Denis J. Evans, Debra J. Searles, andStephen R. Williams

Fundamentals of Classical StatisticalThermodynamics

Denis J. Evans, Debra J. Searles, and Stephen R. Williams

Fundamentals of Classical StatisticalThermodynamics

Dissipation, Relaxation and Fluctuation Theorems

Authors

Prof. Denis J. EvansAustralian National UniversityDepartment of Applied MathematicsResearch School of Physicsand EngineeringCanberra ACT, 2601Australia

Prof. Debra J. SearlesThe University of QueenslandAustralian Institute for Bioengineeringand Nanotechnology Centre forTheoretical and ComputationalMolecular ScienceSchool of Chemistry andMolecular BiosciencesBrisbane, Qld 4072Australia

Dr. Stephen R. WilliamsAustralian National UniversityResearch School of ChemistryBuilding 35Research School of ChemistryCanberra, ACT 2601Australia

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V

Contents

List of Symbols IXAcknowledgments XIII

1 Introduction 1References 14

2 Introduction to Time-Reversible, Thermostatted Dynamical Systems,and Statistical Mechanical Ensembles 17

2.1 Time Reversibility in Dynamical Systems 172.2 Introduction to Time-Reversible, Thermostatted Dynamical

Systems 192.3 Example: Homogeneously Thermostatted SLLOD Equations for

Planar Couette Flow 302.4 Phase Continuity Equation 322.5 Lyapunov Instability and Statistical Mechanics 352.6 Gibbs Entropy in Deterministic Nonequilibrium

Macrostates 422.A Appendix: Phase Space Expansion Calculation 44

References 46

3 The Evans–Searles Fluctuation Theorem 493.1 The Transient Fluctuation Theorem 493.2 Second Law Inequality 533.3 Nonequilibrium Partition Identity 553.4 Integrated Fluctuation Theorem 563.5 Functional Transient Fluctuation Theorem 583.6 The Covariant Dissipation Function 593.7 The Definition of Equilibrium 603.8 Conclusion 62

References 63

VI Contents

4 The Dissipation Theorem 654.1 Derivation of the Dissipation Theorem 654.2 Equilibrium Distributions are Preserved by Their Associated

Dynamics 684.3 Broad Characterization of Nonequilibrium Systems: Driven,

Equilibrating, and T-Mixing Systems 704.3.1 Two Corollaries of the Dissipation Theorem 74

References 75

5 Equilibrium Relaxation Theorems 775.1 Introduction 775.2 Relaxation toward Mixing Equilibrium: The Umbrella Sampling

Approach 785.3 Relaxation of Autonomous Hamiltonian Systems under

T-Mixing 835.4 Thermal Relaxation to Equilibrium: The Canonical Ensemble 875.5 Relaxation to Quasi-Equilibrium for Nonergodic Systems 945.6 Aside: The Thermodynamic Connection 945.7 Introduction to Classical Thermodynamics 985.A Appendix: Entropy Change for a Cyclic Temperature Variation 104

References 107

6 Nonequilibrium Steady States 1096.1 The Physically Ergodic Nonequilibrium Steady State 1096.2 Dissipation in Nonequilibrium Steady States (NESSs) 1116.3 For T-Mixing Systems, Nonequilibrium Steady-State Averages are

Independent of the Initial Equilibrium Distribution 1186.4 In the Linear Response Steady State, the Dissipation is Minimal with

Respect to Variations of the Initial Distribution 1206.5 Sum Rules for Dissipation in Steady States 1216.6 Positivity of Nonlinear Transport Coefficients 1226.7 Linear Constitutive Relations for T-Mixing Canonical

Systems 1246.8 Gaussian Statistics for T-Mixing NESS 1246.9 The Nonequilibrium Steady-State Fluctuation Relation 1256.10 Gallavotti–Cohen Steady-State Fluctuation Relation 1296.11 Summary 130

References 131

7 Applications of the Fluctuation, Dissipation, and RelaxationTheorems 133

7.1 Introduction 1337.2 Proof of the Zeroth “Law” of Thermodynamics 1347.3 Steady-State Heat Flow 1377.4 Dissipation Theorem for a Temperature Quench 144

Contents VII

7.5 Color Relaxation in Color Blind Hamiltonian Systems 1477.6 Instantaneous Fluctuation Relations 1497.7 Further Properties of the Dissipation Function 151

References 153

8 Nonequilibrium Work Relations, the Clausius Inequality, andEquilibrium Thermodynamics 155

8.1 Generalized Crooks Fluctuation Theorem (GCFT) 1578.2 Generalized Jarzynski Equality (GJE) 1618.3 Minimum Average Generalized Work 1648.4 Nonequilibrium Work Relations for Cyclic Thermal Processes 1678.5 Clausius’ Inequality, the Thermodynamic Temperature, and Classical

Thermodynamics 1718.6 Purely Dissipative Generalized Work 1768.7 Application of the Crooks Fluctuation Theorem (CFT), and the

Jarzynski Equality (JE) 1798.8 Entropy Revisited 1828.9 For Thermostatted Field-Free Systems, the Nonequilibrium

Helmholtz Free Energy is a Constant of the Motion 183References 184

9 Causality 1879.1 Introduction 1879.2 Causal and Anti-causal Constitutive Relations 1899.3 Green–Kubo Relations for the Causal and Anti-causal Response

Functions 1909.4 Example: The Maxwell Model of Viscosity 1939.5 Phase Space Trajectories for Ergostatted Shear Flow 1949.6 Simulation Results 1979.7 Summary and Conclusion 200

References 202

Index 203

IX

List of Symbols

Microscopic Dynamics

N Number of particles in the system.DC Cartesian dimensions of the system – usually three.D Accessible phase space domain𝐪 NDC-dimensional vector, representing the particle positions.𝐩 NDC-dimensional vector, representing the particle momenta.𝚪 2NDC-dimensional phase space vector, representing all q’s

and p’s.𝛿V𝚪(St𝚪) Very small volume element of phase space centered on

St𝚪 ≡ exp[iL(𝚪)t]𝚪.p(𝛿V𝚪(𝚪); t) Probability of observing sets of trajectories inside 𝛿V𝚪(𝚪) at

time t.d(St𝚪) Infinitesimal phase space volume centered on St𝚪.p+∕−(t) Probability that the time-integrated dissipation function is

plus/minus over the time interval (0,t).MT Time reversal map MT (𝐪,𝐩) ≡ (𝐪,−𝐩)MT

.MK Kawasaki or K-map of phase space vector for planar Couette

flow, MK (x, y, z, px, py, pz, �̇�) ≡ (x,−y, z,−px, py,−pz, �̇�)MK,

where �̇� ≡ 𝜕ux∕𝜕y is the xy component of the strain ratetensor.

L f -Liouvillean.exp[−iL(𝚪)t] … f -propagator.L p-Liouvillean.exp[iL(𝚪)t] p-propagator.St p-propagator.K(𝐩) Peculiar kinetic energy.Φ(𝐪) Interparticle potential energy.𝜙i,j(rij) Pair potential of particle i with particle j.𝐫ij ≡ 𝐫j − 𝐫i Position vector from particle i to particle j.rij ≡ |𝐫j − 𝐫i| Distance between particles i and j.𝐅ij Force on particle i due to particle j

X List of Symbols

∇𝐪 ≡ (𝜕∕𝜕𝐪1, … , 𝜕∕𝜕𝐪N .H0(𝚪) Internal energy, H0 = K + Φ.H(𝚪) Hamiltonian at phase vector 𝚪.g(𝚪) Deviation function – even in the momenta.HE Extended Hamiltonian for Nosé–Hoover dynamics.Kth Peculiar kinetic energy of thermostatted particles

= DCNthkBTth∕2 + O(1), where Tth is the kinetic temperate ofthe thermostat. If the system is isokinetic, Tth = T – seethermodynamic variables below.

Nth Number of thermostatted particles.𝛼 Gaussian or Nosé–Hoover thermostat or ergostat

multiplier.𝜏 Time constant.Q̇th∕soi Rate of transfer of heat to the thermostat/system of

interest.Λ Phase space expansion factor.Si Switch function.𝐉(𝚪) Dissipative flux.𝐅e Dissipative external field.m Particle mass.T ≡ 𝜕�̇�(𝚪)∕𝜕𝚪 Stability matrix.expL Time-ordered exponential operator, latest times to left.

𝚵(𝚪; t) Tangent vector propagator ≡ expL

(∫

t

0ds T (Ss𝚪)

).

𝜆i ith Lyapunov exponent.𝜆max∕min Largest/smallest Lyapunov exponent for steady or

equilibrium state.

Statistical Mechanics

At Time average of some phase variable, A(𝚪).⟨A(t)⟩ Ensemble average of A at time t, on a time-evolved path.f (𝚪; t) Time-dependent phase space distribution function.⟨· · ·⟩𝜇c Equilibrium microcanonical ensemble average.⟨· · ·⟩c Equilibrium canonical ensemble average,fc(𝚪) Equilibrium canonical distribution.f𝜇c(𝚪) Equilibrium microcanonical distribution.Λ Phase space expansion factor.Ω(St1𝚪; t2) The instantaneous dissipation function, at time t1 on a phase

space trajectory that started at phase 𝚪 and defined withrespect to the distribution function at time t2.Ω(St1𝚪; 0) ≡ Ω(St1𝚪),

Ω(St1𝚪) Ω(St1𝚪) ≡ Ω(St1𝚪; 0) .𝐫 Three-dimensional position vector.

List of Symbols XI

𝐮(𝐫, t) Three-dimensional local fluid streaming velocity, at Cartesianposition r and time t.

SG(t) Fine-grained Gibbs entropy,

≡ −kB∫Dd𝚪 f (𝚪; t) ln(f (𝚪; t))

equilibrium⇒ kB ln(Z𝜇c).

Z Partition function – normalization for the equilibrium phasespace distribution.

Zc Canonical partition function.Z𝜇c Microcanonical partition function.

Mechanical Variables

Q Heat of thermostat.V Volume of system of interest.U Internal energy, U = ⟨H0⟩ of the system of interest.W Work performed on system of interest.Y Purely dissipative generalized dimensionless work.X Generalized dimensionless work.

Thermodynamic Variables

T Equilibrium temperature the system will relax to if it is soallowed.

𝛽 Boltzmann factor (reciprocal temperature) ≡ 1∕kBT .Sir Irreversible calorimetric entropy, defined by

ΔSir ≡ ∫ dt Q̇(t)∕T(t), where T(t) is the instantaneousequilibrium temperature the system would relax to if it was soallowed. In Section 5.7, we show that the Gibbs entropy andthe irreversible calorimetric entropy are equal, up to anadditive constant.

Seq ≡ S The calorimetric entropy defined in classical

thermodynamics as ΔSeq ≡ ΔS≡qs∫2

1dt Q̇(t)∕T(t), where T

is equilibrium temperature of the system. This entropy is astate function.

A Helmholtz free energy; = −kBT ln (Zc) = U − TSeq.Ane Nonequilibrium Helmholtz free energy; = U − TSir. This is

not a state function.⟨Σ(t)⟩ Total entropy production – only defined in the weak fieldlimit close to equilibrium.

G0 Zero-frequency elastic shear modulus.G∞ Infinite-frequency shear modulus.

XII List of Symbols

Transport

𝛾 Strain (note: 𝛾 is sometimes used to fix the system’s totalmomentum).

𝛿𝛾 Small strain..γ Strain rate.Pxy xy element of the pressure tensor.−⟨Pxy⟩ xy element of the ensemble averaged stress tensor.𝜂0+ Limiting zero-frequency shear viscosity of a solid.𝜂 Zero-frequency shear viscosity of a fluid.𝜏M Maxwell relaxation time.𝐉(𝚪) Dissipative flux.J⊥(ky, t) Wavector dependent transverse momentum density.𝜂M(t) Maxwell model memory function for shear viscosity.𝜂M Zero-frequency shear viscosity of a Maxwell fluid.

Mathematics

Θ(t) Heaviside step function at t = 0.∀ For all.∀! For almost all. The exceptions have zero measure.𝜆 Arbitrary scaling parameter.F̃(s) Laplace transform of F(t).F̂(s) Anti-Laplace transform of F(t).

∮PCyclic integral of a periodic function.

qs∫b

aQuasi-static integral from a to b.

DKY Kaplan–Yorke dimension of a nonequilibrium steady state.

Note: Upper case subscripts/superscripts indicate people. Lower case is used in mostother cases. Subscripts are preferred to superscripts so as to not confuse powerswith superscripts. Italics are used for algebraic initials. Nonitalics for word ini-tials. (e.g., T-mixing not T-mixing because T stands for Transient, N-particle notN-particle.)

XIII

Acknowledgments

DJE, DJS, and SRW would like to thank the Australian Research Council forthe long-term support of the research projects that ultimately led to the writingof this book. DJE would also like to acknowledge the assistance of his manyfourth-year honors, science, and mathematics students who over the last decadetook his course: “The Mathematical Foundations of Statistical Thermodynamics.”The authors also wish to thank their former PhD students for their contributionsto some of the material described in this book: especially, Dr Charlotte Petersen,Dr Owen Jepps, Dr James Reid, and Dr David Carberry. The first experimentalverification of a fluctuation theorem was carried out in Professor Edith Sevick’slaboratory at ANU with excellent technical support provided by Dr GenmaioWang. The authors would also like to thank Professor Lamberto Rondoni whohelped them understand some of the elements of ergodic theory.