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

 

Fundamentals of Classical Statistical Thermodynamics

Dissipation, Relaxation and Fluctuation Theorems

 

 

 

Authors

Prof. Denis J. Evans

Australian National University

Department of Applied Mathematics

Research School of Physics and Engineering

Canberra ACT, 2601

Australia

Prof. Debra J. Searles

The University of Queensland

Australian Institute for Bioengineering and Nanotechnology Centre for

Theoretical and Computational

Molecular Science

School of Chemistry and Molecular Biosciences

Brisbane, Qld 4072

Australia

Dr. Stephen R. Williams

Australian National University

Research School of Chemistry

Building 35

Research School of Chemistry

Canberra, ACT 2601

Australia

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List of Symbols

Microscopic Dynamics
N	Number of particles in the system.
	Cartesian dimensions of the system – usually three.
D	Accessible phase space domain
	-dimensional vector, representing the particle positions.
	-dimensional vector, representing the particle momenta.
	-dimensional phase space vector, representing all q's and p's.
	Very small volume element of phase space centered on .
	Probability of observing sets of trajectories inside at time t.
	Infinitesimal phase space volume centered on .
	Probability that the time-integrated dissipation function is plus/minus over the time interval (0,t).
	Time reversal map .
	Kawasaki or K-map of phase space vector for planar Couette flow, , where is the xy component of the strain rate tensor.
	f-Liouvillean.
	f-propagator.
	p-Liouvillean.
	p-propagator.
	p-propagator.
	Peculiar kinetic energy.
	Interparticle potential energy.
	Pair potential of particle i with particle j.
	Position vector from particle i to particle j.
	Distance between particles i and j.
	Force on particle i due to particle j
	.
	Internal energy, .
	Hamiltonian at phase vector .
	Deviation function – even in the momenta.
	Extended Hamiltonian for Nosé–Hoover dynamics.
	Peculiar kinetic energy of thermostatted particles , where is the kinetic temperate of the thermostat. If the system is isokinetic, – see thermodynamic variables below.
	Number of thermostatted particles.
	Gaussian or Nosé–Hoover thermostat or ergostat multiplier.
	Time constant.
	Rate of transfer of heat to the thermostat/system of interest.
	Phase space expansion factor.
	Switch function.
	Dissipative flux.
	Dissipative external field.
m	Particle mass.
	Stability matrix.
	Time-ordered exponential operator, latest times to left.
	Tangent vector propagator .
	Lyapunov exponent.
	Largest/smallest Lyapunov exponent for steady or equilibrium state.
Statistical Mechanics
	Time average of some phase variable, .
	Ensemble average of A at time t, on a time-evolved path.
	Time-dependent phase space distribution function.
	Equilibrium microcanonical ensemble average.
	Equilibrium canonical ensemble average,
	Equilibrium canonical distribution.
	Equilibrium microcanonical distribution.
	Phase space expansion factor.
	The instantaneous dissipation function, at time t1 on a phase space trajectory that started at phase and defined with respect to the distribution function at time t2. ,
	.
	Three-dimensional position vector.
	Three-dimensional local fluid streaming velocity, at Cartesian position r and time t.
	Fine-grained Gibbs entropy, .
Z	Partition function – normalization for the equilibrium phase space distribution.
	Canonical partition function.
	Microcanonical partition function.
Mechanical Variables
Q	Heat of thermostat.
V	Volume of system of interest.
U	Internal energy, 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 so allowed.
	Boltzmann factor (reciprocal temperature) .
	Irreversible calorimetric entropy, defined by , where is the instantaneous equilibrium temperature the system would relax to if it was so allowed. In Section 5.7, we show that the Gibbs entropy and the irreversible calorimetric entropy are equal, up to an additive constant.
	The calorimetric entropy defined in classical thermodynamics as , where T is equilibrium temperature of the system. This entropy is a state function.
A	Helmholtz free energy; .
Ane	Nonequilibrium Helmholtz free energy; . This is not a state function.
	Total entropy production – only defined in the weak field limit close to equilibrium.
	Zero-frequency elastic shear modulus.
	Infinite-frequency shear modulus.
Transport
	Strain (note: is sometimes used to fix the system's total momentum).
	Small strain.
	Strain rate.
	xy element of the pressure tensor.
	xy element of the ensemble averaged stress tensor.
	Limiting zero-frequency shear viscosity of a solid.
	Zero-frequency shear viscosity of a fluid.
	Maxwell relaxation time.
	Dissipative flux.
	Wavector dependent transverse momentum density.
	Maxwell model memory function for shear viscosity.
	Zero-frequency shear viscosity of a Maxwell fluid.
Mathematics
	Heaviside step function at t = 0.
	For all.
!	For almost all. The exceptions have zero measure.
	Arbitrary scaling parameter.
	Laplace transform of .
	Anti-Laplace transform of .
	Cyclic integral of a periodic function.
	Quasi-static integral from a to b.
	Kaplan–Yorke dimension of a nonequilibrium steady state.

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