Methods of statistical physics

310 Pages·2002·1.925 MB·English

by  Tomoyasu Tanaka

METHODS OF STATISTICAL PHYSICS
Thisgraduate-leveltextbookonthermalphysicscoversclassicalthermodynamics,
statistical mechanics, and their applications. It describes theoretical methods to
calculate thermodynamic properties, such as the equation of state, specific heat,
Helmholtzpotential,magneticsusceptibility,andphasetransitionsofmacroscopic
systems.
Inadditiontothemorestandardmaterialcovered,thisbookalsodescribesmore
powerful techniques, which are not found elsewhere, to determine the correlation
effects on which the thermodynamic properties are based. Particular emphasis is
giventotheclustervariationmethod,andanovelformulationisdevelopedforits
expressionintermsofcorrelationfunctions.Applicationsofthismethodtotopics
suchasthethree-dimensionalIsingmodel,BCSsuperconductivity,theHeisenberg
ferromagnet, the ground state energy of the Anderson model, antiferromagnetism
within the Hubbard model, and propagation of short range order, are extensively
discussed.ImportantidentitiesrelatingdifferentcorrelationfunctionsoftheIsing
modelarealsoderived.
Although a basic knowledge of quantum mechanics is required, the mathe-
matical formulation is accessible, and the correlation functions can be evaluated
either numerically or analytically in the form of infinite series. Based on courses
in statistical mechanics and condensed matter theory taught by the author in the
UnitedStatesandJapan,thisbookisentirelyself-containedandallessentialmath-
ematicaldetailsareincluded.Itwillconstituteanidealcompaniontextforgraduate
studentsstudyingcoursesonthetheoryofcomplexanalysis,classicalmechanics,
classical electrodynamics, and quantum mechanics. Supplementary material is
alsoavailableontheinternetathttp://uk.cambridge.org/resources/0521580560/
TOMOYASU TANAKA obtainedhisDoctorofSciencedegreeinphysicsin1953
from the Kyushu University, Fukuoka, Japan. Since then he has divided his time
betweentheUnitedStatesandJapan,andiscurrentlyProfessorEmeritusofPhysics
and Astronomy at Ohio University (Athens, USA) and also at Chubu University
(Kasugai,Japan).Heistheauthorofover70researchpapersonthetwo-timeGreen’s
function theory of the Heisenberg ferromagnet, exact linear identities of the Ising
model correlation functions, the theory of super-ionic conduction, and the theory
ofmetalhydrides.ProfessorTanakahasalsoworkedextensivelyondevelopingthe
clustervariationmethodforcalculatingvariousmany-bodycorrelationfunctions.

METHODS OF
STATISTICAL PHYSICS
TOMOYASU TANAKA
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
 
The Edinburgh Building, Cambridge , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521580564
© Tomoyasu Tanaka 2002
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2002
- ----
eBook (NetLibrary)
- ---
eBook (NetLibrary)
- ----
hardback
- ---
hardback
- ----
paperback
- ---
paperback
Cambridge University Press has no responsibility for the persistence or accuracy of

s for external or third-party internet websites referred to in this book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
TothelateProfessorAkiraHarasima

Contents
Preface pagexi
Acknowledgements xv
1 Thelawsofthermodynamics 1
1.1 Thethermodynamicsystemandprocesses 1
1.2 Thezerothlawofthermodynamics 1
1.3 Thethermalequationofstate 2
1.4 Theclassicalidealgas 4
1.5 Thequasistaticandreversibleprocesses 7
1.6 Thefirstlawofthermodynamics 7
1.7 Theheatcapacity 8
1.8 Theisothermalandadiabaticprocesses 10
1.9 Theenthalpy 12
1.10 Thesecondlawofthermodynamics 12
1.11 TheCarnotcycle 14
1.12 Thethermodynamictemperature 15
1.13 TheCarnotcycleofanidealgas 19
1.14 TheClausiusinequality 22
1.15 Theentropy 24
1.16 Generalintegratingfactors 26
1.17 Theintegratingfactorandcyclicprocesses 28
1.18 Hausen’scycle 30
1.19 Employmentofthesecondlawofthermodynamics 31
1.20 Theuniversalintegratingfactor 32
Exercises 34
2 Thermodynamicrelations 38
2.1 Thermodynamicpotentials 38
2.2 Maxwellrelations 41
vii
Contents
Preface pagexi
Acknowledgements xv
1 Thelawsofthermodynamics 1
1.1 Thethermodynamicsystemandprocesses 1
1.2 Thezerothlawofthermodynamics 1
1.3 Thethermalequationofstate 2
1.4 Theclassicalidealgas 4
1.5 Thequasistaticandreversibleprocesses 7
1.6 Thefirstlawofthermodynamics 7
1.7 Theheatcapacity 8
1.8 Theisothermalandadiabaticprocesses 10
1.9 Theenthalpy 12
1.10 Thesecondlawofthermodynamics 12
1.11 TheCarnotcycle 14
1.12 Thethermodynamictemperature 15
1.13 TheCarnotcycleofanidealgas 19
1.14 TheClausiusinequality 22
1.15 Theentropy 24
1.16 Generalintegratingfactors 26
1.17 Theintegratingfactorandcyclicprocesses 28
1.18 Hausen’scycle 30
1.19 Employmentofthesecondlawofthermodynamics 31
1.20 Theuniversalintegratingfactor 32
Exercises 34
2 Thermodynamicrelations 38
2.1 Thermodynamicpotentials 38
2.2 Maxwellrelations 41
vii
Contents ix
5.6 Thefour-sitereduceddensitymatrix 114
5.7 TheprobabilitydistributionfunctionsfortheIsingmodel 121
Exercises 125
6 Theclustervariationmethod 127
6.1 Thevariationalprinciple 127
6.2 Thecumulantexpansion 128
6.3 Theclustervariationmethod 130
6.4 Themean-fieldapproximation 131
6.5 TheBetheapproximation 134
6.6 Four-siteapproximation 137
6.7 Simplifiedclustervariationmethods 141
6.8 Correlationfunctionformulation 144
6.9 ThepointandpairapproximationsintheCFF 145
6.10 ThetetrahedronapproximationintheCFF 147
Exercises 152
7 Infinite-seriesrepresentationsofcorrelationfunctions 153
7.1 Singularityofthecorrelationfunctions 153
7.2 Theclassicalvaluesofthecriticalexponent 154
7.3 Aninfinite-seriesrepresentationofthepartitionfunction 156
7.4 ThemethodofPade´ approximants 158
7.5 Infinite-seriessolutionsoftheclustervariationmethod 161
7.6 Hightemperaturespecificheat 165
7.7 Hightemperaturesusceptibility 167
7.8 Lowtemperaturespecificheat 169
7.9 Infiniteseriesforothercorrelationfunctions 172
Exercises 173
8 Theextendedmean-fieldapproximation 175
8.1 TheWentzelcriterion 175
8.2 TheBCSHamiltonian 178
8.3 Thes–d interaction 184
8.4 ThegroundstateoftheAndersonmodel 190
8.5 TheHubbardmodel 197
8.6 Thefirst-ordertransitionincubicice 203
Exercises 209
9 TheexactIsinglatticeidentities 212
9.1 Thebasicgeneratingequations 212
9.2 Linearidentitiesforodd-numbercorrelations 213
9.3 Star-triangle-typerelationships 216
9.4 Exactsolutiononthetriangularlattice 218