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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
4.2 P.O.S of Henon Heiles oscillator at E = 18
. . . . . . . . . . . . . . . 36
4.3 P.O.S of Henon Heiles oscillator at E = 16
. . . . . . . . . . . . . . . 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
Hamiltonian Mechanics 15
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
Hamiltonian Mechanics 16
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
Ergodicity and Chaos 19
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
Ergodicity and Chaos 20
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
Ergodicity and Chaos 21
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
Ergodicity and Chaos 22
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
Ergodicity and Chaos 23
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
Ergodicity and Chaos 24
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
Ergodicity and Chaos 25
(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
Ergodicity and Chaos 26
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
Ergodicity and Chaos 27
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
Ergodicity and Chaos 28
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
Ergodicity and Chaos 29
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
Ergodicity and Chaos 30
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
Statistical Mechanics in Small Systems 33
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
Statistical Mechanics in Small Systems 34
∂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
Statistical Mechanics in Small Systems 35
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
Statistical Mechanics in Small Systems 36
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
Figure 4.2: The Poincare map for HenonHeiles oscillator at E = 1
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
Figure 4.3: The Poincare map for HenonHeiles oscillator at E = 1
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
Statistical Mechanics in Small Systems 37
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
Fluctuation theorems 40
∆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
Fluctuation theorems 41
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
Fluctuation theorems 42
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
Fluctuation theorems 43
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
Methodology and objectives 50
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
Methodology and objectives 51
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
Methodology and objectives 52
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
Methodology and objectives 53
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
Methodology and objectives 54
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
Methodology and objectives 55
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
Numerical simulations and results 58
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
Numerical simulations and results 59
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
)
=
∫∫dpxdpy ρpxρpy e
−β p2
m
(Ai
Af−1
)
=
∫∫dpxdpy ρpxρpy e
−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
Numerical simulations and results 60
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
Numerical simulations and results 61
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
Numerical simulations and results 62
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
Numerical simulations and results 63
〈 e−2βW 〉square =
∫∫dpxdpy ρpxρpy e−2βW
=
∫∫dpxdpy ρpxρpy e
−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
Numerical simulations and results 64
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
Numerical simulations and results 65
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
Numerical simulations and results 66
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
Numerical simulations and results 67
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
Numerical simulations and results 68
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
Numerical simulations and results 69
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
Numerical simulations and results 70
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
Numerical simulations and results 71
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
Numerical simulations and results 72
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)
=
∫ ∫dpx dpyρpxρpye
−2β p2
m(
AiAf−1)
=
∫ ∫dpx dpyρpxρpye
−2β(p2x+p2
ym
)(AiAf−1)
=β
2πm
∫ ∫dpx dpy × e−β
p2x
2m × eβ
p2y
2m × e−2βp2y
mγ
(8.15)
72
Numerical simulations and results 73
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
Numerical simulations and results 74
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 =
∫∫dpxdpy ρpxρpy e−4βW
=
∫∫dpxdpy ρpxρpy e
−4βEi
(L2
i
L2f−1
)
=
∫∫dpxdpy ρpxρpy e
−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
Numerical simulations and results 75
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
Numerical simulations and results 76
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
Numerical simulations and results 77
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
Numerical simulations and results 78
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
Numerical simulations and results 79
Comparison at β = 0.01
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
Figure 8.12: Comparison of work fluctuation across different models of the billiardsystem at β=0.01 for expansion protocol.
Comparison at β = 0.001
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
Figure 8.13: Comparison of work fluctuation across different models of the billiardsystem at β=0.001 for expansion protocol.
79
Numerical simulations and results 80
8.4.2 Comparison of work fluctuations across different mod-
els: Contraction protocol
Comparison at β = 0.1
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
Numerical simulations and results 81
Comparison at β = 0.01
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
Figure 8.15: Comparison of work fluctuation across different models of the billiardsystems at β=0.01 for contraction protocol.
Comparison at β = 0.001
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
Figure 8.16: Comparison of work fluctuation across different models of the billiardsystems at β=0.001 for contraction protocol.
81
Numerical simulations and results 82
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
Numerical simulations and results 84
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
Numerical simulations and results 86
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
66 for i=1:nparticles;
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);
163 d2 = sqrt((rootx(2)-x(n))^2 + (rooty(2)-y(n))^2);
164 if min(d1,d2)>=s
165 else
166 if d1 < d2
167 impact = [rootx(1) rooty(1)];
168 else
169 impact = [rootx(2) rooty(2)];
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
33 for i=1:nparticles;
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
64 for i=1:nparticles;
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
164 d1 = sqrt((rootx(1)-x(n))^2 + (rooty(1)-y(n))^2);
165 d2 = sqrt((rootx(2)-x(n))^2 + (rooty(2)-y(n))^2);
166 if min(d1,d2)>=s
167 else
168 if d1 < d2
169 impact = [rootx(1) rooty(1)];
170 else
171 impact = [rootx(2) rooty(2)];
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);
103 position[1] = (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
134 for(i=0; i<num_plane; i++)
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
153 //printf("checking object 10\n");
154 plane_d1.input_start_coord(current_phase_coord);
155 plane_d1.check_collision();
156 if((plane_d1.collision==1)&&(plane_d1.end_coord.t<min_t_coll))
157
158 which_wall = 10;
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;
96 for(i=0; i<num_plane; i++)
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
163 position[0] = (2.0*drand64()-1.0)*(half_L-epsilon_distance);
164 position[1] = (2.0*drand64()-1.0)*(half_L-epsilon_distance);
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
190 //printf("checking object 20\n");
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
197 which_wall = 20;
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
208 for(i=0; i<num_plane; i++)
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
232 //printf("checking object 10\n");
233 plane_d1.input_start_coord(current_phase_coord);
234 plane_d1.check_collision();
235
236 if((plane_d1.collision==1)&&(plane_d1.end_coord.t<min_t_coll))
237
238 which_wall = 10;
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
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