Characteristics of Work Fluctuations in Chaotic Systems

Transcription

1 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

2 ABSTRACT Miniaturisation has become a trademark of our modern society. Technological devices 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 freedom, 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 trajectories from the microcanonical and canonical ensemble and expose it to adiabatic protocols in an attempt to study the behaviour of work fluctuations. Simultaneously, 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

3 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

4 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 document. 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 possible by your understanding and patience. The purpose of university is to unlearn what we have learn. These 4 years, knowing 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

5 CONTENTS Abstract 1 Acknowledgements 2 1 Introduction Motivation Thesis overview Hamiltonian Mechanics The concept of phase space Hamilton s equations and Liouville s theorem Invariant measure in Liouville s dynamics: Phase space volume 15 3 Ergodicity and Chaos On Ergodicity The Ergodic Hypothesis On Microcanonical ensemble (MCE) Ergodic Adiabatic Invariant Physical interpretation of the ergodic adiabatic invariant Chaos theory Visualizing Chaos : The Poincaré Surface of Section (P.O.S) 28 4

6 Contents CONTENTS 4 Statistical mechanics in small system Meaning of temperature in statistical mechanics Relationship between the surface and volume entropy The Hénon Heiles Oscillator: An application Fluctuation theorems Crook s fluctuation theorem Crook s relation for MCE Jarzynski Equality Jarzynski Equality in classical system Work fluctuations The Sinai billiard Adiabatic invariant of Sinai billiard Methodology and objectives Objectives Methodology Generating the ensembles Adiabatic variation of the Hamiltonian Adiabatic expansion of wall Numerical simulations and results Work fluctuations in MCE Work fluctuations in Canonical ensemble Derivation of e βw and e 2βW for Sinai and modified Sinai billiards Derivation of e 2βW square for Square Expansion protocol Determination of δl for expansion protocol: φ = Comparison of work fluctuations for each model at different β: Expansion protocol Contraction protocol

7 Contents CONTENTS Determination of δl for contraction protocol : φ = Comparison of work fluctuations for each model at different β: Contraction protocol Analysis of results: The work fluctuation of each model at different β Analysis of results: On the smoothness of convergence for the expansion and contraction protocol Work fluctuations in chaotic and non chaotic model: MCE Work fluctuations in chaotic and non chaotic models: Canonical ensemble Comparison of work fluctuation across different models: Expansion protocol Comparison of work fluctuations across different models: Contraction protocol Analysis: Work fluctuation for chaotic and non chaotic model by canonical sampling Conclusion 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 B.2 Modified Sinai billiard: MCE B.3 Poincaré surface of section for Hénon Heiles oscillators C C++ Codes 105 C.1 Sinai billiard: Canonical C.2 Modified Sinai billiard: Canonical

8 List of Figures CONTENTS Bibliography 122 7

9 LIST OF FIGURES 3.1 Microcanonical representation in phase space Evolution of energy surface for 1D integrable system Evolution of energy surface for MCE Sample trajectory for Poincaré surface of section P.O.S (Low energy) P.O.S (Medium energy) P.O.S (High energy) P.O.S of Hénon Heiles oscillator at E = P.O.S of Hénon Heiles oscillator at E = P.O.S of Hénon Heiles oscillator at E = The Sinai billiard Matlab simulation for Sinai billiard Matlab simulation for modified Sinai billiard Sinai billiard: Circle Modified Sinai billiard Square billiard Expansion protocol Work fluctuations in MCE Relative work fluctuations in MCE

10 List of Tables LIST OF FIGURES 8.3 Comparison of work fluctuation for square: Expansion Comparison of work fluctuation for modified Sinai billiard: Expansion Comparison of work fluctuation for Sinai billiard: Expansion Contraction protocol Comparison of work fluctuation for square: Contraction Comparison of work fluctuation for semi-circle model: Contraction Comparison of work fluctuation for Sinai model: Contraction Chaotic vs non-chaotic models in MCE Comparison of work fluctuation for expansion at β= Comparison of work fluctuation for expansion at β= Comparison of work fluctuation for expansion at β= Comparison of work fluctuation for contraction at β= Comparison of work fluctuation for contraction at β= Comparison of work fluctuation for contraction at β=

11 LIST OF TABLES 8.1 Determination of δl for expansion protocol Determination of δl for contraction protocol Table of test for δ e 2βW for ergodic systems: circle and semi-circle Table of test for δ e 2βW for the non- ergodic system: Square

12 Chapter 1 INTRODUCTION 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 cannot 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 statistical 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 pressure [5]. This very property is often not found in small system where fluctuations due to work and heat are dominant [6]. 11

13 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 fluctuations 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

14 Chapter 2 HAMILTONIAN 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 by { q,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

15 Hamiltonian Mechanics Hamilton s equations and Liouville s theorem 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. ṗ = H q q = H p dh dt = H t (2.1) (2.2) (2.3) with generalized momentum p = (p 1...p N ), coordinate q = (q 1...q N ) 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 popularly known Liouville s theorem, first formulated by the German physicist Joseph Liouville in The theorem gives an invariant measure to our Hamiltonian system 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 incompressibility 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

16 Hamiltonian Mechanics 15 Liouville s theorem states that By application of the chain rule dρ dt = 0 (2.4) dρ dt = ρ n ( (ρ t + qi ) + (ρṗ i) q i i=1 = ρ t + n i=1 = ρ t + n = ρ t i=1 + {ρ, H} ) p i ( ) ρ ρ q i + ṗ i q i p i ( H ρ H ρ p i q i q i p i ) (2.5) Where i is the number of independent equations born out of the constraints sustained by the dynamical system and the poisson bracket {ρ, H} = ( n H ρ i=1 p i q i H q i ρ p i ). Thus the evolution of the phase space density in Hamiltonian mechanics is given by the compact form ρ t = {ρ, H} (2.6) 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 MJM T = J (2.7) 15

17 Hamiltonian Mechanics 16 where M is the Jacobian matrix and J is defined as The volume element will undergo a canonical transformation from (dη) = dq 1 dq 2...dq n dp 1 dp 2...dp n (2.8) to a new volume element (dζ) = dq 1 dq 2...dQ n dp 1 dp 2...dP n (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

18 Chapter 3 ERGODICITY AND CHAOS 3.1 On Ergodicity The study of ergodicity is abstract and usually restricted to that of pure mathematics. A multidisciplinary approach has to be taken as the concept of ergodicity involves ideas from probability theory, number theory and vector fields on manifold etc. Simply put, ergodic theory is the mathematical theory of dynamical system provided 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

19 Ergodicity and Chaos The Ergodic Hypothesis The Ergodic Theorem is a central concept in the study of ergodicity. Mathematicians 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 straightforward 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) = d N q d N pρ(q, p)a(q, p) (3.1) Hence A ( q(t), p(t) ) 1 T t = lim A(q(t), p(t)) dt (3.2) T T 0 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

20 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 statistical 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 ɛ = (q, p) and ɛ, 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. p i = 1 Ξ (3.4) subjected to the constraint of k p i = 1 i=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 thermodynamics, the Entropy. 19

21 Ergodicity and Chaos 20 The entropy of the system is given by k S(p i ) = p i ln p i = k ln Ξ (3.5) i=1 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 + ɛ and E. E E + ε Phase space volume Figure 3.1: The volume occupied by the micocanonical ensembles will just be a thin 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(ɛω) (3.6) where ɛ 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 ɛω = Ξ will give the total number of microstates for the given constraint. 20

22 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 fundamental 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 invariant 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 H 0 [?]. 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 mechanics. 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

23 Ergodicity and Chaos 22 p p H = E(0) H = E(τ) q q t = 0 t = τ Figure 3.2: During an adiabatic evolution, the energy surface evolve from E(0) to E(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 compared 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 ergodic adiabatic invariant Ω [11]. 22

24 Ergodicity and Chaos 23 For a conservative dynamical system characterised by a time dependent Hamiltonian, 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) ) = U[E H ( p, q, λ(t) ) ] d N p d N q (3.10) where U is the unit step function. V 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

25 Ergodicity and Chaos 24 other. Meaning to say that for a value of E there will only be one unique Ω corresponding to it Physical interpretation of the ergodic adiabatic invariant 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, (q n, p n ), 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 E 0 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

26 Ergodicity and Chaos 25 Adiabatic protocol (q 1, p 1 ) Δλ (q 1,p 1 ) (q 2, p 2 ) (q 2,p 2 ) (q 3, p 3 ) (q 3,p 3 ) E 0 E τ Figure 3.3: During an adiabatic evolution, the energy surface evolve from E 0 to E τ. The phase space volume remains a constant during the protocol hence playing the role as an adiabatic invariant. The ergodic adiabatic invariant is then the phase space volume which is bounded by E 0 and E τ. 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 E τ, 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 final 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

27 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 λ λ (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 λ (3.14) λ Taking the appropriate time derivative, we have Where F λ = H λ λ E final E initial = λ(τ) λ(0) is the microcanonical average. F λ dλ (3.15) From Eq. (3.15), if the initial sampled state was microcanonical then the final energy 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

28 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 trademark 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

29 Ergodicity and Chaos Visualizing Chaos : The Poincaré 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 Poincaré surface of section representation. The Poincaré surface of section is to provide an analysis using a 2D slice through a 3D energy surface given by H(p x, p y, q x, q y ) = H 0. One always have a choice as to decide on which parameters to fix, for instance if we decide to fix q y then we will be studying motions in the (q x, p x ) 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 Poincaré surface of section for a quasi-periodic orbit, notice that the trajectory will still intersection that same section of state space after some finite time Some characteristics from for the P.O.S map are: 1. Characteristics of Poincaré map ˆ A unique point or multiple points: System is periodic ˆ A closed curved: System is quasi-periodic 28

30 Ergodicity and Chaos 29 ˆ A cloud of points: System is chaotic The poincaré 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: The Poincaré map for low energy. Figure 3.6: The Poincaré map for medium energy. Figure 3.7: The Poincaré map for high energy. As observed, as the energy of the system is increased there are fewer periodic orbits and more random points on the Poincaré 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

31 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

32 Chapter 4 STATISTICAL MECHANICS IN SMALL SYSTEM 4.1 Meaning of temperature in statistical mechanics Statistical mechanics is an asymptotic theory valid in the limit of an infinite degrees 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(ɛω) (4.1) where ɛ 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

33 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 temperatures for the MCE which we shall call it the Gibb s (T G ) and Boltzmann s (T B ) temperature respectively. where ν(e) = ω(e) E. T G = Ω(E) ω(e) T B = ω(e) ν(e) (4.3) (4.4) As mentioned, T G and T B 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 Relationship between the surface and volume entropy 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

34 Statistical Mechanics in Small Systems 33 understanding of the meaning of temeprature. unique definition of temperature [16]. In fact, in an MCE there is no 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 ɛ 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(ɛω f ) ln(ɛω i ) = ln(ɛ Ω f E ) ln(ɛ Ω i E ) ( ) e (ln(ɛω f) ln(ɛω i )) ln = e Ω f E / Ω i E (4.6) = Ω f E / Ω i E Let s revert our attention to the Gibbs temperature, T G. From Eq. (4.3), 1 = ln Ω T G 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

35 Statistical Mechanics in Small Systems 34 Ω f E / Ω i E = T i T f Ω f Ω i (4.8) Therefore with reference to Eq. (4.6) and Eq. (4.8) we have the following relation: e (ln(ɛω f) ln(ɛω i )) = T i T f Ω f Ω i (4.9) Now we have the conversion formula which relates the surface to volume entropy by a temperature factor of T i T f. To simplify matters, we can write Eq. (4.9) as e ( S f S i ) = T i T f e (S f S i ) (4.10) Hence we will have a clean relation connecting the surface and the volume entropy as described The Hénon Heiles Oscillator: An application One classical example of non-linear dynamics will be that of celestial mechanics. The Hénon Heiles model,developed by Michel Hénon and Carl Heiles while working 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 Hénon 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 (p2 x + p 2 y) (x2 + y 2 ) + λ(x 2 y y3 3 ) (4.11) The non linear term in λ give rise to chaotic motions. 34

36 Statistical Mechanics in Small Systems 35 The Hamiltonian equation has the following form H(q, p, λ) ṗ = 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 1 6 the motion can be approximately ergodic. This can be seen from the Poincaré s surface of section at different energy levels for the Hénon Heiles oscillator. The following simulation has been done for this system. The Poincaré s plane is fixed at q 1 = 0 hence we will be studying the dynamics in the q 2 and p 2 plane. 35

37 Statistical Mechanics in Small Systems Poincare surface of section at E=1/ Poincare surface of section at E=1/ p 2 0 p q q 2 Figure 4.1: The Poincaré map for HénonFigure 4.2: The Poincaré map for Hénon Heiles oscillator at E = 1 Heiles oscillator at E = Poincare surface of section at E=1/ p q 2 Figure 4.3: The Poincaré map for Hénon Heiles oscillator at E = 1 6 It can be seen that as the energy of the Hénon 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 = 1 the motion of the oscillator is chaotic as represented by 6 the clouds of points in the Poincaré s map [18]. An adiabatic variation of λ introduces an adiabatic protocol to the Hénon 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

38 Statistical Mechanics in Small Systems 37 to perform thermodynamical calculations, one such example will be to derive the entropy of the ergodic Hénon Heiles Hamiltonian: S(E, λ) = ln Ω(E, λ) + Constant (4.14) Following which, thermodynamical quantities like temperature (T) can also be defined in its usual form. S E = 1 T (4.15) 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

39 Chapter 5 FLUCTUATION 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 P F (W ) P R ( W ) = e S k (5.1) Here S is the entropy production of the driven system over some time interval, P F (W ) is the probability distribution of the forward protocol and P R (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 useful 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

40 Fluctuation theorems 39 degrees of freedom where non-equilibrium statistical mechanics plays a dominant role 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 H i = E, the work obtained is W = H f (x f ) H i (x i ). Where x f and x i is the final and initial position respectively. Work(W) being a random variable is given by [21] P E (W ) = dxi δ(h i (x i ) E)δ(W H f (x f ) + H i (x i )) Ω i (E) (5.2) Since the system is microscopically reversible we could also write the reverse probability distribution as P E+W ( W ) = d xf δ(h i ( x f ) E W )δ(h f ( x f ) W H i ( x i )) Ω f (E + W ) (5.3) The Jacobian for the transformation from dx i to d x f is 1. Therefore we can equate Eqs. (5.2) to (5.3) and we have P E (W ) P E+W ( W ) = Ω f(e + W ) Ω i (E) = e S f (E+W ) S i (E) k B (5.4) To further extract information from Eq. (5.4) we can adopt the 1st and 2nd law of thermodynamics. df = du T ds (5.5) For an isolated system du = W thus the change in entropy of a system for a non adiabatic process is 39

41 Fluctuation theorems 40 S = W F T (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 S k B = e β(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 equivalent to the exponential of Helmholtz free energy difference F between the two thermally equilibrated states. This powerful insight allows us to relate the nonequilibrium 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 Jarzynski Equality in classical system The Jarzynski equality relates work statistics with the Helmholtz free energy difference. 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

42 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 W τ = 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 where ρ(λ(0), z(0)) = e βh(λ(0),z(0)) Z 0, (5.10) Z t = 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 = = ρ(λ(0), z(0))e βwτ dz(0) Ω e βh(λ(0),z(0)) Ω = Z τ Z 0 Z 0 e β[h(λ(τ),z(z(0),τ)) H(λ(0),z(0))] dz(0) The Helmholtz free energy expressed by partition function is F = 1 ln Z. Together with the expression in Eq. (5.12), we obtained the Jarzynski equality β in classical system: e βw = e βfτ e βf 0 = 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

43 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. The relation will follow naturally from Eq. (5.12), f(x) > f( x ) (5.13) 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 fluctuations 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 δ 2 (W ) = 1 N N [W i W ] 2 (5.15) i=1 For a canonical ensemble the work fluctuation is expressed as δ 2 (e βw ) = e 2βW e βw 2 (5.16) 42

44 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

45 Chapter 6 THE SINAI BILLIARD We have chosen the Sinai billiard to be our ergodic system of study. The Sinai billiard is a well studied ergodic model characterised by motion which is highly nonlinear. [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 specular 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

46 Sinai billiard 45 Φ V = 0 if within Φ, at boundary. Figure 6.1: Sinai billiard has been proven to be highly ergodic due to the dispersing nature of the circular domain. In the bounded region Φ the potential is zero [25]. 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 time scale of 80 45

47 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 using n=5 and time scale of 80 The modified Sinai billiard shown in Fig 6.3 is less chaotic compared to the circular 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 compared 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) ) = U(E H) ( p, q, t ) d N p d N q (6.1) V 46