Computational Statistical Mechanics

320 Pages·1991·38.129 MB·English

by  Wm.G. HOOVER (Eds.)

STUDIES IN MODERN THERMODYNAMICS 11
Computational
Statistical
Mechanics
Wm.G. HOOVER
Department of Applied Science, University of California at Davis/Livermore and
Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A.
Amsterdam-Oxford-New York-Tokyo 1991
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Library of Congress Cataloging-in-Publication Data
Hoover, William G. (William Graham), 1936-
Computat i ona 1 statistical mechanics / Vim. G. Hoover.
p. cm. — (Studies in modern thermodynamics ; 11)
Includes bibliographical references and index.
ISBN 0-444-88192-1
1. Statistical mechanics. 2. Molecular dynamics. 3. Numerical
calculations. I. Title. II. Series.
QC174.8.H66 1991
530.1'3—dc20 90-27140
CIP
ISBN 0-444-88192-1
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STUDIES IN MODERN THERMODYNAMICS
1 Biochemical Thermodynamics edited by M.N. Jones
2 Principles of Thermodynamics by J. A. Beattie and I. Oppenheim
3 Phase Theory by H. A.J. Oonk
4 Thermodynamics. Principles Characterizing Physical and Chemical
Processes by J.M. Honig
5 Studies in Network Thermodynamics by L. Peusner
6 Measurement, Evaluation and Prediction of Phase Equilibria edited by
H.V. Kehiaian and H. Renon
7 Liquid-Liquid Equilibria by J.P. Nov£k, J. Matous and J. Pick
8 Biochemical Thermodynamics (second edition) edited by M.N. Jones
9 Heats of Vaporization of Fluids by V. Majer, V. Svoboda and J. Pick
10 Kinetic Phase Diagrams edited by Z. Chvoj, J. Sestak and A. Triska
11 Computational Statistical Mechanics by Wm. G. Hoover
vii
Computational Statistical Mechanics
Preface:
The fundamental goal of Statistical Mechanics is to link the detailed
determinism of many-body microscopic dynamics to the phenomenological averaged
description of macroscopic behavior. During my own 30-year study of statistical
mechanics computers have completely transformed the field, substantially widening
the scope of this goal by making it possible to follow the motion of millions, soon to
be billions, of particles. The corresponding coupled nonlinear differential equations
of motion can be solved, numerically, for time intervals including millions of
discrete time steps.
By now computers are readily available instructional tools for learning by doing.
Computers are now firmly established in our high schools and low-cost transputers
have made the speed of a CRAY available at the cost of an automobile. One can hope
that this continuing growth of computational power will play a role in promoting a
healthy diffusion of knowledge throughout the world.
But with our relatively newfound ability to compute comes a challenging
responsibility. The challenge today is not so much generating results, as it is
discovering imaginative ways to display results in comprehensible forms designed to
promote understanding. Because the subject of this book, computational statistical
mechanics, is now feasible, the goal of statistical mechanics has had to grow from the
days when experiment and theory were the complementary alternative approaches to
understanding. Our goal now is to achieve correspondence among three alternative
descriptions of natural phenomena: theory, experiment, and computer simulation.
This wider goal has led to new methods providing a more intimate degree of
understanding than was accessible to Boltzmann 100 years ago when he linked
reversible mechanics to irreversible thermodynamics with his H Theorem. This new
understanding is fundamentally related to another classical subject reinvigorated by
computation, chaos, the exponentially sensitive dependence of the future on the past.
This sensitive dependence has made a qualitative change in what it means to "solve" a
problem. Chaos forces us to take a "statistical" approach. In a real sense chaos now
provides not only an understanding of the fundamental origin of macroscopic
irreversibility, but also a conceptually useful microscopic framework showing that our
approach must be both computational and statistical. Long-time solutions for
individual chaotic systems are out of the question, inaccessible to any approach,
analytic or numerical. Chaos requires ensemble averages.
Most real macroscopic systems are nonequilibrium systems. In them, flows of
mass, momentum, or energy respond to differences in composition, velocity, and
temperature. Fast computers make it practical to generate for study "simulations," the
computational analogs of these nonequilibrium flows. To predict accurate future states
viii
from the present state of a system by computer simulation requires new combinations
of microscopic dynamics with macroscopic thermodynamics, in which equilibrium
equations of motion are augmented to deal with nonequilibrium constitutive
relations, while still retaining their deterministic, time-reversible character. With
these new dynamic equations, and with initial and boundary conditions specified, the
simulated time development generally follows from approximate solutions of ordinary
differential equations. This approach has proved to be useful and educational, for both
small and large systems.
Our view of such nonequilibrium systems has been profoundly changed by
computer simulation. Computers make it possible now to study nonequilibrium
systems in their own right, not just as linear perturbations or equilibrium fluctuations.
Accordingly, the nonequilibrium systems which are all around us make up more of
today's computational statistical mechanics than was possible in the older statistical
mechanics of Tolman, the Mayers, and Hill.
We can still formulate and understand the bases of both equilibrium and
nonequilibrium many-body behavior by computing the solutions of systems of classical
ordinary differential equations, using the old techniques of Gauss, Euler, and Newton
on modern computing machines. But the cumbersome analytic approaches of a
generation ago have either changed or been discarded. New ways, particularly the
direct simulation methods, are much broader in scope, richer in the detail they
provide, and more definite in their predictions. The new have replaced the old.
To keep pace with the continuing improvements in computer capacity and speed
requires the continuous development of new methods and viewpoints. The
interaction of computation, experiment, and theory has led to rapid qualitative gains in
our ability to predict the future reliably and to resolve the classical paradox that
reversible motion equations lead to patently irreversible solutions.
This book aims to use the tools of computation and simulation and the useful
concepts from nonlinear dynamics to link the basis of statistical mechanics established
by Maxwell, Boltzmann, and Gibbs to the concepts and computational tools available,
and still being developed, a century later. Despite the antiquity of the foundations and
the numerical techniques new discoveries are commonplace, so that computational
statistical mechanics promises to be a fertile research area as it is applied to increasingly
complicated problems. The present work describes the fundamentals underlying
numerical simulation of nonequilibrium systems as well as the numerical techniques
needed to apply these principles. I believe that physics can only be learned by first-hand
exploration. Most physicists learn inductively, by example, rather than deductively.
This book is for them. It consciously avoids vague formalism and whenever possible
substitutes restricted particular examples for general propositions and theorems. It is
necessary to measure, calculate, or compute in order to understand. With this in mind
IX
I have included a selection of problems intended to encourage learning and
understanding.
This book began in 1972, shortly after I began teaching in the University of
California's Graduate Department of Applied Science at Davis and Livermore. This
Department was conceived by Edward Teller in order to make available to students the
Livermore Laboratory's unparalleled experimental, computational, and theoretical
facilities. In the Department of Applied Science we teach the skills required to describe
equilibrium and nonequilibrium systems, microscopic and macroscopic, from a
fundamental viewpoint. The year-long graduate course described in the present book
evolved over a 20-year period. I was first exposed to statistical mechanics in graduate
courses at the University of Michigan, taught by De Rocco and Uhlenbeck, and built
from this base through texts I have used in teaching, the Physical Chemistry of Moore
and Adamson and the Statistical Mechanics of Reif and McQuarrie. Recently my
teaching has more closely followed my current research interest in nonequilibrium
statistical mechanics. This book reflects my own interests in achieving an
understanding of macroscopic processes, particularly nonequilibrium processes, from a
computational basis. I have included an account of equilibrium simulations and
simulation techniques too, despite my feeling that this area is by now farther from the
frontier and therefore of less interest in research.
This rewarding work has been generously supported by Universities in Canberra,
Davis, Vienna, and Yokohama, as well as by the Livermore National Laboratory. It has
been stimulated and encouraged by a host of friends around the world. Though such
research is never finished it appears to me that now is a good time to pass on some of
our accumulated present-day knowledge in order to speed future advances by those
people privileged to enjoy the pleasures of solving puzzles in physics.
Livermore and Yokohama
October 1987- June 1990.
xi
Acknowledgment
My intellectual debts are many. My main sources of inspiration were Peter
Debye, Andy De Rocco, George Duvall, Joe Ford, Toshio Kawai, Ed Saibel, George
Uhlenbeck, and Billy Waid. My research efforts have been generously supported and
encouraged by Taisuke Boku, Tony De Groot, Jean-Pierre Hansen, Gianni Iacucci,
Marvin Ross, Shuichi Nose, Irv Stowers, Bob Watts, and Fred Wooten. I shared many
discoveries with Bill Ashurst, Brad Holian, Tony Ladd, Bill Moran, Harald Posch, and
Francis Ree. Besides furnishing the conventional wifely support Carol also read the
book and worked through the problems. Without her, and my parents, Mary Wolfe
Hoover and Edgar Malone Hoover, this work could not have been. It is dedicated to
them.
1
1. Mechanics
1 Introduction; 2 Mechanical States; 3 Newtonian Mechanics; 4 Trajectory Stability; 5 Trajectory
Reversibility; 6 Stoermer and Runge-Kutta Integration; 7 Lagrangian Mechanics; 8 Least Action Principle;
9 Gauss' Principle and Nonholonomic Constraints; 10 Hamiltonian Mechanics; 11 Liouville's Theorem;
12 Mechanics of Ideal-Gas Temperature; 13 Thermostats and Nose-Hoover Mechanics; 14 Summary and
References
1.1 Introduction
Because the basis of macroscopic behavior lies in the reproducible microscopic
motion of particles governed by the simple reversible deterministic laws of mechanics,
we begin our study by exploring classical mechanics. Our fundamental motivating
desire is to find a microscopic mechanical basis for macroscopic phenomena. So we
must start with mechanics. We will illustrate all the basic concepts with simple
examples free of the obscuring shroud of mathematics.
Although Schroedinger's quantum mechanics is apparently more fundamental
than Newton's classical mechanics, quantum mechanics has not yet been developed
into a useful tool for dealing with nonequilibrium systems. We deal with quantum
systems only in those few cases where classical mechanics is useless (phonons at low
temperature, photons, electrons).
Classical mechanics has come far since Newton's 300-year-old formulation of the
basic laws. Newton's work was based on gravitational forces because Kepler's analysis
of planetary data had led to precise conclusions. But gravity is negligibly weak on the
microscopic scale. It affects atomic trajectories only at the double-precision level.
On the atomic level, a variety of short-ranged few-body force laws containing a
few adjustable parameters have been introduced to replicate the behavior of simple
gases, liquids, and solids. And statistical mechanics has been developed to link these
microscopic forces to macroscopic thermodynamic quantities, temperature, energy, and
entropy. Numerical techniques can explore and characterize the link with high
accuracy, and the stability and predictability of the solutions can be described in the
relatively new language of nonlinear dynamics and chaos.
We begin by reviewing Newton's laws of motion first and commenting on the
reversibility and stability of Newtonian trajectories. Numerical integration of the
motion equations is emphasized because it is fundamental to our subject. We then
describe Lagrange's and Gauss' formulations of the mechanics of constrained systems,
and discuss also Hamilton's mechanics, which is a necessary prerequisite to the
treatment of quantum systems and is as well useful in understanding Nose's recent
(1984) work. This last development makes it possible to incorporate temperature and
pressure into microscopic reversible equations of motion.
Nose's dynamical link between the microscopic and macroscopic points of view
recurs repeatedly, in discussing both equilibrium and nonequilibrium systems. This
work has made it possible to advance beyond Boltzmann in understanding the
connection between microscopic reversibility and macroscopic irreversibility.
2
1.2 Mechanical States
Mechanics treats the time development of mechanical systems and furnishes the
rules from which the future motion can be computed for any mechanical state. By the
term "mechanical state" of a microscopic system we mean the list of present
coordinates {r} and velocities {f = v) of the degrees of freedom. The number of such
pairs required depends on the complexity of the system described, and is called the
number of degrees of freedom.
For instance, the number of coordinates required to locate a rigid tetrahedron in
three-dimensional space is six, so that the tetrahedron has six degrees of freedom.
Notice that a "tetrahedron" made up by joining together four point masses with six stiff
springs would require 12 coordinates to locate it in three-dimensional space and 12
more velocities to completely specify its mechanical state. The constraint of making
the tetrahedron rigid removes six degrees of freedom.
For this state information to be useful we must have equations of motion capable
of predicting the future. We must know the accelerations, that is, the change of the
velocities with time. Typically the accelerations and the underlying forces present in
the equations of motion depend upon the types of the particles being described. Thus
the composition of the system, the corresponding equations of motion, as well as any
necessary boundary conditions must be given too before any calculations can be
performed. The analog of such a microscopic "mechanical state" for a macroscopic
system is the macroscopic "hydrodynamic state," in which the thermodynamic state
and velocities are specified, and from which the future behavior can be determined.
Given equations of motion, accelerations or forces, and boundary conditions,
knowing the current mechanical state makes it possible to simulate future behavior.
Mathematicians call this an "initial value problem." The initial value is the current
mechanical state. Consider a simple example. The mechanical state of a one-
dimensional harmonic oscillator is specified by giving its coordinate x, and velocity v =
x. The type of oscillator must further be specified by giving the mass m and the
force constant K. Finally either Newtonian or Lagrangian mechanics leads to the same
second-order differential equation of motion for the acceleration,
x = - (K/m)x ,
for which the solution satisfying the initial values of x and x is unique.
At a particular time t the state of a quantum mechanical system can likewise be
characterized by the real and imaginary parts of its wave function. The Schroedinger
equation then provides two first-order partial differential equations for the time-
development of the real and imaginary parts of the wave function.
3
Likewise the state of a macroscopic hydrodynamic system, given by local values of
density, velocity, and energy, can be propagated by solving the corresponding five
partial differential equations (one each for density, the three components of velocity,
and energy). In principle the partial differential equations of quantum mechanics and
continuum mechanics are superficially more complicated than the ordinary differential
equations of classical mechanics. But by averaging these partial differential equations
over spatial zones or by representing the solutions as sums of orthogonal functions,
both quantum systems and hydrodynamic systems can likewise be converted into sets
of ordinary differential equations.
The mechanical state of a system can be followed in time either by following a
trajectory x(t) in coordinate space or in phase space, where both the coordinates {q} and
the momenta {p} are simultaneously specified. Individual coordinate-momenta points
in phase space give the complete dynamical state. In coordinate space the dynamical
state requires also velocity, the rate at which the trajectory is traced out.
It is worthwhile to emphasize that the accuracy of any numerical solution is
strictly limited, so that all of our computer-generated solutions are necessarily
approximate. We will see that the typical Lyapunov instability of the equations of
motion rapidly and relentlessly destroys computational accuracy, so that the expense of
high accuracy is not feasible for long calculations. Generally eight-digit or twelve-digit
single-precision accuracy is sufficient. So long as we are interested in reproducing
averaged macroscopic behavior there is no evidence that this limited accuracy has
practical consequences. If such evidence were to be found it would indicate a new and
interesting underlying law of physics.
1.3 Newtonian Mechanics
Newton's mechanics furnishes a second-order
differential-equation description of the time
development of particle coordinates through the
coordinate-dependent Forces {F(r)}. The forces depend
upon the locations {r} of the masses {m} and the
boundary conditions, with the boundary contribution to
the forces usually specified as a time-varying function of
the coordinates:
mr = F(r) + F (r,t) .
Boundary
The velocity-dependent Lorentz Forces which prescribe
the motion of charged particles in magnetic fields depenc