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STATISTICAL MECHANICS PDF

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Giovanni Gal lavotti ST A TISTICAL MECHANICS Short T reatise R oma ���� � Short tr e atise of Statistic al Me chanics Gio v anni Galla v otti Dipartimen to di Fisica Univ ersit � a di Roma L a Sapienza ����� Roma Roma ���� � i A Daniela� sempr e Preface This b o ok is the end result of a long story that started with m y in v olv emen t as Co ordinator of the Statistical Mec hanics section of the Italian Encyclo� p edia of Ph ysics� An Italian edition collecting sev eral pap ers that I wrote for the Encyclop e� dia app eared in Septem b er ����� with the p ermission of the Encyclop edia and the sp onsorship of Consiglio Nazionale delle Ricerc he �CNR�GNFM�� The presen t w ork is not a translation of the Italian v ersion but it o v erlaps with it� an imp ortan t part of it �Ch�I�I I�I I I�VI I I� is still based on three arti� cles written as en tries for the �it Encicopledia della Fisica �namely� �Me c� c anic a Statistic a�� �T e oria de gli Insiemi� and �Moto Br owniano�� whic h mak e up ab out ��� of the presen t b o ok and� furthermore� it still con tains �with little editing and up dating� m y old review article on phase transitions �Ch�VI� published in La Rivista del Nuo v o Cimen to�� In translating the ideas in to English� I in tro duced man y revisions and c hanges of p ersp ectiv e as w ell as new material �while also suppressing some other material�� The aim w as to pro vide an analysis� in ten tionally as non tec hnical as I w as able to mak e it� of man y fundamen tal questions of Statistical Mec hanics� ab out t w o cen turies after its birth� Only in a v ery few places ha v e I en� tered in to really tec hnical details� mainly on sub jects that I should kno w rather w ell or that I consider particularly imp ortan t �the con v ergence of the Kirkw o o d�Salsburg equations� the existence of the thermo dynamic limit� the exact soltution of the Ising mo del� and in part the exact solution of the six v ertex mo dels�� The p oin ts of view expressed here w ere presen ted in inn umerable lectures and talks mostly to m y studen ts in Roma during the last �� y ears� They are not alw a ys �mainstream views�� but I am con�den t that they are not to o far from the con v en tionally accepted �truth� and I do not consider it appropriate to list the di�erences from other treatmen ts� I shall consider this b o ok a success if it prompts commen ts �ev en if dictated b y strong disagreemen t or dissatisfaction� on the �few� p oin ts that migh t b e con tro v ersial� This w ould mean that the w ork has attained the goal of b eing noticed and of b eing w orth y of criticism� I hop e that this w ork migh t b e useful to studen ts b y bringing to their at� ten tion problems whic h� b ecause of �c oncr eteness ne c essities� �i�e� b ecause suc h matters seem useless� or sometimes simply b ecause of lack of time�� are usually neglected ev en in graduate courses� This do es not mean that I in tend to encourage studen ts to lo ok at questions dealing with the foundations of Ph ysics� I rather b eliev e that y oung studen ts should r efr ain from suc h activities� whic h should� p ossibly � b ecome a sub ject ii of in v estigation after gaining an exp erience that only activ e and adv anced researc h can pro vide �or at least the attempt at pursuing it o v er man y y ears�� And in an y ev en t I hop e that the con ten ts and the argumen ts I ha v e selected will con v ey m y appreciation for studies on the foundations that k eep a strong c haracter of concreteness� I hop e� in fact� that this b o ok will b e considered concrete and far from sp eculativ e� Not that studen ts should not dev elop their o wn philosophic al b eliefs ab out the problems of the area of Ph ysics that in terests them� Although one should b e a w are that an y philosophical b elief on the foundations of Ph ysics �and Science�� no matter ho w clear and irrefutable it migh t app ear to the p erson who dev elop ed it after long meditations and unending vigils� is v ery unlik ely to lo ok less than ob jectionable to an y other p erson who is giv en a c hance to think ab out it� it is nev ertheless necessary � in order to gro w original ideas or ev en to just p erform w ork of go o d tec hnical qualit y � to p ossess precise philosophical con victions on the r erum natur a� Pro vided one is alw a ys willing to start afresh� a v oiding� ab o v e all� thinking one has �nally reac hed the truth� unique� unchange able and obje ctive �in to whose existence only vain hop e can b e laid�� I am grateful to the Enciclop e dia Italiana for ha ving stim ulated the b egin� ning and the realization of this w ork� b y assigning me the task of co ordinat� ing the Statistical Mec hanics pap ers� I w an t to stress that the �nancial and cultural supp ort from the Enciclop e dia ha v e b een of in v aluable aid� The atmosphere created b y the Editors and b y m y colleagues in the few ro oms of their facilities stim ulated me deeply � It is imp ortan t to remark on the rather un usual editorial en terprise they led to� it w as not immediately an� imated b y the logic of pro�t that mo v es the scien ti�c b o ok industry whic h is v ery concerned� at the same time� to a v oid p ossible costly risks� I w an t to thank G� Alippi� G� Altarelli� P � Dominici and V� Capp elletti who made a �rst v ersion in Italian p ossible� mainly con taining the Encyclop edia articles� b y allo wing the collection and repro duction of the texts of whic h the Encyclop edia retains the righ ts� I am indebted to V� Capp elletti for gran ting p ermission to include here the three en tries I wrote for the Enciclop e dia del le Scienze Fisiche �whic h is no w published�� I also thank the Nuo v o Cimen to for allo wing the use of the ���� review pap er on the Ising mo del� I am indebted for critical commen ts on the v arious drafts of the w ork� in particular� to G� Gen tile whose commen ts ha v e b een an essen tial con� tribution to the revision of the man uscript� I am also indebted to sev eral colleagues� P � Carta� E� J� arv enp� a� a� N� Nottingham and� furthermore� M� Campanino� V� Mastropietro� H� Sp ohn whose in v aluable commen ts made the b o ok more readable than it w ould otherwise ha v e b een� Gio v anni Galla v otti Roma� Jan uary ���� iii Index I � Classical Statistical Mec hanics � � � � � � � � � � � � � � � � � � ��� In tro duction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� Microscopic Dynamics � � � � � � � � � � � � � � � � � � � � � � � � � � ��� Time Av erages and the Ergo dic Hyp othesis � � � � � � � � � � � � � � � �� ��� Recurrence Times and Macroscopic Observ ables � � � � � � � � � � � � � � �� ��� Statistical Ensem bles or �Mono des� and Mo dels of Thermo dynamics� Thermo dy� namics without Dynamics � � � � � � � � � � � � � � � � � � � � � � � � � �� ��� Mo dels of Thermo dynamics� Micro canonical and Canonical Ensem bles and the Er� go dic Hyp othesis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� ��� Critique of the Ergo dic Hyp othesis � � � � � � � � � � � � � � � � � � � �� ��� Approac h to Equilibrium and Boltzmann�s Equation� Ergo dicit y and Irrev ersibilit y �� ��� A Historical Note� The Et ymology of the W ord �Ergo dic� and the Heat Theorems �� App endix ��A�� Mono cyclic Systems� Keplerian Motions and Ergo dic Hyp othesis � �� App endix ��A�� Grad�Boltzmann Limit and Loren tz�s Gas � � � � � � � � � � �� I I � Statistical Ensem bles � � � � � � � � � � � � � � � � � � � � � � �� ��� Statistical Ensem bles as Mo dels of Thermo dynamics� � � � � � � � � � � � �� ��� Canonical and Micro canonical Ensem bles� Ortho dicit y � � � � � � � � � � � �� ��� Equiv alence b et w een Canonical and Micro canonical Ensem bles � � � � � � � �� ���� Non Equiv alence of the Canonical and Micro canonical Ensem bles� Phase T ransitions� Boltzmann�s Constan t � � � � � � � � � � � � � � � � � � � � � � � � � � �� ��� The Grand Canonical Ensem ble and Other Ortho dic Ensem bles � � � � � � � �� ��� Some T ec hnical Asp ects � � � � � � � � � � � � � � � � � � � � � � � � �� iv I I I � Equipartition and Critique � � � � � � � � � � � � � � � � � � �� ���� Equipartition and Other P arado xes and Applications of Statistical Mec hanics �� ��� Classical Statistical Mec hanics when Cell Sizes Are Not Negligible � � � � � � �� ��� In tro duction to Quan tum Statistical Mec hanics � � � � � � � � � � � � � ��� ��� Philosophical Outlo ok on the F oundations of Statistical Mec hanics � � � � ��� I V � Thermo dynamic Limit and Stabilit y � � � � � � � � � � � � ��� ���� The Meaning of the Stabilit y Conditions � � � � � � � � � � � � � � � ��� ���� Stabilit y Criteria � � � � � � � � � � � � � � � � � � � � � � � � � � ��� ��� Thermo dynamic Limit � � � � � � � � � � � � � � � � � � � � � � � � ��� V � Phase T ransitions � � � � � � � � � � � � � � � � � � � � � � � ��� ���� Virial Theorem� Virial Series and v an der W aals Equation � � � � � � � � ��� ���� The Mo dern In terpretation of v an der W aals� Appro ximation � � � � � � ��� ���� Wh y a Thermo dynamic F ormalism� � � � � � � � � � � � � � � � � � ��� ���� Phase Space in In�nite V olume and Probabilit y Distributions on it� Gibbs Distribu� tions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� ���� V ariational Characterization of T ranslation In v arian t Gibbs Distributions � ��� ���� Other Characterizations of Gibbs Distributions� The DLR Equations � � � ��� ���� Gibbs Distributions and Sto c hastic Pro cesses � � � � � � � � � � � � � ��� ���� Absence of Phase T ransitions� d � �� Symmetries� d � � � � � � � � � � ��� ���� Absence of Phase T ransitions� High T emp erature and the KS Equations � � ��� ����� Phase T ransitions and Mo dels � � � � � � � � � � � � � � � � � � � ��� App endix ��A�� Absence of Phase T ransition in non Nearest Neigh b or One�Dimensional Systems � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� V I � Co existence of Phases � � � � � � � � � � � � � � � � � � � � ��� ���� The Ising Mo del� Inequiv alence of Canonical and Grand Canonical Ensem bles ��� ���� The Mo del� Grand Canonical and Canonical Ensem bles� Their Inequiv alence ��� ���� Boundary Conditions� Equilibrium States � � � � � � � � � � � � � � � ��� ���� The Ising Mo del in One and Tw o dimensions and zero �eld � � � � � � � ��� ��� Phase T ransitions� De�nitions � � � � � � � � � � � � � � � � � � � � ��� ���� Geometric Description of the Spin Con�gurations � � � � � � � � � � � ��� ���� Phase T ransitions� Existence � � � � � � � � � � � � � � � � � � � � � ��� ���� Microscopic Description of the Pure Phases � � � � � � � � � � � � � � ��� ���� Results on Phase T ransitions in a Wider Range of T emp erature � � � � � ��� v ����� Separation and Co existence of Pure Phases� Phenomenological Considerations ��� ����� Separation and Co existence of Phases� Results � � � � � � � � � � � � ��� ����� Surface T ension in Tw o Dimensions� Alternativ e Description of the Separation Phenomena � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� ����� The Structure of the Line of Separation� What a Straigh t Line Really is � ��� ����� Phase Separation Phenomena and Boundary Conditions� F urther Results � ��� ����� F urther Results� Some Commen ts and Some Op en Problems � � � � � � ��� V I I � Exactly Soluble Mo dels � � � � � � � � � � � � � � � � � � � ��� ���� T ransfer Matrix in the Ising Mo del� Results in d � �� � � � � � � � � � � ��� ���� Meaning of Exact Solubilit y and the Tw o�Dimensional Ising Mo del � � � � ��� ���� V ertex Mo dels � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� ���� A Non trivial Example of Exact Solution� the Tw o�Dimensional Ising Mo del ��� ���� The Six V ertex Mo del and Bethe�s Ansatz � � � � � � � � � � � � � � � ��� V I I I � Bro wnian Motion � � � � � � � � � � � � � � � � � � � � � � ��� ���� Bro wnian Motion and Einstein�s Theory � � � � � � � � � � � � � � � � ��� ���� Smoluc ho wski�s Theory � � � � � � � � � � � � � � � � � � � � � � � � ��� ���� The Uhlen b ec k�Ornstein Theory � � � � � � � � � � � � � � � � � � � ��� ���� Wiener�s Theory � � � � � � � � � � � � � � � � � � � � � � � � � � ��� I X � Coarse Graining and Nonequilibrium � � � � � � � � � � � ��� ���� Ergo dic Hyp othesis Revisited � � � � � � � � � � � � � � � � � � � � ��� ���� Timed Observ ations and Discrete Time � � � � � � � � � � � � � � � � ��� ���� Chaotic Hyp othesis� Anoso v Systems � � � � � � � � � � � � � � � � � ��� ���� Kinematics of Chaotic Motions� Anoso v Systems � � � � � � � � � � � � ��� ���� Sym b olic Dynamics and Chaos � � � � � � � � � � � � � � � � � � � � ��� ���� Statistics of Chaotic A ttractors� SRB Distributions � � � � � � � � � � � ��� ���� En trop y Generation� Time Rev ersibilit y and Fluctuation Theorem� Exp erimen tal T ests of the Chaotic Hyp othesis � � � � � � � � � � � � � � � � � � � � � ��� ���� Fluctuation P atterns � � � � � � � � � � � � � � � � � � � � � � � � ��� ���� �Conditional Rev ersibilit y� and �Fluctuation Theorems� � � � � � � � � ��� ����� Onsager Recipro cit y and Green�Kub o�s F orm ula � � � � � � � � � � � � ��� ����� Rev ersible V ersus Irrev ersible Dissipation� Nonequilibrium Ensem bles� � � ��� App endix ��A�� M � ecanique statistique hors � equilibre� l�h � eritage de Boltzmann � ��� App endix ��A�� Heuristic Deriv ation of the SRB Distribution � � � � � � � � ��� vi App endix ��A�� Ap erio dic Motions Can b e Begarded as P erio dic with In�nite P erio d� ��� App endix ��A�� Gauss� Least Constrain t Principle � � � � � � � � � � � � � ��� Biblograph y � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� Names index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� Analytic index � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� Citations index � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��� � Chapter I � Classic al Statistic al Me chanics � � I � Classical Statistical Mec hanics � x ���� In tro duction Statistical mec hanics p oses the problem of deducing macroscopic prop erties of matter from the atomic hyp othesis� According to the h yp othesis matter consists of atoms or molecules that mo v e sub ject to the la ws of classical mec hanics or of quan tum mec hanics� Matter is therefore though t of as consisting of a v ery large n um b er N of particles� essen tially p oin t masses� in teracting via simple conserv ativ e forces� � A micr osc opic state is describ ed b y sp ecifying� at a giv en instan t� the v alue of p ositions and momen ta �or� equiv alen tly � v elo cities� of e ach of the N particles� Hence one has to sp ecify �N � �N co ordinates that determine a p oin t in phase sp ac e� in the sense of mec hanics� It do es not seem that in the original viewp oin t Boltzmann particles w ere really though t of as susceptible of assuming a �N dimensional con tin uum of states� ��Bo���� p� ����� Ther efor e if we wish to get a pictur e of the c ontinuum in wor ds� we �rst have to imagine a lar ge� but �nite numb er of p articles with c ertain pr op erties and investigate the b ehavior of the ensemble of such p articles� Certain pr op� erties of the ensemble may appr o ach a de�nite limit as we al low the numb er of p articles ever mor e to incr e ase and their size ever mor e to de cr e ase� Of these pr op erties one c an then assert that they apply to a c ontinuum� and in my opinion this is the only non�c ontr adictory de�nition of a c ontinuum with c ertain pr op erties and lik ewise the phase space itself is really though t of as divided in to a �nite n um b er of v ery small c el ls of essen tially equal dimensions� eac h of whic h determines the p osition and momen tum of eac h particle with a maximum pr e cision� This should mean the maxim um precision that the most p erfect measure� men t apparatus can p ossibly pro vide� And a matter of principle arises� can w e supp ose that ev ery lac k of precision can b e impro v ed b y impro ving the instrumen ts w e use� If w e b eliev e this p ossibilit y then phase space cells� represen ting microscopic states with maximal precision� m ust b e p oin ts and they m ust b e conceiv ed of as a �N dimensional con tin uum� But since atoms and molecules are not directly observ able one is legitimized in his doubts ab out b eing allo w ed to assume p erfect measurabilit y of momen tum and p osition co ordinates� In fact in �recen t� times the foundations of classical mec hanics ha v e b een � N � ���� � �� �� particles p er mole � �Av ogadro�s n um b er�� this implies� for instance� that � cm � of Hydrogen� or of an y other �p erfect� gas� at normal conditions �� atm at � � C� con tains ab out ��� � �� �� molecules� � I � Classical Statistical Mec hanics sub ject to in tense critique and the indetermination principle p ostulates the theoretical imp ossibilit y of the sim ultaneous measuremen t of a comp onen t p of a particle momen tum and of the corresp onding comp onen t q of the p osition with resp ectiv e precisions � p and � q without the constrain t� � p� q � h ������� ����� where h � ���� � �� ��� er g � sec is the Planck�s c onstant� Without attempting a discussion of the conceptual problems that the ab o v e brief and sup er�cial commen ts raise it is b etter to pro ceed b y imagining that the micr osc opic states of a N particles system are represen ted b y phase space cells consisting in the p oin ts of R �N with co ordinates� �e�g� �Bo����� � p � � � � p�� � p � � p � � � � p�� q � � � � q �� � q � � q � � � � q �� � � �� � � � � �N ������� ����� if p � � p � � p � are the momen tum co ordinates of the �rst particle� p � � p � � p � of the second� etc� and q � � q � � q � are the p osition co ordinates of the �rst particle� q � � q � � q � of the second� etc��� The co ordinate p � � and q � � are used to iden tify the cen ter of the cell� hence the cell itself� The cell size will b e supp osed to b e suc h that� � p� q � h ������� ����� where h is an a priori arbitrary constan t� whic h it is con v enien t not to �x b ecause it is in teresting �for the reasons just giv en� to see ho w the theory dep ends up on it� Here the meaning of h is that of a limitation to the preci� sion that is assumed to b e p ossible when measuring a pair of corresp onding p osition and momen tum co ordinates� Therefore the space of the microscopic states is the collection of the cubic cells �� with v olume h �N in to whic h w e imagine that the phase space is divided� By assumption it has no meaning to p ose the problem of attempting to determine the microscopic state with a greater precision� The optimistic viewp oint of ortho do x statistical mec hanics �whic h admits p erfect sim ultaneous measuremen ts of p ositions and momen ta as p ossible� will b e obtained b y considering� in the more general theory with h � �� the limit as h � �� which wil l me an � p � �p � � � q � �q � � with p � � q � �xe d and � � �� Ev en if w e wish to ignore �one should not�� the dev elopmen t of quan� tum mec hanics� the real p ossibilit y of the situation in whic h h � � cannot b e directly c hec k ed b ecause of the practical imp ossibilit y of observing an individual atom with in�nite precision �or just with �great� precision�� x ���� Microscopic Dynamics The atomic h yp othesis� apart from supp osing the existence of atoms and molecules� assumes also that their motions are go v erned b y a deterministic law of motion� I � Classical Statistical Mec hanics � This h yp othesis can b e imp osed b y thinking that there is a map S � S � � � � ������� ����� transforming the phase space cells in to eac h other and describing the system dynamics� If at time t the state of the system is microscopically determined b y the phase space cell �� then at a later time t � � it will b e determined b y the cell � � � Here � is a time step extremely small compared to the macroscopic time in terv als o v er whic h the system ev olution is follo w ed b y an observ er� it is� nev ertheless� a time in terv al directly accessible to observ ation� at least in principle� The ev olution la w S is not arbitrary� it m ust satisfy some fundamen tal prop erties� namely it m ust agree with the la ws of mec hanics in order to prop erly enact the deterministic principle whic h is basic to the atomic h y� p othesis� This means� in essence� that one can asso ciate with eac h phase space cell three fundamen tal dynamical quan tities� the kinetic ener gy� the p otential ener gy and the total ener gy� resp ectiv ely denoted b y K ���� ����� E ���� F or simplicit y assume the system to consist of N iden tical particles with mass m� pairwise in teracting via a conserv ativ e force with p oten tial energy �� If � is the phase space cell determined b y �see �������� �p � � q � �� then the ab o v e basic quan tities are de�ned resp ectiv ely b y� K �p � � � K ��� � N X i�� �p � i � � ��m p � i � �p � �i�� � p � �i�� � p � �i � ��q � � � ���� � ��N X i�j ��q � i � q � j � q � i � �q � �i�� � q � �i�� � q � �i � E �p � � q � � � E ��� � K �p � � � ��q � � ������� ����� where p � i � �p � �i�� � p � �i�� � p � �i �� q � i � �q � �i�� � q � �i�� � q � �i � are the momen tum and p osition of the i�th particle� i � �� �� ���N � in the micr osc opic state corresp onding to the cen ter �p � � q � � of �� Replacing p � � q � � i�e� the cen ter of �� b y another p oin t �p� q � in � one obtains v alues K �p �� ��q �� E �p� q � for the kinetic� p oten tial and total energies di�eren t from K ���������E ���� however suc h a di�erence has to b e non observable� otherwise the cells � w ould not b e the smallest ones to b e observ able� as supp osed ab o v e� If � is a �xed time in terv al and w e consider the solutions of Hamilton s equations of motion� ! q � � E � p �p� q �� ! p � � � E � q �p� q � ������� ����� with initial data �p � � q � � at time � the p oin t �p � � q � � will ev olv e in time � in to a p oin t S � �p � � q � � � �p � � q � � � �S � �p � � q � ��� One then de�nes S so that

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