Non-equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications

479 Pages·2012·3.848 MB·English

by  Attard P.

Non-Equilibrium Thermodynamics and Statistical Mechanics
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Non-Equilibrium Thermodynamics and
Statistical Mechanics
Foundations and Applications
Phil Attard
1
3
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Preface
Fate, Time, Occasion, Chance, and Change,
To these all things are subject
P. B. Shelley, ‘Prometheus Unbound’ (1820)
All things change in time
Motionandchangearethenatureoftheworld. Thescaleofmotionrangesfrom
the orbit of galaxies and planets, to the winds, ocean currents, and river flows,
and even to the sub-microscopic, with the ceaseless movement of atoms and
molecules. Inthelivingworldthereisthelocomotionofanimals,themovement
of limbs and muscles, and the flows of blood, sap, and other vital fluids. There
is also human made motion in the form of transport by planes, trains, and
automobiles,the mechanicalmovement ofmotorisedtools, and controlledflows
such as those of electricity, heat, and fluids.
There are as well other forms of change. Birth, growth, and death is a
familiar trajectory for any individual living organism. But these also describe
the evolution of a species as a whole, as well as that of the universe, planets,
eco-systems, and even social structures. One can also include other physical
phenomena that are the subject of more traditional scientific study such as the
progress of chemical reactions, the development of heat and current fluxes, the
formation of patterns by aerodynamic and hydrodynamic flows, the processes
of self-assembly and physical organisation, and the dynamic deformation and
response of materials. These of course occur in nature, industry, technology,
and under controlled laboratory conditions.
The roˆleofchanceinourworldshouldnotbeunderestimated. Whenthings
occur, their exact trajectory, and their ultimate fate are not perfectly pre-
dictable. In general the complexity of systems, unknown initial conditions,
and the influence of uncontrolled external forces all contribute a degree of ran-
domness that leads to an uncertain future unconstrained by a strict fatalism.
Chancemeansthatchangesintimeoccurnotwithpuredeterminismbutrather
with statistical probability.
v
vi Preface
Time changes everything
The point belabouredaboveis that time dependent phenomena areubiquitous.
Thetechnicalwordfortheseis‘non-equilibrium’. Itisareflectionofthechrono-
logical development of science that this, the most common class of systems, is
described in the negative.
Initially,thermodynamicsandstatisticalmechanicsweredevelopedforequi-
librium systems, which do not change macroscopically with time. Thermody-
namicsisthescienceofmacroscopicsystems,anditprovidesuniversallawsand
relationships that all equilibrium systems must obey. Statistical mechanics en-
ablestheprobabilisticdescriptionofequilibriumsystemsatthemolecularlevel.
It gives the mathematical basis for the empirical laws of thermodynamics and
it provides quantitative values for measured thermodynamic parameters.
It is one of the great ironies of science that the word ‘dynamics’ in ‘ther-
modynamics’, and the word ‘mechanics’ in ‘statistical mechanics’, both imply
motion, when in fact both disciplines have been strictly formulated for static
or equilibrium systems. Of course as an approximation they are often applied
to time dependent systems, either instantaneously or else over time intervals
small enough that any change is negligible, or they can be combined with an
empirical theory such as hydrodynamics. But in terms of an exact treatment,
thermodynamicsandstatisticalmechanicsarerestrictedtoequilibriumsystems.
Thisraisesthequestion: Howdoestimechangethermodynamicsandstatistical
mechanics?
This book
Thisbookseekstoanswerthatdisarminglysimple question. Acoherentformu-
lationofnon-equilibriumthermodynamicsisgiven. Theapproachisbasedupon
aparticularformofentropy,andithastheadvantagethatalmostallofthecon-
cepts ofequilibriumthermodynamicscarryoverto the non-equilibriumfield. It
also enables a consistent derivation of most of the known non-equilibrium the-
orems and results, which exhibits their inter-relationships and places them in
the context of a bigger picture.
Thenon-equilibriumprobabilitydistributionisalsodeveloped,andthispro-
vides a basis for the field of non-equilibrium statistical mechanics. Again, this
enables a unified derivation of known and previously unknown theorems. Im-
portantly, it also enables the development of computer simulation algorithms
for non-equilibrium systems, which are used to test quantitatively the results
and to illustrate them at the molecular level.
Because of the significance of time dependent phenomena, there are many
books and scientific papers concerned with the formulation of non-equilibrium
thermodynamics and non-equilibrium statistical mechanics, and with their ap-
plication to specific systems. The selection of topics, underlying approach,and
method ofpresentationvary enormously,althoughcertainnon-equilibriumthe-
orems and results for which there is broad consensus commonly recur. Others
results lie at the cutting edge of current research,and for these detailed justifi-
Preface vii
cation and explanation are required.
As mentioned above, this book proceeds from the very fundamental prin-
ciples that determine the optimum non-equilibrium thermodynamic state, and
also from the equations of motion and probability distributions appropriate for
non-equilibriumstatisticalmechanics. The strategy employedhere is to set out
thephysicalbasisoftheaxioms,thecloseanalogybetweennon-equilibriumand
equilibriumprinciples,and,mostimportantly,thetheoremsanddetailedresults
that follow as a consequence. In general an attempt is made to provide quanti-
tative tests, experimental or computational, and detailed comparisons between
different approaches, and alternative, independent derivations of the same re-
sult. Itishopedthatsuchconcreteevidenceandtheconsistencyoftheapproach
will give some confidence in the fundamental principles that the book is based
upon.
In the present book, most of the traditional topics in the non-equilibrium
field are covered, and some new ones besides. What is perhaps unique here
is that a single underlying approach suffices to derive and to describe all these
results. Thefieldsofnon-equilibriumthermodynamicsandnon-equilibriumsta-
tisticalmechanicsarehereregardedasacontinuumthatrangesfromthemacro-
scopic to the sub-microscopic, with Brownian motion and stochastic processes
lying in the boundary region where they merge.
In a sense, this book is one long argument for non-equilibrium thermody-
namicsandstatisticalmechanics. Thereareseveralreasonswhythereadermay
find the present approach useful and may have confidence in the results. First,
is the simplicity of the concepts, examples, and equations. Stripping away all
that is unnecessary removes the possibility of confusion masquerading as com-
plexity, and displays the results in a clear and unambiguous light. Second, is
the physical basis of the approach. Thermodynamics and statistical mechanics
are derived from, and designed for the real world, and here is emphasised the
physical basis and interpretation of all the terms that occur in each equation.
This removes the likelihood of inadvertent non-physical behaviour due to arti-
ficial assumptions, it gives an intuitive feel to the equations and results, and it
enablesthecommonsensetesttobereadilyapplied. Third,isthecoherenceand
self-consistencyoftheapproach. Thosetheoremsandresultsinnon-equilibrium
thermodynamics and statistical mechanics that are widely accepted are all de-
rivedherefromasingleapproachbasedonentropy. This consiliencegivessome
confidence in both the approach itself and the new results also generated by
it. Fourth, the results of a number of computer simulations are given in the
text, both to illustrate the procedures and to test quantitatively the results. In
addition, certain experimental measurements are used, again quantitatively, to
test predictions of the theory. Such tests should prove convincing, both of the
individual results and of the formulation as a whole.
The fields of thermodynamics and statistical mechanics have grown over
the years. This book is part of that evolution; it is intended to be timely
rather than timeless. The principles for non-equilibrium thermodynamics and
statistical mechanics set out herein consolidate the present state of knowledge
and provide a basis for future growth and new applications.
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Contents
1 Prologue ..............................................................1
2 Fluctuation Theory ..................................................33
3 Brownian Motion ....................................................61
4 Heat Conduction ....................................................97
5 Second Entropy for Fluctuating Hydrodynamics ....................121
6 Heat Convection and Non-Equilibrium Phase Transitions ...........145
7 Equilibrium Statistical Mechanics ..................................173
8 Non-Equilibrium Statistical Mechanics ..............................233
9 Statistical Mechanics of Steady Flow: Heat and Shear ..............295
10 Generalised Langevin Equation .....................................329
11 Non-Equilibrium Computer Simulation Algorithms .................389
References ..........................................................451
Index ...............................................................455
ix