DOCSLIB.ORG

Statistical Mechanics of Nonequilibrium Liquids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Statistical Mechanics of Nonequilibrium Liquids statistical mechanics of nonequilibrium liquids statistical mechanics of nonequilibrium liquids Denis J. evans | Gary P. Morriss Published by ANU E Press The Australian National University Canberra ACT 0200, Australia Email: [email protected] This title is also available online at: http://epress.anu.edu.au/sm_citation.html Previously published by Academic Press Limited National Library of Australia Cataloguing-in-Publication entry Evans, Denis J. Statistical mechanics of nonequilibrium liquids. 2nd ed. Includes index. ISBN 9781921313226 (pbk.) ISBN 9781921313233 (online) 1. Statistical mechanics. 2. Kinetic theory of liquids. I. Morriss, Gary P. (Gary Phillip). II. Title. 530.13 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior permission of the publisher. Cover design by Teresa Prowse Printed by University Printing Services, ANU First edition © 1990 Academic Press Limited This edition © 2007 ANU E Press Contents Preface ..................................................................................................... ix Biographies ............................................................................................... xi List of Symbols ....................................................................................... xiii 1. Introduction ......................................................................................... 1 References ........................................................................................ 10 2. Linear Irreversible Thermodynamics ................................................ 11 2.1 The Conservation Equations ........................................................ 11 2.2 Entropy Production .................................................................... 17 2.3 Curie's Theorem .......................................................................... 20 2.4 Non-Markovian Constitutive Relations: Viscoelasticity .................. 27 References ........................................................................................ 32 3. The Microscopic Connection .............................................................. 33 3.1 Classical Mechanics ..................................................................... 33 3.2 Phase Space ................................................................................ 42 3.3 Distribution Functions and the Liouville Equation ........................ 43 3.4 Ergodicity, Mixing and Lyapunov Exponents ............................... 51 3.5 Equilibrium Time Correlation Functions ....................................... 56 3.6 Operator Identities ...................................................................... 59 3.7 The Irving-Kirkwood Procedure .................................................. 63 3.8 Instantaneous Microscopic Representation of Fluxes ..................... 69 3.9 The Kinetic Temperature ............................................................. 73 References ........................................................................................ 74 4. The Green Kubo Relations ................................................................. 77 4.1 The Langevin Equation ............................................................... 77 4.2 Mori-Zwanzig Theory ................................................................. 80 4.3 Shear Viscosity ........................................................................... 84 4.4 Green-Kubo Relations for Navier-Stokes Transport Coefficients ...................................................................................... 89 References ........................................................................................ 92 5. Linear Response Theory ..................................................................... 93 5.1 Adiabatic Linear Response Theory ............................................... 93 5.2 Thermostats and Equilibrium Distribution Functions .................... 98 5.3 Isothermal Linear Response Theory ............................................ 108 5.4 The Equivalence of Thermostatted Linear Responses ................... 112 References ...................................................................................... 115 6. Computer Simulation Algorithms .................................................... 117 6.1 Introduction ............................................................................. 117 v Statistical Mechanics of Nonequilibrium Liquids 6.2 Self Diffusion ............................................................................ 124 6.3 Couette Flow and Shear Viscosity ............................................... 128 6.4 Thermostatting Shear Flows ....................................................... 142 6.5 Thermal Conductivity ............................................................... 145 6.6 Norton Ensemble Methods ........................................................ 148 6.7 Constant-Pressure Ensembles ..................................................... 152 6.8 Constant Stress Ensemble .......................................................... 155 References ...................................................................................... 163 7. Nonlinear Response Theory ............................................................. 167 7.1 Kubo's Form for the Nonlinear Response .................................... 167 7.2 Kawasaki Distribution Function ................................................. 168 7.3 The Transient Time Correlation Function Formalism .................... 172 7.4 Trajectory Mappings ................................................................. 176 7.5. Numerical Results for the Transient Time-Correlation Function ........................................................................................ 183 7.6. Differential Response Functions ................................................ 188 7.7 Numerical Results for the Kawasaki Representation ..................... 194 7.8 The Van Kampen Objection to Linear Response Theory ............... 198 References ...................................................................................... 207 8. Time Dependent Response Theory ................................................... 209 8.1 Introduction ............................................................................. 209 8.2 Time Evolution of Phase Variables .............................................. 209 8.3 The Inverse Theorem ................................................................. 211 8.4 The Associative Law and Composition Theorem .......................... 214 8.5 Time Evolution of the Distribution Function ............................... 216 8.6 Time Ordered Exponentials ........................................................ 217 8.7 Schrödinger and Heisenberg Representations ............................. 218 8.8 The Dyson Equation .................................................................. 220 8.9 Relation Between p- and f- Propagators ...................................... 221 8.10 Time Dependent Response Theory ............................................ 223 8.11 Renormalisation ...................................................................... 226 8.12 Discussion ............................................................................... 228 References ...................................................................................... 229 9. Steady State Fluctuations .................................................................. 231 9.1 Introduction ............................................................................. 231 9.2 The Specific Heat ...................................................................... 232 9.3 The Compressibility and Isobaric Specific Heat ........................... 237 9.4 Differential Susceptibility .......................................................... 239 9.5 The Inverse Burnett Coefficients ................................................ 242 References ...................................................................................... 244 vi Statistical Mechanics of Nonequilibrium Liquids 10. Towards a Thermodynamics of Steady States .................................. 245 10.1 Introduction ............................................................................ 245 10.2 Chaotic Dynamical Systems ...................................................... 247 10.3 The Characterization of Chaos .................................................. 258 10.4 Chaos in Planar Couette Flow ................................................... 267 10.5 Green's Expansion for the Entropy ........................................... 281 References ...................................................................................... 291 Index ..................................................................................................... 295 vii Preface During the 1980's there have been many new developments regarding the nonequilibrium statistical mechanics of dense classical systems. These developments have had a major impact on the computer simulation methods used to model nonequilibrium fluids. Some of these new algorithms are discussed in the recent book by Allen and Tildesley, Computer Simulation of Liquids. However that book was never intended to provide a detailed statistical mechanical backdrop to the new computer algorithms. As the authors commented in their preface, their main
Load more

Recommended publications
  • Arxiv:1910.10745V1 [Cond-Mat.Str-El] 23 Oct 2019 2.2 Symmetry-Protected Time Crystals
    A Brief History of Time Crystals Vedika Khemania,b,∗, Roderich Moessnerc, S. L. Sondhid aDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA bDepartment of Physics, Stanford University, Stanford, California 94305, USA cMax-Planck-Institut f¨urPhysik komplexer Systeme, 01187 Dresden, Germany dDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA Abstract The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the per- petuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization (MBL). Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics — that of an intrinsically out-of-equilibrium many-body phase of matter with no equilibrium counterpart. We include a compendium of the necessary background on the statistical mechanics of phase structure in many- body systems, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. In particular, we provide precise definitions that formalize the notion of a time-crystal as a stable, macroscopic, conservative clock — explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. Our discussion emphasizes that TTSB in a time-crystal is accompanied by the breaking of a spatial symmetry — so that time-crystals exhibit a novel form of spatiotemporal order.
    [Show full text]
  • How Nonequilibrium Thermodynamics Speaks to the Mystery of Life
    CORE CONCEPTS How nonequilibrium thermodynamics speaks to the mystery of life CORE CONCEPTS Stephen Ornes, Science Writer In his 1944 book What is Life?, Austrian physicist Erwin Schrödinger argued that organisms stay alive pre- cisely by staving off equilibrium. “How does the living organism avoid decay?” he asks. “The obvious answer is: By eating, drinking, breathing and (in the case of plants) assimilating. The technical term is metabolism” (1). However, the second law of thermodynamics, and the tendency for an isolated system to increase in entropy, or disorder, comes into play. Schrödinger wrote that the very act of living is the perpetual effort to stave off disor- der for as long as we can manage; his examples show how living things do that at the macroscopic level by taking in free energy from the environment. For example, people release heat into their surroundings but avoid running out of energy by consuming food. The ultimate source of “negative entropy” on Earth, wrote Schrödinger, is the Sun. Recent studies suggest something similar is hap- pening at the microscopic level as well, as many cellular processes—ranging from gene transcription to intracellular transport—have underlying nonequi- librium drivers (2, 3). Indeed, physicists have found that nonequilibrium systems surround us. “Most of the world around us is in this situation,” says theoretical physicist Michael Cross at the California Institute of Technology, in Pasa- Attributes of living organisms, such as the flapping hair-like flagella of these single- dena. Cross is among many theorists who spent de- celled green algae known as Chlamydomonas, are providing insights into the cades chasing a general theory of nonequilibrium thermodynamics of nonequilibrium systems.
    [Show full text]
  • Lecture 6: Entropy
    Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 6: Entropy 1 Introduction In this lecture, we discuss many ways to think about entropy. The most important and most famous property of entropy is that it never decreases Stot > 0 (1) Here, Stot means the change in entropy of a system plus the change in entropy of the surroundings. This is the second law of thermodynamics that we met in the previous lecture. There's a great quote from Sir Arthur Eddington from 1927 summarizing the importance of the second law: If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equationsthen so much the worse for Maxwell's equations. If it is found to be contradicted by observationwell these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of ther- modynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. Another possibly relevant quote, from the introduction to the statistical mechanics book by David Goodstein: Ludwig Boltzmann who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. There are many ways to dene entropy. All of them are equivalent, although it can be hard to see. In this lecture we will compare and contrast dierent denitions, building up intuition for how to think about entropy in dierent contexts. The original denition of entropy, due to Clausius, was thermodynamic.
    [Show full text]
  • Dirty Tricks for Statistical Mechanics
    Dirty tricks for statistical mechanics Martin Grant Physics Department, McGill University c MG, August 2004, version 0.91 ° ii Preface These are lecture notes for PHYS 559, Advanced Statistical Mechanics, which I’ve taught at McGill for many years. I’m intending to tidy this up into a book, or rather the first half of a book. This half is on equilibrium, the second half would be on dynamics. These were handwritten notes which were heroically typed by Ryan Porter over the summer of 2004, and may someday be properly proof-read by me. Even better, maybe someday I will revise the reasoning in some of the sections. Some of it can be argued better, but it was too much trouble to rewrite the handwritten notes. I am also trying to come up with a good book title, amongst other things. The two titles I am thinking of are “Dirty tricks for statistical mechanics”, and “Valhalla, we are coming!”. Clearly, more thinking is necessary, and suggestions are welcome. While these lecture notes have gotten longer and longer until they are al- most self-sufficient, it is always nice to have real books to look over. My favorite modern text is “Lectures on Phase Transitions and the Renormalisation Group”, by Nigel Goldenfeld (Addison-Wesley, Reading Mass., 1992). This is referred to several times in the notes. Other nice texts are “Statistical Physics”, by L. D. Landau and E. M. Lifshitz (Addison-Wesley, Reading Mass., 1970) par- ticularly Chaps. 1, 12, and 14; “Statistical Mechanics”, by S.-K. Ma (World Science, Phila., 1986) particularly Chaps.
    [Show full text]
  • Arxiv:1708.03422V2 [Cond-Mat.Stat-Mech]
    Fluctuation Theorem and Central Limit Theorem for the Time-Reversible Nonequilibrium Baker Map William Graham Hoover Ruby Valley Research Institute Highway Contract 60, Box 601 Ruby Valley, NV 8983 (Dated: February 13, 2018) Abstract The nonequilibrium Time-Reversible Baker Map provides simple illustrations of the Fluctua- tion Theorem, the Central Limit Theorem, and the Biased Random Walk. This is material in preparation for the Book form of Carol’s and my 2016 Kharagpur Lectures. Comments welcome. Keywords: Fluctuation Theorem, Chaos, Lyapunov Exponents, Irreversibility, Second Law, Baker Map arXiv:1708.03422v2 [cond-mat.stat-mech] 14 Aug 2017 1 I. INTRODUCTION In 1993 Denis Evans, Eddie Cohen, and Gary Morriss discovered an interesting symmetry, by now “well-known”, in their studies of the periodic shear flow of 56 hard disks1. They kept track of the time-averaged Gibbs’ entropy changes associated with the flow as a function of the averaging time τ. At a strainrateǫ ˙ and over a time window τ the distribution of entropy 2 production rates approaches a smooth curve with a mean value, h (S/k˙ )=(V/T )ηǫ˙ iτ . The fluctuations about this mean necessarily satisfy the Central Limit Theorem for large τ. Evans, Cohen, and Morriss stressed that both positive and negative values of the entropy production can be observed if the system is not too small ( 56 soft disks in their case ) and τ is not too large ( a few collision times ). At equilibrium the positive and negative values even out over time. Away from equilibrium the positive values win out. The relatively few time intervals with negative values correspond to periods of entropy decrease.
    [Show full text]
  • Generalized Molecular Chaos Hypothesis and the H-Theorem: Problem of Constraints and Amendment of Nonextensive Statistical Mechanics
    Generalized molecular chaos hypothesis and the H-theorem: Problem of constraints and amendment of nonextensive statistical mechanics Sumiyoshi Abe1,2,3 1Department of Physical Engineering, Mie University, Mie 514-8507, Japan*) 2Institut Supérieur des Matériaux et Mécaniques Avancés, 44 F. A. Bartholdi, 72000 Le Mans, France 3Inspire Institute Inc., McLean, Virginia 22101, USA Abstract Quite unexpectedly, kinetic theory is found to specify the correct definition of average value to be employed in nonextensive statistical mechanics. It is shown that the normal average is consistent with the generalized Stosszahlansatz (i.e., molecular chaos hypothesis) and the associated H-theorem, whereas the q-average widely used in the relevant literature is not. In the course of the analysis, the distributions with finite cut-off factors are rigorously treated. Accordingly, the formulation of nonextensive statistical mechanics is amended based on the normal average. In addition, the Shore- Johnson theorem, which supports the use of the q-average, is carefully reexamined, and it is found that one of the axioms may not be appropriate for systems to be treated within the framework of nonextensive statistical mechanics. PACS number(s): 05.20.Dd, 05.20.Gg, 05.90.+m ____________________________ *) Permanent address 1 I. INTRODUCTION There exist a number of physical systems that possess exotic properties in view of traditional Boltzmann-Gibbs statistical mechanics. They are said to be statistical- mechanically anomalous, since they often exhibit and realize broken ergodicity, strong correlation between elements, (multi)fractality of phase-space/configuration-space portraits, and long-range interactions, for example. In the past decade, nonextensive statistical mechanics [1,2], which is a generalization of the Boltzmann-Gibbs theory, has been drawing continuous attention as a possible theoretical framework for describing these systems.
    [Show full text]
  • LECTURE 9 Statistical Mechanics Basic Methods We Have Talked
    LECTURE 9 Statistical Mechanics Basic Methods We have talked about ensembles being large collections of copies or clones of a system with some features being identical among all the copies. There are three different types of ensembles in statistical mechanics. 1. If the system under consideration is isolated, i.e., not interacting with any other system, then the ensemble is called the microcanonical ensemble. In this case the energy of the system is a constant. 2. If the system under consideration is in thermal equilibrium with a heat reservoir at temperature T , then the ensemble is called a canonical ensemble. In this case the energy of the system is not a constant; the temperature is constant. 3. If the system under consideration is in contact with both a heat reservoir and a particle reservoir, then the ensemble is called a grand canonical ensemble. In this case the energy and particle number of the system are not constant; the temperature and the chemical potential are constant. The chemical potential is the energy required to add a particle to the system. The most common ensemble encountered in doing statistical mechanics is the canonical ensemble. We will explore many examples of the canonical ensemble. The grand canon- ical ensemble is used in dealing with quantum systems. The microcanonical ensemble is not used much because of the difficulty in identifying and evaluating the accessible microstates, but we will explore one simple system (the ideal gas) as an example of the microcanonical ensemble. Microcanonical Ensemble Consider an isolated system described by an energy in the range between E and E + δE, and similar appropriate ranges for external parameters xα.
    [Show full text]
  • H Theorems in Statistical Fluid Dynamics
    J. Phys. A: Math. Gen. 14 (1981) 1701-1718. Printed in Great Britain H theorems in statistical fluid dynamics George F Camevaleit, Uriel FrischO and Rick Salmon1 $National Center for Atmospheric Researchq, Boulder, Colorado 80307, USA §Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA and CNRS, Observatoire de Nice, Nice, France SScripps Institution of Oceanography, La Jolla, California 92093, USA Received 28 April 1980, in final form 6 November 1980 Abstract. It is demonstrated that the second-order Markovian closures frequently used in turbulence theory imply an H theorem for inviscid flow with an ultraviolet spectral cut-off. That is, from the inviscid closure equations, it follows that a certain functional of the energy spectrum (namely entropy) increases monotonically in time to a maximum value at absolute equilibrium. This is shown explicitly for isotropic homogeneous flow in dimensions d 3 2, and then a generalised theorem which covers a wide class of systems of current interest is presented. It is shown that the H theorem for closure can be derived from a Gibbs-type H theorem for the exact non-dissipative dynamics. 1. Introduction In the kinetic theory of classical and quantum many-particle systems, statistical evolution is described by a Boltzmann equation. This equation provides a collisional representation for nonlinear interactions, and through the Boltzmann H theorem it implies monotonic relaxation toward absolute canonical equilibrium. Here we show to what extent this same framework formally applies in statistical macroscopic fluid dynamics. Naturally, since canonical equilibrium applies only to conservative systems, a strong analogy can be produced only by confining ourselves to non-dissipative idealisations of fluid systems.
    [Show full text]
  • Statistical Mechanics I: Lecture 26
    VII.G Superfluid He4 Interesting examples of quantum fluids are provided by the isotopes of helium. The two electrons of He occupy the 1s orbital with opposite spins. The filled orbital makes this noble gas particularly inert. There is still a van der Waals attraction between two He atoms, but the interatomic potential has a shallow minimum of depth 9◦K at a separation of roughly 3A.˚ The weakness of this interaction makes He a powerful wetting agent. As the He atoms have a stronger attraction for practically all other molecules, they easily spread over the surface of any substance. Due to its light mass, the He atom undergoes large zero point fluctuations at T = 0. These fluctuations are sufficient to melt a solid phase, and thus He remains in a liquid state at ordinary pressures. Pressures of over 25 atmosphere are required to sufficiently localize the He atoms to result in a solid phase at T = 0. A remarkable property of the quantum liquid at zero temperature is that, unlike its classical counterpart, it has zero entropy. The lighter isotope of He3 has three nucleons and obeys fermi statistics. The liquid phase is very well described by a fermi gas of the type discussed in sec.VII.E. The inter­ actions between atoms do not significantly change the non-interacting picture described earlier. By contrast, the heavier isotope of He4 is a boson. Helium can be cooled down by a process of evaporation: Liquid helium is placed in an isolated container, in equilibrium with the gas phase at a finite vapor density.
    [Show full text]
  • Lecture 7: Ensembles
    Matthew Schwartz Statistical Mechanics, Spring 2019 Lecture 7: Ensembles 1 Introduction In statistical mechanics, we study the possible microstates of a system. We never know exactly which microstate the system is in. Nor do we care. We are interested only in the behavior of a system based on the possible microstates it could be, that share some macroscopic proporty (like volume V ; energy E, or number of particles N). The possible microstates a system could be in are known as the ensemble of states for a system. There are dierent kinds of ensembles. So far, we have been counting microstates with a xed number of particles N and a xed total energy E. We dened as the total number microstates for a system. That is (E; V ; N) = 1 (1) microstatesk withsaXmeN ;V ;E Then S = kBln is the entropy, and all other thermodynamic quantities follow from S. For an isolated system with N xed and E xed the ensemble is known as the microcanonical 1 @S ensemble. In the microcanonical ensemble, the temperature is a derived quantity, with T = @E . So far, we have only been using the microcanonical ensemble. 1 3 N For example, a gas of identical monatomic particles has (E; V ; N) V NE 2 . From this N! we computed the entropy S = kBln which at large N reduces to the Sackur-Tetrode formula. 1 @S 3 NkB 3 The temperature is T = @E = 2 E so that E = 2 NkBT . Also in the microcanonical ensemble we observed that the number of states for which the energy of one degree of freedom is xed to "i is (E "i) "i/k T (E " ).
    [Show full text]
  • Section 2 Introduction to Statistical Mechanics
    Section 2 Introduction to Statistical Mechanics 2.1 Introducing entropy 2.1.1 Boltzmann’s formula A very important thermodynamic concept is that of entropy S. Entropy is a function of state, like the internal energy. It measures the relative degree of order (as opposed to disorder) of the system when in this state. An understanding of the meaning of entropy thus requires some appreciation of the way systems can be described microscopically. This connection between thermodynamics and statistical mechanics is enshrined in the formula due to Boltzmann and Planck: S = k ln Ω where Ω is the number of microstates accessible to the system (the meaning of that phrase to be explained). We will first try to explain what a microstate is and how we count them. This leads to a statement of the second law of thermodynamics, which as we shall see has to do with maximising the entropy of an isolated system. As a thermodynamic function of state, entropy is easy to understand. Entropy changes of a system are intimately connected with heat flow into it. In an infinitesimal reversible process dQ = TdS; the heat flowing into the system is the product of the increment in entropy and the temperature. Thus while heat is not a function of state entropy is. 2.2 The spin 1/2 paramagnet as a model system Here we introduce the concepts of microstate, distribution, average distribution and the relationship to entropy using a model system. This treatment is intended to complement that of Guenault (chapter 1) which is very clear and which you must read.
    [Show full text]
  • 4 Pressure and Viscosity
    4 Pressure and Viscosity Reading: Ryden, chapter 2; Shu, chapter 4 4.1 Specific heats and the adiabatic index First law of thermodynamics (energy conservation): d = −P dV + dq =) dq = d + P dV; (28) − V ≡ ρ 1 = specific volume [ cm3 g−1] dq ≡ T ds = heat change per unit mass [ erg g−1] − s ≡ specific entropy [ erg g−1 K 1]: The specific heat at constant volume, @q −1 −1 cV ≡ [ erg g K ] (29) @T V is the amount of heat that must be added to raise temperature of 1 g of gas by 1 K. At constant volume, dq = d, and if depends only on temperature (not density), (V; T ) = (T ), then @q @ d cV ≡ = = : @T V @T V dT implying dq = cV dT + P dV: 1 If a gas has temperature T , then each degree of freedom that can be excited has energy 2 kT . (This is the equipartition theorem of classical statistical mechanics.) The pressure 1 ρ P = ρhjw~j2i = kT 3 m since 1 1 3kT h mw2i = kT =) hjw~j2i = : 2 i 2 m Therefore kT k P V = =) P dV = dT: m m Using dq = cV dT + P dV , the specific heat at constant pressure is @q dV k cP ≡ = cV + P = cV + : @T P dT m Changing the temperature at constant pressure requires more heat than at constant volume because some of the energy goes into P dV work. 19 For reasons that will soon become evident, the quantity γ ≡ cP =cV is called the adiabatic index. A monatomic gas has 3 degrees of freedom (translation), so 3 kT 3 k 5 k 5 = =) c = =) c = =) γ = : 2 m V 2 m P 2 m 3 A diatomic gas has 2 additional degrees of freedom (rotation), so cV = 5k=2m, γ = 7=5.
    [Show full text]