SSttaattiissttiiccaall MMeecchhaanniiccss oo ff NNoonneeqquuiilliibbrriiuumm LLiiqquuiiddss by Denis J Evans and Gary P Morriss Research School of Chemistry, Australian National University, Canberra, ACT Australia Academic Press, London 1990, Theoretical Chemistry Monograph Series Copyright © 1990 Denis J. Evans and Gary P. Morriss 2 Contents. 1. Introduction 2. Linear Irreversible thermodynamics 2.1 The Conservation Equations 2.2 Entropy production 2.3 Curie's theorem 2.4 Non-Markovian Constitutive Relations: Viscoelasticity 3. The Microscopic Connection 3.1 Classical mechanics 3.2 Phase space 3.3 Distribution functions and the Liouville equation 3.4 Ergodicity, mixing and Lyapunov exponents 3.5 Equilibrium time correlation functions 3.6 Operator identities 3.7 Irving-Kirkwood procedure. 3.8 Instantaneous microscopic representation of fluxes 3.9 The kinetic temperature 4. Green-Kubo Relations for Thermal Transport Coefficients 4.1 The Langevin Equation 4.2 Mori-Zwanzig Theory 4.3 Shear Viscosity 4.4 Green-Kubo Relations for Navier-Stokes Transport Coefficients 5. Linear Response Theory 5.1 Adiabatic linear response theory 5.2 Thermostats and thermostatted distribution functions 5.3 Isothermal linear response theory 5.4 The equivalence of the thermostatted linear responses 3 6. Computer Simulation Algorithms 6.1 Introduction 6.2 Self Diffusion 6.3 Couette Flow and Shear Viscosity 6.4 Thermostatting shear flows 6.5 Thermal conductivity 6.6 Norton ensemble methods - Self Diffusion 6.7 Constant Pressure Ensembles 6.8 The Constant Stress Ensemble 7. Nonlinear Response Theory 7.1 Kubo’s form for the nonlinear response 7.2 Kawasaki Distribution Function 7.3 The Transient Time Correlation Formalism 7.4 Trajectory Mapping 7.5 Numerical Results for the Transient Time Correlation Functions 7.6 The Differential Response Function 7.7 Numerical results for the Kawasaki representation 7.8 The van Kampen Objection to Linear Response theory 8. Time Dependent Response Theory 8.1 Introduction 8.2 Time Evolution of Phase Variables 8.3 The Inverse Theorem 8.4 The Associative Law and Composition Theorem 8.5 Time Evolution of the Distribution Function 8.6 Time Ordered Exponentials 8.7 Schrödinger and Heisenberg Representations 8.8 The Dyson Equation 8.9 Morriss' Lemma 8.10 Time Dependent Response Theory 8.11 Renormalisation 8.12 Discussion 4 9. Steady State Fluctuations 9.1 Introduction 9.2 The Specific Heat 9.3 The compressibili 9.4 Differential susceptibilities 9.5 The Burnett Coefficients 10. Towards a Thermodynamics of Steady States 10.1 Introduction 10.2 Chaotic Dynamical Systems 10.3 The Characterization of Chaos 10.4 Chaos in planar Couette flow 10.5 Green's expansion for the entropy 5 List of Symbols Transport Coefficients η shear viscosity λ thermal conductivity ηV bulk viscosity ζ Brownian friction coefficient D self diffusion coefficient Thermodynamic Fluxes P the pressure tensor JQ heat flux Π viscous pressure tensor Thermodynamic forces ∇u strain rate tensor γ shear rate, =∂ux/∂y ∇T temperature gradient ∇... u dilation rate u streaming velocity ε elastic deformation ∇ ε strain tensor dot(ε) dilation rate, =1/3(∇... ε ) Thermodynamic State Variables T temperature kB Boltzmann's Constant β1/kBT V volume p hydrostatic pressure, =1/3 tr(P) N number of particles ρ mass density n number density Thermodynamic Constants G shear modulus CV constant volume specific heat Cp constant pressure, specific heat cV constant volume, specific heat per unit mass cp constant pressure, specific heat per unit mass DT isochoric thermal diffusivity 6 Thermodynamic Potentials E internal energy U(r,t) internal energy per unit mass S entropy s(r,t) internal energy per unit volume σ entropy source strength = rate of spontaneous entropy production per unit volume I enthalpy Q heat Mechanics L Lagrangian, seldom used H Hamiltonian H0 phase variable whose average is the internal energy I0 phase variable whose average is the enthalpy J(Γ) dissipative flux Fe external field α thermostatting multiplier iL p-Liouvillean iL f-Liouvillean A† Hermitian adjoint of A Λ phase space compression factor expR right time-ordered exponential expL left time-ordered exponential UR(t1,t2) incremental p-propagator t1 to t2 UR(t1,t2)-1 inverse of UR(t1,t2)., take phase variables from t2 to t1 UR(t2,t1) = UL( t1,t2 ) UR(t1,t2)† incremental f-propagator t1 to t2 TR right time-ordering opertator TL left time-ordering opertator CAB(t) equilibrium time correlation function, = < A(t)B*> [A,B] commutator bracket {A,B} Poisson bracket δ(t) Dirac delta function (of time) δ(r-ri) Kirkwood delta function = 0, | r-ri | > an 'infinitesmal' macroscopic distance, l = 1/l3, | r-ri | < an 'infinitesmal' macroscopic distance, l f(k) =∫ dr eik.r f(r) spatial Fourier transform 7 ∞ ω f(ω) =0∫ dr e-i t f(t) temporal Fourier-Laplace trasform transform dS infinitesmal vector area element J⊥ transverse momentum current Φ total intermolecular potential energy φ ij potential energy of particle i,j K total kinetic energy fc canonical distribution function fT isokinetic distribution function m particle mass ri position of particle i rij=rj - ri v i velocity of particle i Chapter 1 - 1 Chapter 1. Introduction Chapter 1 - 2 Mechanics provides a complete microscopic description of the state of a system. When the equations of motion are combined with initial conditions and boundary conditions, the subsequent time evolution of a classical system can be predicted. In systems with more than just a few degrees of freedom such an exercise is impossible. There is simply no practical way of measuring the initial microscopic state of for example, a glass of water, at some instant in time. In any case, even if this was possible we could not then solve the equations of motion for a coupled system of 1023 molecules. In spite of our inability to fully describe the microstate of a glass of water, we are all aware of useful macroscopic descriptions for such systems. Thermodynamics provides a theoretical framework for correlating the equilibrium properties of such systems. If the system is not at equilibrium, fluid mechanics is capable of predicting the macroscopic nonequilibrium behaviour of the system. In order for these macroscopic approaches to be useful their laws must be supplemented not only with a specification of the appropriate boundary conditions but with the values of thermophysical constants such as equation of state data and transport coefficients. These values cannot be predicted by macroscopic theory. Historically this data has been supplied by experiments. One of the tasks of statistical mechanics is to predict these parameters from knowledge of the interactions of the system's constituent molecules. This then is a major purpose for statistical mechanics. How well have we progressed? Equilibrium classical statistical mechanics is relatively well developed. The basic ground rules - Gibbsian ensemble theory - have been known for the best part of a century (Gibbs, 1902). The development of electronic computers in the 1950's provided unambiguous tests of the theory of simple liquids leading to a consequently rapid development of integral equation and perturbation treatments of liquids (Barker and Henderson 1976). With the possible exceptions of phase equilibria and interfacial phenomena (Rowlinson and Widom, 1982) one could say that the equilibrium statistical mechanics of atomic fluids is a solved problem. Much of the emphasis has moved to molecular, even macromolecular liquids. The nonequilibrium statistical mechanics of dilute atomic gases - kinetic theory - is likewise, essentially complete (Ferziger and Kaper, 1972). However attempts to extend kinetic theory to higher densities have been fraught with severe difficulties. One might have imagined being able to develop a power series expansion of the transport coefficients in much the same way that one expands the equilibrium equation of state in the virial series. In 1965 Cohen and Dorfman (1965 and 1972) proved that such an expansion does not exist. The Navier-Stokes transport coefficients are nonanalytic functions of density. It was at about this time that computer simulations began to have an impact on Chapter 1 - 3 the field. In a celebrated 1957 paper, Kubo (1957) showed that linear transport coefficients could be calculated from a knowledge of the equilibrium fluctuations in the flux associated with the particular transport coefficient. For example the shear viscosity η, is defined as the ratio of the shear stress, -Pxy, to the strain rate, ∂ux/∂y ≡ γ, ≡ η γ ( 1 . 1) Pxy - The Kubo relation predicts that the limiting, small shear rate, viscosity, is given by ∞ η = βV ds P (0) P (s) (1.2) ∫ < xy xy > 0 where β is the reciprocal of the absolute temperature T, multiplied by Boltzmann's constant kB, V is the system volume and the angle brackets denote an equilibrium ensemble average. The viscosity is then the infinite time integral of the equilibrium, autocorrelation function of the shear stress . Similar relations are valid for the other Navier-Stokes transport coefficients such as the self diffusion coefficient, the thermal conductivity and the bulk viscosity (see Chapter 4). Alder and Wainwright(1956) were the first to use computer simulations to compute the transport coefficients of atomic fluids. What they found was unexpected. It was believed that at sufficiently long time, equilibrium autocorrelation functions should decay exponentially. Alder and Wainwright discovered that in two dimensional systems the velocity autocorrelation function which determines the self-diffusion coefficient, only decays as 1/t. Since the diffusion coefficient is thought to be the integral of this function, we were forced to the reluctant conclusion that the self diffusion coefficient does not exist for two dimensional systems. It is presently believed that each of the Navier-Stokes transport coefficients diverge in two dimensions (Pomeau and Resibois, 1975). This does not mean that two dimensional fluids are infinitely resistant to shear flow. Rather, it means that the Newtonian constitutive relation (1.1), is an inappropriate definition of viscosity in two dimensions.