.4)

Thus for a 2D system Eq. (6.4) allows us to relate the final energy of the system

given the initial energy and the corresponding areas. This must be re-emphasized

that the above relation only holds true in the adiabatic regime.

47

Chapter 7METHODOLOGY AND OBJECTIVES

7.1 Objectives

The objective of this projection is to gain a deeper understanding of work fluctua-

tions in ergodic systems for both the Sinai billiard and the modified Sinai billiard,

which we shall touch on in the subsequent section. We will also be using a non-

ergodic variant to benchmark our results, more will be explained in the following

section. In this paper, we will be investigating the following 4 main areas :

Work flucuations of MCE

For this part we will investigate how an adiabatic protocol will affect the work

fluctuations of our initial sampled states from MCE. Various parameters such

as the types billiards and initial energy of the sample will be changed to study

the behaviour of the system.

Work fluctuations of Canonical Ensemble

This part is an extension of the previous part except that we will obtain

our samples from a bath of fixed β. We will be testing work fluctuations for

different values of β across the different models of the billiards.

Work fluctuations in chaotic and non chaotic regime

For this part of the discussion, we will be studying the work fluctuation

between the chaotic and non chaotic models. This can be done by comparing

48

Methodology and objectives 49

between the ergodic models: Sinai, modified Sinai billiard and our non-

ergodic model which is the square billiard. These models will be discussed in

details at a later part of this chapter. The purpose of this section is to find

out how work fluctuation differs in the above mentioned regimes and to set a

benchmark for what is the accepted values of work fluctuation for practical

purposes in future works.

7.2 Methodology

For the case of this project we will be using Matlab and C++ for our coding and

to generate the results for the numerical simulations. This will be done for 100000

particles, one at a time through our ‘Loop’ iteration.

This section will present a general outline of the computational methods with more

details being provided in the subsequent section.

1. Generating the Ensembles

Generate samples from Microcanonical ensemble

Generate samples from Canonical ensemble

2. Adiabatic variation of the Hamiltonian

The Hamiltonian of the billiard systems can be varied by adjusting the

velocity of expansion of a side of our wall. The different velocities of

the wall (~w) will be generated through an iterative command, which

will be discussed in details in the following section.

3. General algorithm

Generate 100000 sets of particles with respect to MCE or the canonical

ensemble.

Create a ‘loop’ command to expose each particle, for 100000 particles,

through the protocol under a particular value of (~w). This is considered

to be one cycle.

49


After one cycle is achieved, i.e 100000 particles for a particular (~w).

Proceed with the same protocol but with a different (~w).

Tabulate the work statistics

4. Variables to consider

We will also investigate the effects of work fluctuations on variations of some

parameters namely :

Initial energy

Configuration of set up: 1) Sinai billiard, 2) Modified Sinai billiard and

3) square billiard.

50


The 3 types of billiard systems are :

Figure 7.1: The Sinai bil-liard, named as ‘Sinai’

Figure 7.2: The modifiedSinai billiard model, namedas ‘Semi’

Figure 7.3: The square bil-liard, named as ‘Square’

We shall refer these configurations as ‘Sinai’, ‘Semi’ and ‘Square’ respectively

for easy referencing when we discuss the simulations results.

7.2.1 Generating the ensembles

Microcanonical ensemble

The general method of sampling the ensemble is by adopting the Monte Carlo

random number generator. To generate the position (x,y) randomly, we use the

uniformly distributed random number generator. Of which we need to satisfy our

constraint i.e the dimension of the billiard domain.

51


For a given interval (a,b), a uniform distributed random variable X, is given by:

X = (b− a). t+a (7.1)

where t =[0,1], is the standard normal distribution.

For our billiard, which has a dimension of a square, with length L. We will fixed

the origin at the centre of the square thus we will have

X = −L2

+ [L× rand(1, 1)] (7.2)

where the function rand is the random number generator in built in Matlab.

To generate samples from the MCE we will need to fix the initial energy of our

samples, E = E0. The energy term for our system is carried by the velocities of

the particles. For particles traversing in the domain Φ they are experiencing free

motion we have:

E0 = (v2x + v2

y)/2 = |V | /2 (7.3)

where we set m= 1.

With vx and vy as:

vx = |V | cos θ (7.4)

vy = |V | sin θ (7.5)

where θ is also generated from a random uniform distribution [0, 2π].

Canonical ensemble

The canonical ensemble can be thought of as a system, made up by infinite number

of subsystems, coupled to an external infinite heat bath at a constant temperature

T . Unlike the microcanonical ensembles, the subsystems can transfer energy so as

to keep the temperature constant.

52


The probability of finding a microstate with a certain energy E is given by the

Gibb’s distribution

P =1

Ze−Ei/(kT ), (7.6)

where Z =∑N

i=1 e−Ei/(kT ) is the canonical partition function.

For the canonical ensemble, the positions (x,y) are generated by a uniform distri-

bution as well, discussed in the previous section. The 2 velocity variables vx and

vy will be sampled from a Gaussian distribution. For a Gaussian distribution:

ρGaussian =1

σ√

2πe−

(x−µ)2

2σ2 (7.7)

We will adopt the NORMRND function in Matlab and generate normally distributed

random variables, R according to R ∼ N (0,√

mβ

), where σ =√

mβ

.

7.2.2 Adiabatic variation of the Hamiltonian

Initialization of parameters

The Hamiltonian of the system will be varied by changing the dimension of the

area in which the billiard is free to traverse.

That is to say we will expand one side of our wall, in our case it is wall B, see Fig

7.4 below, but keeping the final area constant after every loop and cycle. This can

be done by keeping the change in the length of the billiard box constant for every

expansion. Do note that this will be the standard protocol for the MCE case. For

the canonical case, the protocol will be slightly different, more details will be

provided in Chapter 8.

Before going through the details of the adiabatic expansion, we need to be clear

about some of the initial parameters of our system.

53


By initializing our parameters (dimensionless units);

1. Dimensions

Length of square box= 40

Radius of circle= 15

2. Number of particles

100000

3. Change in length for billiard domain (δL)

We will fixed this value to be 10.

We will then study the statistics of the sample at different generalised time scale

T . For the canonical setting, as we shall discuss in the later sections, δL is not

fixed across the 3 types of models.

7.2.3 Adiabatic expansion of wall

The generalised time scale T can be used to determine ~w by:

~w =δL

T(7.8)

We first generate 2 parameters to fix the upper and lower bound of T . We will

identify them as Tfast and Tslow.

For our case, we set:

Tfast= 10

Tslow= 100000

Following the algorithm:

54


for k = 1 : n+ 1

T =

(Tslow

Tfast

)K−1n

× Tfast (7.9)

where k is the loop index and we let n = 16, be the number of data points needed

for plotting purposes. a diagrammatic representation will help us to visualise the

expansion process. This method to generate T is applicable for the canonical case

as well. Thus to sum it up, T is inversely proportional to ~w. The behaviour of

work fluctuations can then be studied at the adiabatic limit, i.e in the large T

regime.

δL

A

B

C

D

A

B

C

D

Figure 7.4: The figure shows the expansion protocol to vary the hamiltonian. Forour case we only allow wall B to expand. An example of a trajectory during theexpansion process is included in the figure as well.

55

Chapter 8NUMERICAL SIMULATIONS AND RESULTS

8.1 Work fluctuations in MCE

The following results are generated for the MCE. We will be testing the work fluc-

tuations of the MCE for both initial staring energy of 250 and 500 respectively,

for both configurations Sinai and the modified Sinai billiards.

0

10

20

30

40

50

60

70

80

90

0 0.02 0.04 0.06 0.08 0.1 0.12

Wo

rk f

luct

ua

tio

n (δW

)

T-1

Sinai_250

Sinai_500

Semi_250

Semi_500

Comparison of Sinai and modified Sinai (Semi) at E= 250 and E= 500 for MCE

Figure 8.1: The variation of work fluctuations with time is shown here. For thelarge T regime the fluctuations will tends towards zero.

56

Numerical simulations and results 57

From Fig 8.1, it can be observed that in the large T regime where the expansion

is adiabatic. The work fluctuations will approach zero. Secondly it is also inter-

esting to note that the rate of convergence is faster for the Sinai compared to the

modified Sinai case at a fixed energy.

This can be explained by the fact that the Sinai billiard is more chaotic compared

to the modified Sinai billiard. Refer to Fig 6.2 and Fig 6.3. Thus a more chaotic

system will aid in the accleration of the process of convergence to the minimal

work fluctuation.

To have a more quantitative view of the results, we will consider the relative work

fluctuation (RWF) for the MCE ensemble as well.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.02 0.04 0.06 0.08 0.1 0.12

Re

lati

ve w

ork

flu

ctu

ati

on

δW/W

T-1

Sinai_500

Sinai_250

Semi_500

Semi_250

RWF of Sinai and modified Sinai (Semi) at E= 250 and E= 500 for MCE

Figure 8.2: The relative work fluctuations clearly shows that no matter what arethe values of the initial energy, the work fluctuation of the modified Sinai billiardis always greater than that of the Sinai billiard as it is less chaotic

From Fig 8.2, it is clear that a more chaotic system is able to suppress the work

fluctuations to a greater extent. The convergence of the work fluctuation to zero

57


in the adiabatic limit signifies that the energy of the ensemble is a constant if the

Hamiltonian is varied adiabatically.

The convergence of the work fluctuation to zero, in the adiabatic limit, implies

that the final state of the ensemble will be a microcanonical state as well. Thus

for an ergodic system whose initial state was prepared at a microcanonical state.

We know that the ensemble will evolve to another micocanonical state under an

adiabatic protocol.

In other words we are able to perform statistical treatment to our small system.

This will definitely provide new insights to the study of statistical mechanics for

systems with finite degrees of freedom.

8.2 Work fluctuations in Canonical ensemble

For any classical dynamical system it is known that on exposure to an adiabatic

protocol the work fluctuation will indeed yield a minimum value [1]. We are

interested to know if this is true for chaotic systems sampled from a canonical

ensemble. Samples were drawn from three temperature bath, β=0.1, 0.01 and

0.001. Here β =1

kTis the inverse temperature. From (5.16), the variance of the

work fluctuation is determined by 2 parts, firstly the expression for relative work

fluctuation is given by 〈e−βW 〉 and 〈e−2βW 〉. We shall derive an explicit expression

for both the above mentioned terms to compare our numerical results with the

theoretical predictions.

58


8.2.1 Derivation of 〈e−βW 〉 and 〈e−2βW 〉 for Sinai and modi-

fied Sinai billiards.

Derivation of 〈e−βW 〉

e−β∆F =ZτZ0

=

∫ ∫d2q d2pe

−βp2x + p2

y

2m

∫ ∫d2q0 d2p0e

−βp2x + p2

y

2m

=AfAi

(8.1)

Where Af and Ai is the final and initial area of the billiard system.

Derivation of 〈e−2βW 〉

〈 e−2βW 〉 =

∫∫dpxdpy ρpxρpy e−2βW

=

∫∫dpxdpy ρpxρpy e

−2βEi

(Ai

Af−1

)

=


−β p2

m

(Ai

Af−1

)

=


−βp2x+p2

y

m

(Ai

Af−1

)(8.2)

For subsequent we will allow (AiAf− 1) be α since it is a constant.

Since our samples are drawn from the canonical ensemble, ρpx and ρpy is none

other than the probability distribution function of the Gaussian distribution as

evident from Eq. (7.7)

ρGaussian =1

σ√

2πe−

(x−µ)2

2σ2 (8.3)

By direct comparison the standard deviation, σ will be characterised by β. Giving

59


us the expression of σ =

√m

β.

By direct substitution of ρ and σ into Eq. (8.2), we will split the integral into

it’s x and y component, note that they are equivalent numerically and it will be

sufficient to take the square of just one component as described below:

〈e−2βW 〉 =β

2πm

[ ∫ ∞−∞

dpxe−β(1+2α×p2

x

2m)

]2

=β

2πm× 2πm

β(1 + 2α)

=1

1 + 2α

(8.4)

By evaluating α in terms of the final and initial areas we have

〈e−2βW 〉 =Af

2Ai − Af(8.5)

To make the expression more compact we introduce φ =Af

Ai

, which is the ratio

between between the final and initial area. Do take note that this ratio gives the

expression of 〈e−βW 〉 as explained in Eq. (8.1).

〈e−2βW 〉 =φ

2− φ(8.6)

Derivation of the work fluctuation for the ergodic models

The work fluctuation of e−βW for the ergodic models is:

60


δ(e−βW) =√〈e−2βW〉 − 〈e−βW〉2

=

√φ

2− φ− φ2

(8.7)

Note that the relative work fluctuation is just a function of the initial and final area.

The term 〈e−βW 〉 is a constant for a predefined protocol. That is to say it is only

dependent on the initial and final Hamiltonian of the system and not on the speed

in which the protocol is implemented. In the following section, we will test our

numerical results on the relative work fluctuation as compared to the theoretical

derivation above.

Extra care has to be taken when we derive the theoretical expression for 〈e−2βW 〉.We made an assumption when we calculate W in the exponential term by adopting

the ergodic adiabatic relation mentioned in Eq. (6.4). This is only true for the

Sinai and modified Sinai models as they are ergodic.

For the square which is non-ergodic and integrable, we will have to approach the

derivation of 〈e−2βW 〉 differently. Note that the expression 〈e−βW 〉 for the square

remains unchanged and follows suit from Eq. (8.1).

8.2.2 Derivation of 〈e−2βW 〉square for Square

To avoid any confusion , We shall rename 〈e−2βW 〉 as 〈e−2βW 〉square for the non-

ergodic variant. To derive the necessary expression for the square’s case. We

make use of the action angle representation in classical mechanics. The classical

adiabatic theorem states that for an integrable system under an adiabatic protocol,

the action I of the system is an invariant. Harking back to Eq. (3.8) we will define

a new set of adiabatic relation unique to the square billiard.

61


I =1

2π

∮pdq

=1

2π

∮ √2mEdL notag

=

√2m

π

∫ Lmax

Lmin

√EdL

Where L is the length of the billiard wall ‘A’.

Hence from Eq. (8.8) it is clear that for a non- ergodic and integrable system the

adiabatic relation connecting the initial and final state is as follows:

√EiLi =

√EfLf (8.8)

Where the subscript ‘i′ and ‘f ′ represent the initial and final state respectively.

Since we have 2 degrees of freedom in the configuration space for the billiard. The

action Ix and Iy are decoupled. An additional point to note is that only the y-

component of the energy will be of concern here as the x-component will always

be conserved, in virtue the protocol that we implemented. Hence following the

relation in Eq. (8.8) we will do a slight modification to Eq. (8.2) and perform the

necessary Gaussian integration.

62


〈 e−2βW 〉square =


=


−2βEi

(L2

i

L2f−1

)

=β

2πm

(∫ ∞−∞

e−βp2x

2mdpx

)(∫ ∞−∞

e−βp2y

1+2γ2m dpy

)=

β

2πm

√2πm

β(1 + 2γ)

√2πm

β

=1√

1 + 2γ,

(8.9)

Where we have replaced (L2i

L2f

− 1) with γ, and it is just a function of the initial

and final length of wall ‘A’.

By substituting in the value of γ we have:

〈e−2βW 〉square =Lf√

2L2i − L2

f

(8.10)

Thus the work fluctuation for the square is given by

δ( e−βW ) =

√Lf√

2L2i − L2

f

− φ2 (8.11)

8.2.3 Expansion protocol

We made an observation that in the canonical setting, the work fluctuations of

the ergodic configurations, modified Sinai and Sinai billiards depend solely on φ.

Whereas for the non-ergodic model, the square, the work fluctuation is a function

of L and φ.

Hence to make the comparison in the canonical setting fair we have decided to set

63


φ to be a constant for all 3 models. Setting φ to be a constant is tantamount to

equating 〈e−βW 〉 to be a constant as well. For our case we have chosen φ =5

4.

Thus by fixing φ, we will then need to have different δL for the 3 kinds of models.

8.2.4 Determination of δL for expansion protocol: φ =5

4A sample calculation will be that we set the initial length of wall ‘A’ to be 40,

similar to what was described in the methodology. We will now calculate δL with

φ =5

4.

A sample calculation for the square billiard will be:

5

4=AfAi

=(40 + δL)× 40)

40× 40

(8.12)

Therefore

δL = 10

With that we derive the respective δL ’s for the Sinai and semi-circle models,to 3

significance figures, as shown in the table below.

Determination of δL for expansion protocol

Model δL

Square 10

Modified Sinai (Semi) 7.80

Sinai 5.58

Table 8.1: Determination of δL for the 3 models for the expansion protocol at

φ =5

4.

64


8.2.5 Comparison of work fluctuations for each model at

different β: Expansion protocol

Square

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14

δex

p(-β

W)

ln T

Comparison of work fluctuation for Square(Expansion)

Square_0.1

Square_0.01

Square_0.001

Theoretical

Figure 8.3: Comparison of work fluctuation of the square configuration at differentvalues of β. The length of ‘A’ is increased by 10 from 40.

65


Modified Sinai

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14

δex

p(-β

W)

ln T

Semi_0.1

Semi_0.01

Semi_0.001

Theoretical

Comparison of work fluctuation for modified Sinai (Semi)(Expansion)

Figure 8.4: Comparison of work fluctuation of the modified Sinai billiard at dif-ferent values of β. The length of ‘A’ is increased by 7.80 from 40.

Sinai

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14

δex

p(-β

W)

ln T

Comparison of work fluctuation for Sinai(Expansion)

Sinai_0.1

Sinai_0.01

Sinai_0.001

Theoretical

Figure 8.5: Comparison of work fluctuation of the Sinai billiard at different valuesof β. The length of ‘A’ is increased by 5.58 from 40.

66


It is important to note that at the adiabatic limit the relative work fluctuation

does indeed converge to the minimum. Unlike the microcanonical case, the work

fluctuation will never converge to zero as our samples are drawn from the canonical

ensembles with a distribution of energies.

Secondly, it can be observed that the convergence is not smooth for the expansion

process. Which is to say that the data points are fluctuating but the eventual

convergence convergence at the adiabatic limit is guaranteed.

As such it will be interesting to study the behaviour of the work fluctuation in the

reversed protocol, i.e the contraction process.

8.2.6 Contraction protocol

Now let us study what will be the effects on the work fluctuations if we reverse the

protocol. i.e we set φ =4

5to determine the different values of δL for convenience

we set the initial length of wall ‘A’ to be 50.

The contraction of wall ‘B’ will thus result in the length of ‘A’ to decrease at the

end of the protocol.

67


This can be shown in the graphical representation below.

δL

A

B

C

D

A

B

C

D

Figure 8.6: The reversed protocol i.e the contraction process. The billiard systemwill contract from its initial area set by δL. This pictorial diagram represent anexample for the Sinai model.

8.2.7 Determination of δL for contraction protocol : φ =4

5We will apply the above protocol to our 3 models. To derive the respective δL for

the each of the variant we adopt the same method mentioned in subsection (8.2.4)

with φ =5

4this time round.

Determination of δL for contraction protocol : φ =4

5

Variant δL

Square -10

Modified Sinai (Semi) -7.80

Sinai -5.58

Table 8.2: Determination of δL for the 3 variants for the expansion protocol at

φ =4

5.

68


8.2.8 Comparison of work fluctuations for each model at

different β: Contraction protocol

Square

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0 2 4 6 8 10 12 14

δe

xp (

-βW

)

ln T

Comparison of work fluctuation for square(Contraction)

Square_0.1

Square_0.01

Square_0.001

Theoretical

Figure 8.7: Comparison of work fluctuation of the square billiard at different valuesof β. The length of ‘A’ is reduced by 10 from 50.

69


Modified Sinai

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0 2 4 6 8 10 12 14

δex

p (

-βW

)

ln T

Comparison of work fluctuation for modified Sinai (Semi)(Contraction)

Semi_0.1

Semi_0.01

Semi_0.001

Theoretical

Figure 8.8: Comparison of work fluctuation of the modified Sinai billiard at dif-ferent values of β. The length of ‘A’ is reduced by 7.80 from 50.

Sinai

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0 2 4 6 8 10 12 14

δex

p (

-βW

)

ln T

Comparison of work fluctuation for Sinai(Contraction)

Sinai_0.1

Sinai_0.01

Sinai_0.001

Theoretical

Figure 8.9: Comparison of work fluctuation of the Sinai billiard at different valuesof β. The length of ‘A’ is reduced by 5.58 from 50.

70


8.2.9 Analysis of results: The work fluctuation of each

model at different β.

It can be observed from subsections (8.2.5) and (8.2.8) the work fluctuation will

converge to the theoretical minimum in the adiabatic limit. Hence it furthers

strengthen our belief that minimal work fluctuation could be achieved by exposing

chaotic systems to adiabatic protocols.

Secondly, it was observed that the ensemble that is drawn from a bath with a

higher value of β will tend to have a higher work fluctuation in the low T regime.

We postulate that it might be due to a longer relaxation time as ensembles sam-

pled at higher β regime have lower initial energy. By relaxation time we meant

the time needed for a trajectory to go back to its initial neighbourhood in state

space. Having a longer relaxation time will imply that the cut off to adibaticity

is stricter as seen from the intersection of the graphs to the theoretical prediction.

Whereby it will need a longer time to attain the adiabatic limit. Hence a system

with a longer relaxation time will, in effect, have higher work fluctuation at the

low T regime.

In the adiabatic limit we understand that the work fluctuation for all sampled

values of β will tend towards the theoretical minimum. A note worthy point to

make is that for the low T regime, initial samples that were drawn from a heat

bath with lower β shows sign of less deviation from the theoretical prediction.

By sampling our initial trajectories from a bath of higher temperature we do not

need that strict of an adiabatic condition to achieve convergence. This information

is helpful from the experimental point of view, when experimenting with small

systems.

71


8.2.10 Analysis of results: On the smoothness of conver-

gence for the expansion and contraction protocol.

It is peculiar that for the expansion process, there is a chance that 〈e−2βW 〉 will

blow thus ill-defined. This is because negative work is done to the system when

we consider an expansion protocol. For the contraction process the convergence to

the theoretical values is smoother as seen. To explain this trend lets refer back to

Eq. (5.16). The only term that can result in a fluctuations of values is the term

〈e−2βW 〉. Hence we will need to study the standard deviation of this value to get

a clearer picture.

Firstly 〈e−2βW 〉 will also follow a normal distribution

〈e−2βW 〉numerical ∼ N [〈e−2βW 〉theoretical,δ√N

] (8.13)

we are interested to find out what is the value of δ , the standard deviation of

〈e−2βW 〉 . In other words we can establish the relation

δ〈e−2βW 〉 =√〈e−4βW 〉 − 〈e−2βW 〉2 (8.14)

To calculate 〈e−4βW 〉 we used similar approach as Eq. (8.2), however we need to

perform this calculation twice, once for the ergodic systems: Sinai and modified

Sinai and the other for the non-ergodic system: square billiard.

Derivation of 〈e−4βW〉 for ergodic system

〈e−4βW 〉 =

∫ ∫dpx dpyρpxρpye−4βW

=

∫ ∫dpx dpyρpxρpye

−4βEi(AiAf−1)

=


−2β p2

m(

AiAf−1)

=


−2β(p2x+p2

ym

)(AiAf−1)

=β

2πm

∫ ∫dpx dpy × e−β

p2x

2m × eβ

p2y

2m × e−2βp2y

mγ

(8.15)

72


as usual we let ( AiAf− 1) = α.

By doing the necessary Gaussian integral and substituting back the value of α ,

we will have an expression that is purely a function of the initial and final area

only :

〈e−4βW 〉 =Af

4Ai − 3Af(8.16)

Now let us bring back expression Eq. (8.5) so that we are able to make a compar-

ison with Eq. (8.15). In order for the evaluated integral of 〈e−2βW 〉 as discussed

in (8.2) to converge we need:

Af2Ai − Af

> 0 (8.17)

Thus a necessary condition will be that of

Af < 2Ai (8.18)

Now we will need the integral for the evaluation of 〈e−4βW 〉 to converge as well

as that would mean that our standard deviation will be of a finite value. This

is important as a finite and small value of δ〈e−2βW 〉 will increase the accuracy of

〈e−2βW 〉numerical being the true mean. i.e 〈e−2βW 〉theoretical

Hence we require

Af4Ai − 3Af

> 0 (8.19)

73


A necessary condition will then be

Af <4

3Ai (8.20)

Derivation of 〈e−4βW〉square for non-ergodic system

We shall rename 〈e−4βW 〉 as 〈e−4βW 〉square to avoid any confusion. The derivation

is as follows with only a slight change in the parameter.

〈 e−4βW 〉square =


=


−4βEi

(L2

i

L2f−1

)

=


−2βp2y

m

(L2

i

L2f−1

)

=β

2πm

∫∫dpxdpy × e−β

p2x

2m × e−βp2y

2m × e−2βγp2y

m

(8.21)

As usual, we let (L2i

Lf

2− 1) = γ.

By doing the necessary Gaussian integration and substituting back the value of γ

, we will have an expression that is purely a function of the initial and final length

of wall ‘A’ only :

〈e−4βW 〉square =Lf

4L2i − 3L2

f

(8.22)

Similar to the above discussion we require

Lf4L2

i − 3L2f

> 0 (8.23)

A necessary condition will then be

Lf <

√4

3Li (8.24)

74


Brining back (8.10), in order for the expectation value not to diverge we need

Lf <√

2Li (8.25)

It is obvious that the Eq. (8.20) and Eq. (8.24) formed a tighter bound as com-

pared to Eq.(8.18) and Eq. (8.25) respectively. We will then check for both the

expansion and contraction process to see if these conditions will be satisfied.

For the ergodic systems we have :

A_f<2 A_i

Sinai

Modified Sinai

Expansion protocol Contraction protocol

𝐴𝑓 < 2 𝐴𝑖 𝐴𝑓 < 2 𝐴𝑖𝐴𝑓 <4

3𝐴𝑖 𝐴𝑓 <

4

3𝐴𝑖

Table 8.3: The table shows that for the contraction protocol, all criteria are sat-isfied in order for the integral of concern to be convergent, thus explaining thesmooth convergence of the work fluctuation as depicted in (8.2.8).

For the non ergodic system we have :

Square

Expansion protocol Contraction protocol

𝐿𝑓 < 2𝐿𝑖 𝐿𝑓 < 2𝐿𝑖𝐿𝑓 <4

3𝐿𝑖 𝐿𝑓 <

4

3𝐿𝑖

Table 8.4: The table shows that for the contraction protocol, all criteria are sat-isfied in order for the integral of concern to be convergent, hence for the squarebilliard the convergence to the theoretical minimum is smoother for the contractionprocess.

For the contraction protocol, all of the conditions are fulfilled, especially that of the

75


tighter bound. This implies that δ 〈e−2βW 〉 will not be divergent for the contraction

protocol, for both ergodic and non-ergodic systems. Thus δ, the standard deviation

as explored in expression Eq. (8.13) in the data values can be kept to a minimal.

Which explains why the convergence of the work fluctuations is smoother for the

contraction protocol.

8.3 Work fluctuations in chaotic and non chaotic

model: MCE

This section will study the relative fluctuations concerning non chaotic system, i.e

the square configuration, compared with that of our Sinai system. We hope to

study how important is the effect of chaotic and ergodic system in curbing work

fluctuations.

The microcanonaical case is first studied in this section. We will consider both the

Sinai and square billiards for both initial energies of 250 and 500.

0

10

20

30

40

50

60

70

80

90

0 0.02 0.04 0.06 0.08 0.1 0.12

Wo

rk f

luct

uat

ion

(δW

)

T-1

Square_250

Sinai_250

Square_500

Sinai_500

Comparison of Sinai and Square billiard at E= 250 and E= 500 for MCE

Figure 8.10: Comparisons of work fluctuations Sinai and square billiard at energyof 250 and 500

76


From Fig 8.10 it is clear that the work fluctuations of the square billiard remains

relatively constant. On the other hand, for the Sinai system, it will minimise the

work fluctuations in the adiabatic limit. For the case of the MCE, the work fluc-

tuations will approach zero in the adiabatic limit for ergodic systems.

The intersection points of the graph clearly sets a benchmark for us to study the

work fluctuations. Below this point the fluctuations in work of the chaotic system

is less than that of the non chaotic system. It provides knowledge as to what

degree should we implement the adiabatic protocol so as to achieve the desirable

amount of work fluctuations.

77


8.4 Work fluctuations in chaotic and non chaotic

models: Canonical ensemble

For this section we will compare the work fluctuation of the 3 different variants at

fixed β. We will like to study if, like the MCE case, a more chaotic system will

be able to suppress the work fluctuations.

8.4.1 Comparison of work fluctuation across different mod-

els: Expansion protocol

Comparison at β = 0.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10 12 14

δex

p (

-βW

)

ln T

Comparison of work fluctuation at β=0.1(Expansion)

square_0.1

Semi_0.1

Sinai_0.1

Figure 8.11: Comparison of work fluctuation across different models of the billiardsystem at β=0.1 for expansion protocol.

78



0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0 2 4 6 8 10 12 14

δex

p (

-βW

)

ln T

Comparison of work fluctuation at β=0.01(Expansion)

Square_0.01

Semi_0.01

Sinai_0.01



0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 2 4 6 8 10 12 14

δex

p(-β

W)

ln T

Comparion of work fluctuation at β=0.001(Expansion)

Square_0.001

Semi_0.001

Sinai_0.001


79


8.4.2 Comparison of work fluctuations across different mod-

els: Contraction protocol


0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10 12 14

δex

p(-β

W)

ln T

Comparison of work fluctuation at β = 0.1(Contraction)

Square_0.1

Semi_0.1

Sinai_0.1

Figure 8.14: Comparison of work fluctuation across different models of the billiardsystems at β=0.1 for contraction protocol.

80



0.12

0.14

0.16

0.18

0.2

0.22

0.24

0 2 4 6 8 10 12 14

δex

p(-β

W)

ln T

Comparison of work fluctuation at β = 0.01(Contraction)

Square_0.01

Semi_0.01

Sinai_0.01



0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0 2 4 6 8 10 12 14

δex

p (-β

W)

ln T

Comparison of work fluctuation at β=0.001(Contraction)

Square_0.001

Semi_0.001

Sinai_0.001


81


8.4.3 Analysis: Work fluctuation for chaotic and non chaotic

model by canonical sampling

In this section we will analyse the results from the comparison of our non chaotic

system,the square, to that of our chaotic systems, the circle and the semi-circle.

Similarly with previous comparison, the work fluctuation tends to its theoretical

minimum in the adiabatic limit.

What is interesting here is that we cannot differentiate between the circle and

semi-circle model at a low temperature regime, such as β=0.1 in Fig 8.14. In-

terestingly as we tune β to a lower value the distribution of the work fluctuation

between the Sinai and the semicircle case starts to show signs of differences, with

it being most obvious in Fig 8.16.

At a low temperature the fully ergodic system is not well optimised to lower the

work fluctuation, and its effect of suppressing the work fluctuation is only as good

as that for the less chaotic one. With increasing temperature, the more ergodic

system will prove to be more superior in suppressing the work fluctuation. Thus

one extension of this research would be to study the extend on the degree of mixed

phase space have on the characteristics of the work fluctuations.

82

Chapter 9CONCLUSION

This research sets a preliminary overview on the role of chaos and ergodicity in

work fluctuations of small systems. We have shown there is a need for us to re

evaluate statistical mechanics when it comes to small systems. Let us also be

reminded that fluctuations is dominant in small system and that to increase the

performance we need to reduce the fluctuations that accompanies it.

We have effectively shown that if we draw our samples, for our small system, from

a MCE and let it evolve under an adiabatic protocol, then the work fluctuation will

approach zero in the adiabatic limit. Secondly from the simulation on the MCE

we showed that the resulting ensemble after the protocol is also a microcanonical

ensemble as well. This result is exciting as it implies that as long as the system is

ergodic enough, we will still be able to apply the laws of statistical mechanics to

our small system provided that the variation in the Hamiltonian is adiabatic.

For the case of the canonical ensemble, we also observed that an adiabatic pro-

tocol, applied to our ergodic models, will result in the minimisation of the work

fluctuation in the adiabatic limit. Recall that this is not the case for the non-

ergodic square billiard. We also showed that convergence is much smoother for

the contraction process as opposed to the expansion process. Recall that for the

contraction process, work is done to the system and vice versa. For the canonical

ensemble we are also aware that a more ergodic system will have the capability to

83


suppress the work fluctuation to a smaller value even in the low T regime.

84

Chapter 10THE STEP FORWARD

This research is still in its nascent stage and there is still much excitement on its

future developments. More tests can be done to our canonical system as it is a

more realistic model from the experimental point of view.

Firstly, from the discussion in subsection (8.2.10), we proposed an explanation to

account for the rough convergence experienced by the expansion protocol. Recall

that this is due to the possible divergence of the value 〈e−2βW 〉. One could argue

that this behaviour is specific only for our case but there is a chance that this

value may be ill-defined for other ergodic systems as well. If that is true, then we

need to really delve deeper into the physics underlying these chaotic systems.

Secondly, we can also ask questions such as what is the characteristics of work fluc-

tuations in dynamical systems that possesses mixed phase space. Currently, we

are still not clear on how to define the degrees of ergodicity in a physical system of

course one could adopt a simplistic view on this definition, whereby a more ergodic

system will have a higher percentage of its phase space covered by the evolving

trajectories. This definition however lacks the mathematical rigour to correctly

understand the true nature of ergodicity.

Lastly, it will be exciting if we can verify the fluctuation relations derived in Chap-

ter 5 to see if it holds for our simulations on the microcanonical ensemble. As such,

85


we can then have a numerical verification against the theory. Simultaneously, we

can determine if the microcanonical Crook’s relations is true for ergodic systems

with finite degrees of freedom.

The nature of the project is extensive. It is really unfortunate that due to time

constraint for this project that we are not able to explore the additional areas as

mentioned above.

86

Appendices

87

Appendix ADERIVATION OF THE ADIABATIC INVARIANT FOR 2D

SINAI SYSTEM

The general expression to calculate the ergodic adiabatic invariant in a system for

N degrees of freedom is represented in (3.10)

µ(E, t) =

∫∫V

U(E −H)(p,q, t

)d2p d2q

= A

∫U(

p2

2m−H

(p,q, t

)) d2p

= Aπ(√

2mE)2

= 2πmEA

(A.1)

88

Appendix BMATLAB CODES

B.1 Sinai billiard: MCE

1 clc;

2 close all;

3

4

5 L = 40;

6 nparticles=100000;

7

8 deltaL = 10;

9 T_fast = 0.0001;

10 sets = 16;

11 T_slow = 1;

12 s = vmod*dt;

13 M = zeros(nparticles,4);

14 I_Results=zeros(nparticles,4);

15 work=zeros(sets+1,3);

16

17

18 A=-L/2;

89

Appendix 90

19 B=L/2;

20 C=L/2;

21 D=-L/2;

22 r = 15;

23

24 Plotting of graphs

25 figure(1);

26 theta =0:0.01:2*pi;

27 x_c=r*cos(theta);

28 y_c=r*sin(theta);

29 patch(x_c,y_c,’b’)

30 axis([A C D B])

31 xlabel(’x axis’),ylabel(’y axis’)

32 hold on

33

34

35 for i=1:nparticles;

36 while (1)

37

38 xMC = -L/2 + (L)*rand(1,1);

39 yMC= -L/2 + (L)*rand(1,1);

40 random_num = 2*pi*rand(1,1);

41 vxMC= vmod*cos(random_num);

42 vyMC =vmod*sin(random_num);

43 if (xMC)^2 + (yMC)^2 > r^2

44 M(i,1)=xMC;

45 M(i,2)=yMC;

46 M(i,3)=vxMC;

47 M(i,4)=vyMC;

48 break

49 else

90

Appendix 91

50 end

51 end

52 end

53

54 for k=1:sets+1

55 T = ((T_slow/T_fast)^((k-1)/sets))*T_fast;

56 w = deltaL/T;

57 work(k,1)=w;

58 work(k,2)=T;

59 nmax = int32(T/dt+1);

60 mu=zeros(k+1,2);

61 x=zeros(nmax,1);

62 y=zeros(nmax,1);

63 vx=zeros(nmax,1);

64 vy=zeros(nmax,1);

65


67

68 x(1)= M(i,1);

69 y(1)= M(i,2);

70 vx(1)=M(i,3);

71 vy(1)=M(i,4);

72 B=L/2;

73

74 for n = 1:nmax-1;

75

76

77 B=L/2 + w*dt;

78 else if

79 x(n) = 2*A- x(n);

80 vx(n) = -vx(n);

91

Appendix 92

81 x(n+1) = x(n) + vx(n)*dt;

82 vx(n+1) = vx(n);

83 if y(n) < D

84 y(n) = 2*D - y(n);

85 vy(n) = -vy(n);

86 y(n+1) = y(n) + vy(n)*dt;

87 vy(n+1) = vy(n);

88 elseif y(n) > B

89 y(n) = 2*B - y(n);

90 vy(n) = -vy(n) +2*w;

91 vy(j,n) = -vy(j,n);

92 y(n+1) = y(n) + vy(n)*dt;

93 s=(vx(n).^2 + vy(n).^2)*dt;

94 elseif y(n) <= B && y(n) >= D

95 y(n+1) = y(n) + vy(n)*dt;

96 vy(n+1) = vy(n);

97 else

98 end

99

100 elseif x(n) > C

101 x(n) = 2*C - x(n);

102 vx(n) = -vx(n);

103 x(n+1) = x(n) + vx(n)*dt;

104 vx(n+1) = vx(n);

105

106 if y(n) < D

107

108 y(n) = 2*D - y(n);

109 vy(n) = -vy(n);

110 y(n+1) = y(n) + vy(n)*dt;

111 vy(n+1) = vy(n);

92

Appendix 93

112

113 elseif y(n) > B

114

115 y(n) = 2*B - y(n);

116 vy(n) = -vy(n) +2*w

117 y(n+1) = y(n) + vy(n)*dt;

118 vy(n+1) = vy(n);

119 s=(vx(n).^2 + vy(n).^2)*dt;

120

121 elseif y(n) <= B && y(n) >= D

122 y(n+1) = y(n) + vy(n)*dt;

123 vy(n+1) = vy(n);

124

125 else

126

127 end

128

129 elseif x(n) <= C && x(n) >= A

130 x(n+1) = x(n) + vx(n)*dt;

131 vx(n+1) = vx(n);

132

133 if y(n) < D

134

135 y(n) = 2*D - y(n);

136 vy(n) = -vy(n);

137 y(n+1) = y(n) + vy(n)*dt;

138 vy(n+1) = vy(n);

139

140 elseif y(n) > B

141 y(n) = 2*B - y(n);

142 vy(n) = -vy(n) +2*w;

93

Appendix 94

143 y(n+1) = y(n) + vy(n)*dt;

144 vy(n+1) = vy(n);

145 s=(vx(n).^2 + vy(n).^2)*dt;

146

147 elseif y(n) <= B && y(n)>= D;

148 y(n+1) = y(n) + vy(n)*dt;

149 vy(n+1) = vy(n);

150

151 else

152 end

153 else

154 end

155 if x(n+1)^2 + y(n+1)^2 < r^2

156 m = (y(n+1) - y(n))/(x(n+1) - x(n));

157 X = (m*x(n)+y(n))/(m+1/m);

158 Y = -1/m*X;

159 d = sqrt(X^2+Y^2);

160 c = -m*x(n)+y(n);

161 rootx = roots([(m^2+1) 2*m*c (c^2-r^2)]);

162 d1 = sqrt((rootx(1)-x(n))^2 + (rooty(1)-y(n))^2);


164 if min(d1,d2)>=s

165 else

166 if d1 < d2

167 impact = [rootx(1) rooty(1)];

168 else


170 end

171 N = 1/norm(impact)*impact;

172 vx(n+1) = -2*(vx(n)*N(1)+vy(n)*N(2))*N(1)+vx(n);

173 vy(n+1) = -2*(vx(n)*N(1)+vy(n)*N(2))*N(2)+vy(n);

94

Appendix 95

174

175 dref = s - sqrt((x(n)-impact(1))^2 + (y(n)-impact(2))^2);

176 x(n+1) = impact(1) + vx(n+1)/vmod*dref;

177 y(n+1) = impact(2) + vy(n+1)/vmod*dref;

178 end

179 end

180 I_Results(i,1)= x(nmax);

181 I_Results(i,2)= y(nmax);

182 I_Results(i,3)= vx(nmax);

183 I_Results(i,4)= vy(nmax);

184

185

186 plot_step = 2;

187 if( rem(n/plot_step,1) == 0 && n>=2*plot_step)

188 plot_x = [x(n);x(n-plot_step)];

189 colorVec=hsv(nparticles);

190 plot(plot_x,plot_y,’color’,colorVec(i,:))

191 hold on

192 pause(0.01)

193

194 end

195 Kinetic= 0.5*(I_Results(:,3).^2 + I_Results(:,4).^2);

196 Q=(Kinetic-mean(Kinetic)).^2;

197 delW=sqrt(sum(Q)/nparticles);

198 work(k,3)=delW;

199

200 end

201 end

B.2 Modified Sinai billiard: MCE

1 clc;

95

Appendix 96

2 close all;

3 L = 40;

4 dt = 0.01;

5 nparticles=100000;

6 deltaL = 10;

7 T_fast = 10;

8 sets = 16;

9 T_slow = 100000;

10

11 s = vmod*dt;

12 M = zeros(nparticles,4);

13 I_Results=zeros(nparticles,4);

14 work=zeros(sets+1 ,3);

15

16

17 A=-L/2;

18 B=L/2;

19 C=L/2;

20 D=-L/2;

21 E=0;

22 r = 15;

23 figure(1);

24 theta = -pi/2:0.01:pi/2;

25 x_c=r*cos(theta);

26 y_c=r*sin(theta);

27 plot(x_c,y_c);

28 patch(x_c,y_c,’b’)

29 axis([A C D B])

30 xlabel(’x axis’),ylabel(’y axis’)

31 hold on

32

96

Appendix 97


34 while 1;

35 xMC = -L/2 + (L)*rand(1,1);

36 yMC= -L/2 + (L)*rand(1,1);

37 random_num = 2*pi*rand(1,1);

38 vxMC= vmod*cos(random_num);

39 vyMC =vmod*sin(random_num);

40 if xMC <0 || xMC >= sqrt(r^2-(yMC)^2);

41 M(i,1)=xMC;

42 M(i,2)=yMC;

43 M(i,3)=vxMC;

44 M(i,4)=vyMC;

45 break

46 else

47 end

48 end

49 end

50

51

52 for k=1:sets+1

53 T = ((T_slow/T_fast)^((k-1)/sets))*T_fast;

54 w = deltaL/T;

55 work(k,1)=w;

56 work(k,2)=T;

57 nmax = int32(T/dt+1);

58 work(k,4)=nmax;

59 x=zeros(nmax,1);

60 y=zeros(nmax,1);

61 vx=zeros(nmax,1);

62 vy=zeros(nmax,1);

63

97

Appendix 98


65 x(1)= M(i,1);

66 y(1)= M(i,2);

67 vx(1)=M(i,3);

68 vy(1)=M(i,4);

69 B=L/2;

70 for n = 1:nmax-1;

71

72

73 B=L/2 + w*dt;

74 if x(n) < A

75 x(n) = 2*A- x(n);

76 vx(n) = -vx(n);

77 x(n+1) = x(n) + vx(n)*dt;

78 vx(n+1) = vx(n);

79 if y(n) < D

80 y(n) = 2*D - y(n);

81 vy(n) = -vy(n);

82 y(n+1) = y(n) + vy(n)*dt;

83 vy(n+1) = vy(n);

84 elseif y(n) > B

85 y(n) = 2*B - y(n);

86 vy(n) = -vy(n) +2*w;

87 y(n+1) = y(n) + vy(n)*dt;

88 s=(vx(n).^2 + vy(n).^2)*dt;

89 elseif y(n) <= B && y(n) >= D

90 y(n+1) = y(n) + vy(n)*dt;

91 vy(n+1) = vy(n);

92 else

93 end

94

98

Appendix 99

95 elseif x(n) > C

96 x(n) = 2*C - x(n);

97 vx(n) = -vx(n);

98 x(n+1) = x(n) + vx(n)*dt;

99 vx(n+1) = vx(n);

100

101 if y(n) < D

102 y(n) = 2*D - y(n);

103 vy(n) = -vy(n);

104 y(n+1) = y(n) + vy(n)*dt;

105 vy(n+1) = vy(n);

106

107 elseif y(n) > B

108

109 y(n) = 2*B - y(n);

110 vy(n) = -vy(n) +2*w;

111 y(n+1) = y(n) + vy(n)*dt;

112 vy(n+1) = vy(n);

113 s=(vx(n).^2 + vy(n).^2)*dt;

114

115 elseif y(n) <= B && y(n) >= D

116 y(n+1) = y(n) + vy(n)*dt;

117 vy(n+1) = vy(n);

118

119 else

120

121 end

122

123 elseif x(n) <= C && x(n) >= A

124 x(n+1) = x(n) + vx(n)*dt;

125 vx(n+1) = vx(n);

99

Appendix 100

126

127 if y(n) < D

128

129 y(n) = 2*D - y(n);

130 vy(n) = -vy(n);

131 y(n+1) = y(n) + vy(n)*dt;

132 vy(n+1) = vy(n);

133

134 elseif y(n) > B

135 y(n) = 2*B - y(n);

136 vy(n) = -vy(n) +2*w;

137 y(n+1) = y(n) + vy(n)*dt;

138 vy(n+1) = vy(n);

139 s=(vx(n).^2 + vy(n).^2)*dt;

140

141 elseif y(n) <= B && y(n)>= D;

142 y(n+1) = y(n) + vy(n)*dt;

143 vy(n+1) = vy(n);

144

145 else

146 end

147 else

148 end

149

150 if x(n+1) < sqrt(r^2-(y(n+1)^2)) && x(n+1) > 0;

151 x(n) = 2*E - x(n);

152 vx(n) = -vx(n);

153 x(n+1) = x(n) + vx(n)*dt;

154 vx(n+1) = vx(n);

155

156 m = (y(n+1) - y(n))/(x(n+1) - x(n));

100

Appendix 101

157 X = (m*x(n)+y(n))/(m+1/m);

158 Y = -1/m*X;

159 d = sqrt(X^2+Y^2);

160 c = -m*x(n)+y(n);

161 rootx = roots([(m^2+1) 2*m*c (c^2-r^2)]);

162 rooty = m*rootx + c;

163



166 if min(d1,d2)>=s

167 else

168 if d1 < d2


170 else


172 end

173 N = 1/norm(impact)*impact;

174 vx(n+1) = -2*(vx(n)*N(1)+vy(n)*N(2))*N(1)+vx(n);

175 vy(n+1) = -2*(vx(n)*N(1)+vy(n)*N(2))*N(2)+vy(n);

176

177 dref = s - sqrt((x(n)-impact(1))^2 + (y(n)-impact(2))^2);

178 x(n+1) = impact(1) + vx(n+1)/vmod*dref;

179 y(n+1) = impact(2) + vy(n+1)/vmod*dref;

180

181 end

182 end

183 end

184 I_Results(i,1)= x(nmax);

185 I_Results(i,2)= y(nmax);

186 I_Results(i,3)= vx(nmax);

187 I_Results(i,4)= vy(nmax);

101

Appendix 102

188 Kinetic= 0.5*(I_Results(:,3).^2 + I_Results(:,4).^2);

189 Q=(Kinetic-mean(Kinetic)).^2;

190 delW=sqrt(sum(Q)/nparticles);

191 work(k,3)=delW;

192

193 plot_step = 2;

194 if( rem(n/plot_step,1) == 0 && n>=2*plot_step)

195 plot_x = [x(n);x(n-plot_step)];

196 plot_y = [y(n);y(n-plot_step)];

197 colorVec=hsv(nparticles);

198 plot(plot_x,plot_y,’color’,colorVec(i,:))

199 hold on

200 pause(0.01)

201 end

202 end

203 end

B.3 Poincare surface of section for Henon Heiles

oscillators

1 function g=henon(energy,tmax,n)

2 --------------------------------------------

3 close all,

4 E=energy;

5 timespan=[0 tmax];

6 e1=E*6;

7

8 if e1 > 1

9 disp(’energy exceeds threshold, motion unbounded’);

10 return

11 end

102

Appendix 103

12

13 s_1 = linspace(eps, 1-eps, 15);

14 p_1 = sqrt(2*E)*sin(s_1*pi/2);

15 p_2 = sqrt(2*E)*cos(s_1*pi/2);

16

17 zz=[];

18 %%----------------------------------------------------------------

19 for iter=1:n

20

21 iv = [0, p_1(iter),0 , p_2(iter)]’;

22

23 options=odeset(’AbsTol’,1e-10,’RelTol’,1e-5,’Events’,@events );

24 [T,Y, TE,YE,IE]=ode45(@ff,timespan,iv,options);

25

26 zz=[zz; YE(:,3:4)];

27 disp(iter),

28

29 end

30 %%----------------------------------------------------------------

31

32 figure(1);

33 plot(zz(:,1), zz(:,2),’.’), hold on,

34 plot(zz(:,1), - zz(:,2),’.’), hold off,

35 axis equal,

36 title(’Poincare surface of section at E=1/10’);

37 xlabel(’q_2’), ylabel(’p_2’),

38 %% ---------------------------------------------------------------

39 function ydot=ff(t,y);

40 ydot = [y(2);

41 - y(1)*(1+ 2*y(3));

42 y(4);

103

Appendix 104

43 - y(1)^2 - y(3) + y(3)^2];

44 % %%--------------------------------------------------------------

45 % function g=energy(y);

46 % g=0.5*sum(y.^2) + y(3)*(y(1)^2 - (y(3)^2)/3);

47 % %%--------------------------------------------------------------

48 function [value,isterminal,direction] = events(t,y)

49 global rho

50 value = y(1);

51 isterminal = 0;

52 direction = 1;

53 %%----------------------------------------------------------------

104

Appendix CC++ CODES

C.1 Sinai billiard: Canonical

1 #include<stdio.h>

2 #include<math.h>

3 #include<time.h>

4 #include <iostream>

5 #include <fstream>

6 #include <string>

7 #include <sstream>

8 #include"normal.h"

9 #include"vector_operation.h"

10 #include"check_collision.h"

11 extern "C"//call C function, for pseudo random number

12

13 void srand64(int, FILE *);

14 double drand64(void);

15

16 //using namespace std;

17 std::string IntToStr(int n)

18

105

Appendix 106

19 std::stringstream result;

20 result << n;

21 return result.str();

22

23 int main()

24

25 //full sphere!!! no semicircle!!!

26 const int dim = 2;

27 int i,j,k,MC_count;

28 const real radius = 15.0;

29 const real half_L = 20.0;

30 const real T_min = 10;

31 const real T_max = 100000;

32 const int total_T = 16;

33 const int MC_num = 100000;

34 //const real K0 = 500;

35 //const real v_mod = sqrt(2*K0);

36 const real beta = 0.1;

37 const real epsilon_distance = 0.000001*half_L;

38 const double area_ratio = 1.25;

39 const double area_i = pow(2*(half_L),2.0) - 3.1415926*pow(radius,2.0);

40 const double delta_L = (area_ratio-1)*area_i/(2*half_L);

41 //temp variables for MC

42 double *position, *velocity;

43 position = new double[dim];

44 velocity = new double[dim];

45 phase_coord current_phase_coord;

46 FILE *work_stats=fopen("work_stats.dat","w");//data location

47 //FILE *test_position=fopen("test_position.dat","w");//data location

48 srand64(time(NULL),NULL);

49 //initialize sphere

106

Appendix 107

50 sphere_static sphere1;

51 real* center = new double[dim];

52 center[0] = 0.0;

53 center[1] = 0.0;

54 sphere1.initialize(center, radius);//radius is 1

55 //initialize static planes

56 const int num_plane = 2*dim - 1;//leave one side for dynamic plane

57 plane_static plane[num_plane];

58 double normal_and_passing[dim];

59 int div;

60 int resid;

61 for(i=0; i<num_plane; i++)

62

63 div = i;

64 resid = div % 2;

65 div = div/2;

66 for(j=0; j<dim; j++)

67

68 normal_and_passing[j] = 0.0;

69

70 normal_and_passing[div] = (double)(2*resid - 1)*half_L;

71 plane[i].initialize(normal_and_passing,normal_and_passing);

72

73 //initialize dynamic plane

74 plane_dynamic plane_d1;

75 double plane_norm[dim];

76 plane_norm[0] = 0;

77 plane_norm[1] = half_L;

78 std::ofstream outFile;

79 std::string filename;

80 int count_T;

107

Appendix 108

81 double T;

82 for(count_T = 0; count_T<=total_T; count_T++)

83

84 T = T_min*pow(T_max/T_min,(((double)count_T)/((double)total_T)));

85 plane_d1.initialize(plane_norm, plane_norm, T, delta_L);

86 plane_d1.

87 double W_average = 0.0;

88 double W_std_dev = 0.0;

89 filename="work_dist_" + IntToStr(count_T) +".dat";

90 outFile.open(filename.c_str());

91 //outFile <<filename<<" : Writing this to a file.\n";

92 outFile << "T = "<<T<<std::endl;

93 for(MC_count=0; MC_count<MC_num; MC_count++)

94

95 double Ki,Kf;

96 //printf("%dth MC\n",MC_count);

97 position[0] = 0.0;

98 position[1] = 0.0;

99 while(vec_dot(position,position)

100 <pow(radius+epsilon_distance,2.0))

101

102 position[0] = (2.0*drand64()-1.0)*(half_L-epsilon_distance);


104

105 //fprintf(test_position,"%7.5f, %7.5f\n",position[0],position[1]);

106 two_normal_rv(1/sqrt(beta),velocity);//set mass to be 1

107 Ki = vec_dot(velocity,velocity)/2.0;//initial energy

108 current_phase_coord.initialize(position,velocity,0.0);

109 plane_d1.input_start_coord(current_phase_coord);

110 //dynamic part

111 const real epsilon = 0.0001;

108

Appendix 109

112 int which_wall = -1;

113 int count_coll = 0;

114 while(current_phase_coord.t < (T - epsilon))

115

116 double min_t_coll = T;

117 count_coll++;

118 int which_wall_last = which_wall;

119 if(which_wall_last != 20)//20 stands for sphere

120

121 //printf("checking object 20\n");

122 sphere1.input_start_coord(current_phase_coord);

123 sphere1.check_collision();

124 if((sphere1.collision==1)&&(sphere1.end_coord.t<min_t_coll))

125

126 which_wall = 20;

127 min_t_coll = sphere1.end_coord.t;

128

129

130 else

131

132 //printf("start from sphere\n");

133


135

136 if(i==which_wall_last)//do not check collision with the wall just hitted

137

138 plane[i].collision = -1;

139 //printf("start from static plane\n");

140 continue;

141

142 //printf("checking object %d\n",i);

109

Appendix 110

143 plane[i].input_start_coord(current_phase_coord);

144 plane[i].check_collision();

145 if((plane[i].collision==1)&&(plane[i].end_coord.t<min_t_coll))

146

147 which_wall = i;

148 min_t_coll = plane[i].end_coord.t;

149

150

151 if(which_wall_last != 10)//10 stands for the moving wall

152



155 plane_d1.check_collision();

156 if((plane_d1.collision==1)&&(plane_d1.end_coord.t<min_t_coll))

157


159 min_t_coll = plane_d1.end_coord.t;

160

161

162 else

163

164 //printf("start from dynamic plane\n");

165

166 if(min_t_coll >= T)

167

168 double temp_x[dim];

169 vec_copy(temp_x,current_phase_coord.x);

170 vec_copy(current_phase_coord.x,current_phase_coord.v);

171 vec_sca(current_phase_coord.x,(T - current_phase_coord.t));

172 vec_add(current_phase_coord.x,temp_x);

173 current_phase_coord.t = T;

110

Appendix 111

174

175 else

176

177 if(which_wall==20)

178

179 copy_phase_coord(current_phase_coord,sphere1.end_coord);

180

181 else if((which_wall>=0)&&(which_wall<num_plane))

182

183 if(fabs(plane[which_wall].end_coord.x[0])>(half_L+epsilon))

184

185 printf("x-direction out of box");

186 return 0;

187

188 copy_phase_coord(current_phase_coord,plane[which_wall].end_coord);

189

190 else if((which_wall==10))

191

192 copy_phase_coord(current_phase_coord,plane_d1.end_coord);

193

194 else

195

196 printf("which_wall does not fall in proper value\n");

197 return 0;

198

199

200

201 Kf = 0.5*vec_dot(current_phase_coord.v,current_phase_coord.v);

202 //printf("K0 = %f, Kf = %f\n",K0,Kf);

203 //fprintf(work_stats,"%f, %f, %f\n",area_i*K0,area_f*Kf,area_f*Kf/(area_i*K0));

204 W_average += Ki - Kf;

111

Appendix 112

205 W_std_dev += (Ki - Kf)*(Ki - Kf);

206 outFile <<Ki<<", "<<Kf<<", "<<Ki-Kf<<std::endl;

207

208 W_average = W_average/MC_num;

209 W_std_dev = W_std_dev/MC_num;

210 W_std_dev = sqrt(W_std_dev - W_average*W_average);

211 printf("for T = %f\n",T);

212 printf("%.7f, %.7f\n",W_average, W_std_dev);

213 fprintf(work_stats, "%.7f, %.7f, %.7f\n",T, W_average, W_std_dev);

214 //close the file!!!!!

215 outFile.close();

216

217 //fclose(test_position);

218 fclose(work_stats);

219 return 0;

220

C.2 Modified Sinai billiard: Canonical

1 #include<stdio.h>

2 #include<math.h>

3 #include<time.h>

4

5

6 #include <iostream>

7 #include <fstream>

8 #include <string>

9 #include <sstream>

10

11

12 #include"normal.h"

13 #include"vector_operation.h"

112

Appendix 113

14 #include"check_collision.h"

15

16 extern "C"//call C function, for pseudo random number

17

18 void srand64(int, FILE *);

19 double drand64(void);

20

21 //using namespace std;

22 std::string IntToStr(int n)

23

24 std::stringstream result;

25 result << n;

26 return result.str();

27

28

29

30 int main()

31

32 //printf("New!\n");

33

34

35 //full sphere!!! no semicircle!!!

36 const int dim = 2;

37 int i,j,k,MC_count;

38

39 const real radius = 15.0;

40 const real half_L = 20.0;

41

42

43 const real T_min = 10;

44 const real T_max = 100000;

113

Appendix 114

45 const int total_T = 16;

46 const int MC_num = 100000;

47 //const real K0 = 500;

48 //const real v_mod = sqrt(2*K0);

49

50 const real beta = 0.001;

51

52 const real epsilon_distance = 0.000001*half_L;

53

54

55 const double area_ratio = 1.25;

56 const double area_i = pow(2*(half_L),2.0) - 0.5*3.1415926*pow(radius,2.0);

57 //modify

58 const double delta_L = area_i*(area_ratio-1)/(2*half_L);

59 //modify

60

61

62

63 //temp variables for MC

64 double *position, *velocity;

65 position = new double[dim];

66 velocity = new double[dim];

67 phase_coord current_phase_coord;

68

69

70 FILE *work_stats=fopen("work_stats.dat","w");//data location

71

72 srand64(time(NULL),NULL);

73

74

75

114

Appendix 115

76 //declare and initialize cutted sphere

77 sphere_static_cutted sphere1;

78 real* center = new double[dim];

79 center[0] = 0.0;

80 center[1] = 0.0;

81 real* cut_plane_passing = new double[dim];

82 cut_plane_passing[0] = 0.0;

83 cut_plane_passing[1] = 0.0;

84 real* cut_plane_normal = new double[dim];

85 cut_plane_normal[0] = 1.0;

86 cut_plane_normal[1] = 0.0;

87 sphere1.initialize(center, radius, cut_plane_passing, cut_plane_normal);

88

89 //initialize static planes

90 const int num_plane = 2*dim - 1;//leave one side for dynamic plane

91 plane_static plane[num_plane];

92 double normal_and_passing[dim];

93

94 int div;

95 int resid;


97

98 div = i;

99 resid = div % 2;

100 div = div/2;

101 for(j=0; j<dim; j++)

102

103 normal_and_passing[j] = 0.0;

104

105 normal_and_passing[div] = (double)(2*resid - 1)*half_L;

106

115

Appendix 116

107 plane[i].initialize(normal_and_passing,normal_and_passing);

108

109

110

111 //initialize dynamic plane

112 //modify

113 plane_dynamic plane_d1;

114 double plane_norm[dim];

115 plane_norm[0] = 0;

116 plane_norm[1] = half_L;

117 //modify

118

119

120

121 std::ofstream outFile;

122 std::string filename;

123

124

125 int count_T;

126 double T;

127 for(count_T = 0; count_T<=total_T; count_T++)

128

129

130 T = T_min*pow(T_max/T_min,(((double)count_T)/((double)total_T)));

131

132

133 //modify

134 //replace check_collision.cpp and check_collision.h

135 plane_d1.initialize(plane_norm, plane_norm, T, delta_L);

136 //modify

137

116

Appendix 117

138 double W_average = 0.0;

139 double W_std_dev = 0.0;

140

141

142 filename="work_dist_" + IntToStr(count_T) +".dat";

143 outFile.open(filename.c_str());

144 //outFile <<filename<<" : Writing this to a file.\n";

145 outFile << "T = "<<T<<std::endl;

146

147

148

149 for(MC_count=0; MC_count<MC_num; MC_count++)

150

151

152 double Ki,Kf;

153 //printf("%dth MC\n",MC_count);

154

155 position[0] = 0.0;

156 position[1] = 0.0;

157

158

159 while((vec_dot(position,position)<pow(radius+epsilon_distance,2.0))

160 &&(position[1]>(-epsilon_distance)))

161

162 //modify



165 //modify

166

167

168 two_normal_rv(1/sqrt(beta),velocity);//set mass to be 1

117

Appendix 118

169 Ki = vec_dot(velocity,velocity)/2.0;//initial energy

170

171

172

173 current_phase_coord.initialize(position,velocity,0.0);

174

175 //dynamic part

176 const real epsilon = 0.0001;

177 int which_wall = -1;

178

179 int count_coll = 0;

180 while(current_phase_coord.t < (T - epsilon))

181

182

183 double min_t_coll = T;

184 count_coll++;

185

186 int which_wall_last = which_wall;

187

188 if(which_wall_last != 20)//20 stands for sphere

189


191

192 sphere1.input_start_coord(current_phase_coord);

193 sphere1.check_collision();

194

195 if((sphere1.collision==1)&&(sphere1.end_coord.t<min_t_coll))

196


198 min_t_coll = sphere1.end_coord.t;

199

118

Appendix 119

200

201 else

202

203 //printf("start from sphere\n");

204

205

206

207


209

210 if(i==which_wall_last)//do not check collision with the wall just hitted

211

212 plane[i].collision = -1;

213 //printf("start from static plane\n");

214 continue;

215

216

217 //printf("checking object %d\n",i);

218 plane[i].input_start_coord(current_phase_coord);

219 plane[i].check_collision();

220

221 if((plane[i].collision==1)&&(plane[i].end_coord.t<min_t_coll))

222

223 which_wall = i;

224 min_t_coll = plane[i].end_coord.t;

225

226

227

228

229

230 if(which_wall_last != 10)//10 stands for the moving wall

119

Appendix 120

231



234 plane_d1.check_collision();

235

236 if((plane_d1.collision==1)&&(plane_d1.end_coord.t<min_t_coll))

237


239 min_t_coll = plane_d1.end_coord.t;

240

241

242 else

243

244 //printf("start from dynamic plane\n");

245

246

247

248

249

250 if(min_t_coll >= T)

251

252 double temp_x[dim];

253

254 vec_copy(temp_x,current_phase_coord.x);

255

256 vec_copy(current_phase_coord.x,current_phase_coord.v);

257 vec_sca(current_phase_coord.x,(T - current_phase_coord.t));

258 vec_add(current_phase_coord.x,temp_x);

259

260 current_phase_coord.t = T;

261

120

Appendix 121

262 else

263

264 if(which_wall==20)

265

266 copy_phase_coord(current_phase_coord,sphere1.end_coord);

267

268 else if((which_wall>=0)&&(which_wall<num_plane))

269

270 if(fabs(plane[which_wall].end_coord.x[0])>(half_L+epsilon))

271

272 printf("x-direction out of box");

273

274 return 0;

275

276 copy_phase_coord(current_phase_coord,plane[which_wall].end_coord);

277

278 else if((which_wall==10))

279

280 copy_phase_coord(current_phase_coord,plane_d1.end_coord);

281

282 else

283

284 printf("which_wall does not fall in proper value\n");

285 return 0;

286

287

288

289

290

291

292 Kf = 0.5*vec_dot(current_phase_coord.v,current_phase_coord.v);

121

Bibilography; 122

293 //printf("K0 = %f, Kf = %f\n",K0,Kf);

294 //fprintf(work_stats,"%f, %f, %f\n",area_i*K0,area_f*Kf,area_f*Kf/(area_i*K0));

295

296

297 W_average += Ki - Kf;

298 W_std_dev += (Ki - Kf)*(Ki - Kf);

299

300 outFile <<Ki<<", "<<Kf<<", "<<Ki-Kf<<std::endl;

301

302

303

304 W_average = W_average/MC_num;

305 W_std_dev = W_std_dev/MC_num;

306 W_std_dev = sqrt(W_std_dev - W_average*W_average);

307

308 printf("for T = %f\n",T);

309 printf("%.7f, %.7f\n",W_average, W_std_dev);

310

311 fprintf(work_stats, "%.7f, %.7f, %.7f\n",T, W_average, W_std_dev);

312 //close the file!!!!!

313 outFile.close();

314

315

316 fclose(work_stats);

317

318

319 return 0;

320

122

BIBLIOGRAPHY

[1] Gaoyang Xiao and Jiangbin Gong. Principle of minimal work fluctuations.

Physical Review E, 92(2):022130, 2015.

[2] Gaoyang Xiao and Jiangbin Gong. Suppression of work fluctuations by opti-

mal control: An approach based on jarzynski’s equality. Physical Review E,

90(5):052132, 2014.

[3] Obinna Abah, Johannes Rossnagel, Georg Jacob, Sebastian Deffner, Ferdi-

nand Schmidt-Kaler, Kilian Singer, and Eric Lutz. Single-ion heat engine at

maximum power. Physical review letters, 109(20):203006, 2012.

[4] Thorsten Hugel, Nolan B Holland, Anna Cattani, Luis Moroder, Markus

Seitz, and Hermann E Gaub. Single-molecule optomechanical cycle. Science,

296(5570):1103–1106, 2002.

[5] Herbert B Callen. Thermodynamics & an Intro. to Thermostatistics. John

Wiley & Sons, 2006.

[6] Gaoyang Xiao and Jiangbin Gong. Construction and optimization of a quan-

tum analog of the carnot cycle. Physical Review E, 92(1):012118, 2015.

[7] VL Berdichevsky and MV Alberti. Statistical mechanics of henon-heiles os-

cillators. Physical Review A, 44(2):858, 1991.

[8] Jerry B Marion and Stephen T Thornton. Classical mechanics of particles

and systems. Saunders Colledge Pub., quarta edicao, 1995.

123

Bibilography; 124

[9] Giovanni Gallavotti, Federico Bonetto, and Guido Gentile. Aspects of ergodic,

qualitative and statistical theory of motion. Springer Science & Business Me-

dia, 2013.

[10] Cesar R de Oliveira and Thiago Werlang. Ergodic hypothesis in classical

statistical mechanics. Revista Brasileira de Ensino de Fısica, 29(2):189–201,

2007.

[11] Reggie Brown, Edward Ott, and Celso Grebogi. Ergodic adiabatic invariants

of chaotic systems. Physical review letters, 59(11):1173, 1987.

[12] Edward Ott. Goodness of ergodic adiabatic invariants. Physical Review Let-

ters, 42(24):1628, 1979.

[13] Herbert Goldstein, Charles P Poole, and John L Safko. Classical Mechanics:

Pearson New International Edition. Pearson Higher Ed, 2014.

[14] The new chaotic pendulum: Study of poincare surface of section for

the chaotic pendulum. http://physique.unice.fr/sem6/2011-2012/

PagesWeb/PT/Pendule/En/more3.html. Accessed: 2016-03-19.

[15] Professor Gong Jiangbin : PC 5210.

[16] Stefan Hilbert, Peter Hanggi, and Jorn Dunkel. Thermodynamic laws in

isolated systems. Physical Review E, 90(6):062116, 2014.

[17] Michel Henon and Carl Heiles. The applicability of the third integral of

motion: some numerical experiments. The Astronomical Journal, 69:73, 1964.

[18] Poincare plots for the Henon-Heiles system : Source code for ref-

erence. http://theorie.physik.uni-konstanz.de/theodorakopoulos/

poincare/henon-heiles.html. Accessed: 2016-03-24.

[19] Gavin E Crooks. Entropy production fluctuation theorem and the nonequilib-

rium work relation for free energy differences. Physical Review E, 60(3):2721,

1999.

124

http://physique.unice.fr/sem6/2011-2012/PagesWeb/PT/Pendule/En/more3.html

http://physique.unice.fr/sem6/2011-2012/PagesWeb/PT/Pendule/En/more3.html

http://theorie.physik.uni-konstanz.de/theodorakopoulos/poincare/henon-heiles.html


Bibilography; 125

[20] Carlos Bustamante, Jan Liphardt, and Felix Ritort. The nonequilibrium ther-

modynamics of small systems. arXiv preprint cond-mat/0511629, 2005.

[21] Bart Cleuren, Christian Van den Broeck, and R Kawai. Fluctuation and dis-

sipation of work in a joule experiment. Physical review letters, 96(5):050601,

2006.

[22] Christopher Jarzynski. Nonequilibrium equality for free energy differences.

Physical Review Letters, 78(14):2690, 1997.

[23] AS Sachrajda, R Ketzmerick, C Gould, Y Feng, PJ Kelly, A Delage, and

Z Wasilewski. Fractal conductance fluctuations in a soft-wall stadium and a

sinai billiard. Physical review letters, 80(9), 1997.

[24] Domokos Szasz and Leonid A Bunimovich. Hard ball systems and the Lorentz

gas. Soringer, 2000.

[25] Scholarpedia :sinai billiards. http://theorie.physik.uni-konstanz.de/

theodorakopoulos/poincare/henon-heiles.html. Accessed: 2016-02-24.

125



characteristics of work fluctuations in chaotic system projects repository 2015... ·...

Documents