De Gruyter Studies in Mathematical Physics 18 Editors Michael Efroimsky, Bethesda, USA Leonard Gamberg, Reading, USA Dmitry Gitman, São Paulo, Brasil Alexander Lazarian, Madison, USA Boris Smirnov, Moscow, Russia Michael V. Sadovskii Statistical Physics De Gruyter (cid:51)(cid:75)(cid:92)(cid:86)(cid:76)(cid:70)(cid:86)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:36)(cid:86)(cid:87)(cid:85)(cid:82)(cid:81)(cid:82)(cid:80)(cid:92)(cid:3)(cid:38)(cid:79)(cid:68)(cid:86)(cid:86)(cid:76)(cid:191)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:21)(cid:19)(cid:20)(cid:19)(cid:29)(cid:3)05.20.-y; 05.20.Dd; 05.20.Gg; 05.30.-d; 05.30.Ch; 05.30.Fk; 05.30.Pr; 05.70.Ph; 68.18.Jk; 68.18.Ph ISBN 978-3-11-027031-0 e-ISBN 978-3-11-027037-2 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek (cid:55)(cid:75)(cid:72)(cid:3)(cid:39)(cid:72)(cid:88)(cid:87)(cid:86)(cid:70)(cid:75)(cid:72)(cid:3)(cid:49)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:69)(cid:76)(cid:69)(cid:79)(cid:76)(cid:82)(cid:87)(cid:75)(cid:72)(cid:78)(cid:3)(cid:79)(cid:76)(cid:86)(cid:87)(cid:86)(cid:3)(cid:87)(cid:75)(cid:76)(cid:86)(cid:3)(cid:83)(cid:88)(cid:69)(cid:79)(cid:76)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:76)(cid:81)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:39)(cid:72)(cid:88)(cid:87)(cid:86)(cid:70)(cid:75)(cid:72)(cid:3)(cid:49)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:69)(cid:76)(cid:69)(cid:79)(cid:76)(cid:82)(cid:74)(cid:85)(cid:68)(cid:191)(cid:72)(cid:30)(cid:3) detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen (cid:136)Printed on acid-free paper Printed in Germany www.degruyter.com Preface This book is essentially based on the lecture course on “Statistical Physics”, which wastaughtbytheauthoratthephysicalfacultyoftheUralStateUniversityinEkater- inburg since 1992. This course was intended for all physics students, not especially forthosespecializingintheoreticalphysics. Inthissensethematerialpresentedhere containsthenecessaryminimumofknowledgeofstatisticalphysics(alsooftencalled statisticalmechanics), whichis inauthor’s opinion necessaryfor everypersonwish- ingtoobtainageneraleducationinthefieldofphysics. Thisposedtheratherdifficult problemofthechoiceofmaterialandcompactenoughpresentation. Atthesametime itnecessarilyshouldcontainallthebasicprinciplesofstatisticalphysics,aswellasits main applicationstodifferentphysicalproblems, mainlyfromthefieldof thetheory of condensed matter. Extended version of these lectures were published in Russian in2003. ForthepresentEnglishedition,someofthematerialwasrewrittenandsev- eralnewsectionsandparagraphswereadded,bringing contentsmore uptodateand addingmorediscussiononsomemoredifficultcases.Ofcourse,theauthorwasmuch influencedbyseveralclassicalbooksonstatisticalphysics[1,2,3],andthisinfluence isobviousinmanypartsofthetext. However,thechoiceofmaterialandtheformof presentationisessentiallyhisown. Still,mostattentionisdevotedtorathertraditional problems and models of statisticalphysics. One of the few exceptions is an attempt to present an elementary and short introduction to the modern quantum theoretical methods of statisticalphysics atthe endof thebook. Also,alittle bitmore attention than usual is given to the problems of nonequilibrium statistical mechanics. Some of the more special paragraphs, of more interest to future theorists, are denoted by asterisksormovedtoAppendices. Ofcourse,thisbookistooshorttogiveacomplete presentation of modern statisticalphysics. Those interestedin further developments shouldaddressmorefundamentalmonographsandmodernphysicalliterature. Ekaterinburg,2012 M.V.Sadovskii Contents Preface v 1 Basicprinciplesofstatistics 1 1.1 Introduction .............................................. 2 1.2 Distributionfunctions ...................................... 2 1.3 Statisticalindependence .................................... 7 1.4 Liouvilletheorem ......................................... 9 1.5 Roleofenergy,microcanonicaldistribution ..................... 13 (cid:2) 1.6 Partialdistributionfunctions ................................ 17 1.7 Densitymatrix ............................................ 21 1.7.1 Pureensemble ...................................... 22 1.7.2 Mixedensemble .................................... 24 1.8 QuantumLiouvilleequation ................................. 26 1.9 Microcanonicaldistributioninquantumstatistics ................. 28 (cid:2) 1.10 Partialdensitymatrices .................................... 29 1.11 Entropy ................................................. 32 1.11.1 Gibbsentropy. Entropyandprobability .................. 32 1.11.2 Thelawofentropygrowth ............................ 35 2 Gibbsdistribution 43 2.1 Canonicaldistribution ...................................... 43 2.2 Maxwelldistribution ....................................... 48 2.3 FreeenergyfromGibbsdistribution ........................... 50 2.4 Gibbsdistributionforsystemswithvaryingnumberofparticles ..... 52 2.5 ThermodynamicrelationsfromGibbsdistribution ................ 55 3 Classicalidealgas 60 3.1 Boltzmanndistribution ..................................... 60 3.2 Boltzmanndistributionandclassicalstatistics ................... 61 3.3 Nonequilibriumidealgas.................................... 63 3.4 FreeenergyofBoltzmanngas ................................ 66 viii Contents 3.5 EquationofstateofBoltzmanngas ............................ 67 3.6 Idealgaswithconstantspecificheat ........................... 69 3.7 Equipartitiontheorem ...................................... 70 3.8 One-atomidealgas ........................................ 72 4 Quantumidealgases 75 4.1 Fermidistribution ......................................... 75 4.2 Bosedistribution .......................................... 76 4.3 NonequilibriumFermiandBosegases ......................... 77 4.4 GeneralpropertiesofFermiandBosegases ..................... 79 4.5 Degenerategasofelectrons .................................. 82 (cid:2) 4.6 Relativisticdegenerateelectrongas ........................... 85 4.7 Specificheatofadegenerateelectrongas ....................... 86 4.8 Magnetismofanelectrongasinweakfields ..................... 88 (cid:2) 4.9 Magnetismofanelectrongasinhighfields .................... 92 4.10 DegenerateBosegas ....................................... 94 4.11 Statisticsofphotons ........................................ 97 5 Condensedmatter 101 5.1 Solidstateatlowtemperature ................................ 101 5.2 Solidstateathightemperature ............................... 104 5.3 Debyetheory ............................................. 105 5.4 QuantumBoseliquid ....................................... 109 5.5 Superfluidity ............................................. 113 (cid:2) 5.6 PhononsinaBoseliquid ................................... 117 5.7 DegenerateinteractingBosegas .............................. 121 5.8 Fermiliquids ............................................. 124 (cid:2) 5.9 Electronliquidinmetals ................................... 130 6 Superconductivity 133 6.1 Cooperinstability ......................................... 133 6.2 Energyspectrumofsuperconductors........................... 135 6.3 Thermodynamicsofsuperconductors .......................... 144 (cid:2) 6.4 Coulombrepulsion ....................................... 148 6.5 Ginzburg–Landautheory .................................... 151 Contents ix 7 Fluctuations 160 7.1 Gaussiandistribution ....................................... 160 7.2 Fluctuationsinbasicphysicalproperties ........................ 164 7.3 Fluctuationsinidealgases ................................... 167 8 Phasetransitionsandcriticalphenomena 170 8.1 Mean-fieldtheoryofmagnetism .............................. 170 (cid:2) 8.2 Quasi-averages ........................................... 177 8.3 Fluctuationsintheorderparameter ............................ 180 8.4 Scaling .................................................. 186 9 Linearresponse 195 9.1 Linearresponsetomechanicalperturbation ..................... 195 9.2 Electricalconductivityandmagneticsusceptibility ............... 201 9.3 Dispersionrelations ........................................ 204 10 Kineticequations 208 10.1 Boltzmannequation ........................................ 208 10.2 H-theorem ............................................... 214 (cid:2) 10.3 Quantumkineticequations ................................. 216 10.3.1 Electron-phononinteraction ........................... 218 10.3.2 Electron–electroninteraction .......................... 222 11 Basicsofthemoderntheoryofmany-particlesystems 225 11.1 QuasiparticlesandGreen’sfunctions .......................... 225 11.2 Feynmandiagramsformany-particlesystems ................... 233 11.3 Dysonequation ........................................... 237 11.4 Effectiveinteractionanddielectricscreening .................... 241 11.5 Green’sfunctionsatfinitetemperatures ........................ 244 A Motioninphasespace,ergodicityandmixing 248 A.1 Ergodicity ............................................... 248 A.2 Poincarerecurrencetheorem ................................. 254 A.3 Instabilityoftrajectoriesandmixing ........................... 256 B Statisticalmechanicsandinformationtheory 259 B.1 RelationbetweenGibbsdistributionsandtheprincipleofmaximal informationentropy ........................................ 259 B.2 PurgingMaxwell’s“demon” ................................. 263