Statistical Physics of Particles - Mehran Kardar - Cambridge University

MEHRAN KARDAR Statistical Physics of Particles CAMBRIDGE wwwcambridgeorg9780521873420 This page intentionally left blank Statistical Physics of Particles Statistical physics has its origins in attempts to describe the thermal properties of matter in terms of its constituent particles and has played a fundamental role in the development of quantum mechanics It describes how new behavior emerges from interactions of many degrees of freedom and as such has found applications outside physics in engineering social sciences and increasingly in biological sciences This textbook introduces the central concepts and tools of statistical physics It includes a chapter on probability and related issues such as the central limit theorem and information theory not usually covered in existing texts The book also covers interacting particles and includes an extensive description of the van der Waals equation and its derivation by meanfield approximation A companion volume Statistical Physics of Fields discusses nonmean field aspects of scaling and critical phenomena through the perspective of renormalization group Based on lectures for a course in statistical mechanics taught by Professor Kardar at Massachusetts Institute of Technology MIT this textbook contains an integrated set of problems with solutions to selected problems at the end of the book It will be invaluable for graduate and advanced undergraduate courses in statistical physics Additional solutions are available to lecturers on a password protected website at wwwcambridgeorg9780521873420 Mehran Kardar is Professor of Physics at MIT where he has taught and researched in the field of statistical physics for the past 20 years He received his BA at Cambridge and gained his PhD at MIT Professor Kardar has held research and visiting positions as a junior Fellow at Harvard a Guggenheim Fellow at Oxford UCSB and at Berkeley as a Miller Fellow In this muchneeded modern text Kardar presents a remarkably clear view of statistical mechanics as a whole revealing the relationships between different parts of this diverse subject In two volumes the classical beginnings of thermodynamics are connected smoothly to a thoroughly modern view of fluctuation effects stochastic dynamics and renormalization and scaling theory Students will appreciate the precision and clarity in which difficult concepts are presented in generality and by example I particularly like the wealth of interesting and instructive problems inspired by diverse phenomena throughout physics and beyond which illustrate the power and broad applicability of statistical mechanics Statistical Physics of Particles includes a concise introduction to the mathematics of probability for physicists an essential prerequisite to a true understanding of statistical mechanics but which is unfortunately missing from most statistical mechanics texts The old subject of kinetic theory of gases is given an updated treatment which emphasizes the connections to hydrodynamics As a graduate student at Harvard I was one of many students making the trip to MIT from across the Boston area to attend Kardars advanced statistical mechanics class Finally in Statistical Physics of Fields Kardar makes his fantastic course available to the physics community as a whole The book provides an intuitive yet rigorous introduction to fieldtheoretic and related methods in statistical physics The treatment of renormalization group is the best and most physical Ive seen and is extended to cover the oftenneglected or not properly explained but beautiful problems involving topological defects in two dimensions The diversity of lattice models and techniques are also wellillustrated and complement these continuum approaches The final two chapters provide revealing demonstrations of the applicability of renormalization and fluctuation concepts beyond equilibrium one of the frontier areas of statistical mechanics Leon Balents Department of Physics University of California Santa Barbara Statistical Physics of Particles is the welcome result of an innovative and popular graduate course Kardar has been teaching at MIT for almost twenty years It is a masterful account of the essentials of a subject which played a vital role in the development of twentieth century physics not only surviving but enriching the development of quantum mechanics Its importance to science in the future can only increase with the rise of subjects such as quantitative biology Statistical Physics of Fields builds on the foundation laid by the Statistical Physics of Particles with an account of the revolutionary developments of the past 35 years many of which were facilitated by renormalization group ideas Much of the subject matter is inspired by problems in condensed matter physics with a number of pioneering contributions originally due to Kardar himself This lucid exposition should be of particular interest to theorists with backgrounds in field theory and statistical mechanics David R Nelson Arthur K Solomon Professor of Biophysics Harvard University If Landau and Lifshitz were to prepare a new edition of their classic Statistical Physics text they might produce a book not unlike this gem by Mehran Kardar Indeed Kardar is an extremely rare scientist being both brilliant in formalism and an astoundingly careful and thorough teacher He demonstrates both aspects of his range of talents in this pair of books which belong on the bookshelf of every serious student of theoretical statistical physics Kardar does a particularly thorough job of explaining the subtleties of theoretical topics too new to have been included even in Landau and Lifshitzs most recent Third Edition 1980 such as directed paths in random media and the dynamics of growing surfaces which are not in any text to my knowledge He also provides careful discussion of topics that do appear in most modern texts on theoretical statistical physics such as scaling and renormalization group H Eugene Stanley Director Center for Polymer Studies Boston University This is one of the most valuable textbooks I have seen in a long time Written by a leader in the field it provides a crystal clear elegant and comprehensive coverage of the field of statistical physics Im sure this book will become the reference for the next generation of researchers students and practitioners in statistical physics I wish I had this book when I was a student but I will have the privilege to rely on it for my teaching Alessandro Vespignani Center for Biocomplexity Indiana University Statistical Physics of Particles Mehran Kardar Department of Physics Massachusetts Institute of Technology CAMBRIDGE UNIVERSITY PRESS Cambridge New York Melbourne Madrid Cape Town Singapore São Paulo Cambridge University Press The Edinburgh Building Cambridge CB2 8RU UK First published in print format ISBN13 9780521873420 ISBN13 9780511289125 M Kardar 2007 2007 Information on this title wwwcambridgeorg9780521873420 This publication 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 ISBN10 051128912X ISBN10 0521873428 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or thirdparty internet websites referred to in this publication and does not guarantee that any content on such websites is or will remain accurate or appropriate Published in the United States of America by Cambridge University Press New York wwwcambridgeorg hardback eBook EBL eBook EBL hardback Contents Preface page ix 1 Thermodynamics 1 11 Introduction 1 12 The zeroth law 2 13 The first law 5 14 The second law 8 15 Carnot engines 10 16 Entropy 13 17 Approach to equilibrium and thermodynamic potentials 16 18 Useful mathematical results 20 19 Stability conditions 22 110 The third law 26 Problems 29 2 Probability 35 21 General definitions 35 22 One random variable 36 23 Some important probability distributions 40 24 Many random variables 43 25 Sums of random variables and the central limit theorem 45 26 Rules for large numbers 47 27 Information entropy and estimation 50 Problems 52 3 Kinetic theory of gases 57 31 General definitions 57 32 Liouvilles theorem 59 33 The BogoliubovBornGreenKirkwoodYvon hierarchy 62 34 The Boltzmann equation 65 35 The Htheorem and irreversibility 71 36 Equilibrium properties 75 37 Conservation laws 78 v vi Contents 38 Zerothorder hydrodynamics 82 39 Firstorder hydrodynamics 84 Problems 87 4 Classical statistical mechanics 98 41 General definitions 98 42 The microcanonical ensemble 98 43 Twolevel systems 102 44 The ideal gas 105 45 Mixing entropy and the Gibbs paradox 107 46 The canonical ensemble 110 47 Canonical examples 113 48 The Gibbs canonical ensemble 115 49 The grand canonical ensemble 118 Problems 120 5 Interacting particles 126 51 The cumulant expansion 126 52 The cluster expansion 130 53 The second virial coefficient and van der Waals equation 135 54 Breakdown of the van der Waals equation 139 55 Meanfield theory of condensation 141 56 Variational methods 143 57 Corresponding states 145 58 Critical point behavior 146 Problems 148 6 Quantum statistical mechanics 156 61 Dilute polyatomic gases 156 62 Vibrations of a solid 161 63 Blackbody radiation 167 64 Quantum microstates 170 65 Quantum macrostates 172 Problems 175 7 Ideal quantum gases 181 71 Hilbert space of identical particles 181 72 Canonical formulation 184 73 Grand canonical formulation 187 74 Nonrelativistic gas 188 75 The degenerate fermi gas 190 Contents vii 76 The degenerate bose gas 194 77 Superfluid He4 198 Problems 202 Solutions to selected problems 211 Chapter 1 211 Chapter 2 224 Chapter 3 235 Chapter 4 256 Chapter 5 268 Chapter 6 285 Chapter 7 300 Index 318 This page intentionally left blank Preface Historically the discipline of statistical physics originated in attempts to describe thermal properties of matter in terms of its constituent particles but also played a fundamental role in the development of quantum mechanics More generally the formalism describes how new behavior emerges from interactions of many degrees of freedom and as such has found applications in engineering social sciences and increasingly in biological sciences This book introduces the central concepts and tools of this subject and guides the reader to their applications through an integrated set of problems and solutions The material covered is directly based on my lectures for the first semester of an MIT graduate course on statistical mechanics which I have been teaching on and off since 1988 The material pertaining to the second semester is presented in a companion volume While the primary audience is physics graduate students in their first semester the course has typically also attracted enterprising undergraduates as well as students from a range of science and engineering departments While the material is reasonably standard for books on statistical physics students taking the course have found my exposition more useful and have strongly encouraged me to publish this material Aspects that make this book somewhat distinct are the chapters on probability and interacting particles Probability is an integral part of statistical physics which is not sufficiently emphasized in most textbooks Devoting an entire chapter to this topic and related issues such as the central limit theorem and information theory provides valuable tools to the reader In the context of interacting particles I provide an extensive description of the van der Waals equation including its derivation by meanfield approximation An essential part of learning the material is doing problems an interesting selection of problems and solutions has been designed and integrated into the text Following each chapter there are two sets of problems solutions to the first set are included at the end of the book and are intended to introduce additional topics and to reinforce technical tools Pursuing these problems should also prove useful for students studying for qualifying exams There ix x Preface are no solutions provided for a second set of problems which can be used in assignments I am most grateful to my many former students for their help in formulating the material problems and solutions typesetting the text and figures and pointing out various typos and errors The support of the National Science Foundation through research grants is also acknowledged 1 Thermodynamics 11 Introduction Thermodynamics is a phenomenological description of properties of macro scopic systems in thermal equilibrium Imagine yourself as a postNewtonian physicist intent on understanding the behavior of such a simple system as a container of gas How would you proceed The prototype of a successful physical theory is classical mechanics which describes the intricate motions of particles starting from simple basic laws and employing the mathematical machinery of calculus By analogy you may proceed as follows Idealize the system under study as much as possible as is the case of a point particle The concept of mechanical work on the system is certainly familiar yet there appear to be complications due to exchange of heat The solution is first to examine closed systems insulated by adiabatic walls that dont allow any exchange of heat with the surroundings Of course it is ultimately also necessary to study open systems which may exchange heat with the outside world through diathermic walls As the state of a point particle is quantified by its coordinates and momenta proper ties of the macroscopic system can also be described by a number of thermodynamic coordinates or state functions The most familiar coordinates are those that relate to mechanical work such as pressure and volume for a fluid surface tension and area for a film tension and length for a wire electric field and polarization for a dielectric etc As we shall see there are additional state functions not related to mechanical work The state functions are well defined only when the system is in equilibrium that is when its properties do not change appreciably with time over the intervals of interest observation times The dependence on the observation time makes the concept of equilibrium subjective For example window glass is in equi librium as a solid over many decades but flows like a fluid over time scales of millennia At the other extreme it is perfectly legitimate to consider the equilibrium between matter and radiation in the early universe during the first minutes of the Big Bang 1 Finally the relationship between the state functions is described by the laws of thermodynamics As a phenomenological description these laws are based on a number of empirical observations A coherent logical and mathematical structure is then constructed on the basis of these observations which leads to a variety of useful concepts and to testable relationships among various quantities The laws of thermodynamics can only be justified by a more fundamental microscopic theory of nature For example statistical mechanics attempts to obtain these laws starting from classical or quantum mechanical equations for the evolution of collections of particles 12 The zeroth law The zeroth law of thermodynamics describes the transitive nature of thermal equilibrium It states If two systems A and B are separately in equilibrium with a third system C then they are also in equilibrium with one another Despite its apparent simplicity the zeroth law has the consequence of implying the existence of an important state function the empirical temperature Θ such that systems in equilibrium are at the same temperature Let the equilibrium state of systems A B and C be described by the coordinates A1 A2 B1 B2 and C1 C2 respectively The assumption that A and C are in equilibrium implies a constraint between the coordinates of A and C that is a change in A1 must be accompanied by some changes in A2 C1 C2 to maintain equilibrium of A and C Denote this constraint by fACA1 A2 C1 C2 0 11 The equilibrium of B and C implies a similar constraint fBCB1 B2 C1 C2 0 12 Note that each system is assumed to be separately in mechanical equilibrium If they are allowed also to do work on each other additional conditions eg constant pressure are required to describe their joint mechanical equilibrium Clearly we can state the above constraint in many different ways For example we can study the variations of C1 as all of the other parameters are changed This is equivalent to solving each of the above equations for C1 to yield 1 C1 FACA1 A2 C2 C1 FBCB1 B2 C2 13 Thus if C is separately in equilibrium with A and B we must have FACA1 A2 C2 FBCB1 B2 C2 14 However according to the zeroth law there is also equilibrium between A and B implying the constraint fABA1 A2 B1 B2 0 15 We can select any set of parameters A B that satisfy the above equation and substitute them in Eq 14 The resulting equality must hold quite independently of any set of variables C in this equation We can then change these parameters moving along the manifold constrained by Eq 15 and Eq 14 will remain valid irrespective of the state of C Therefore it must be possible to simplify Eq 14 by canceling the coordinates of C Alternatively we can select any fixed set of parameters C and ignore them henceforth reducing the condition 15 for equilibrium of A and B to ΘAA1 A2 ΘBB1 B2 16 that is equilibrium is characterized by a function Θ of thermodynamic coordinates This function specifies the equation of state and isotherms of A are described by the condition ΘAA1 A2 Θ While at this point there are many potential choices of Θ the key point is the existence of a function that constrains the parameters of each system in thermal equilibrium There is a similarity between Θ and the force in a mechanical system Consider two onedimensional systems that can do work on each other as in the case of two springs connected together Equilibrium is achieved when the forces exerted by each body on the other are equal Of course unlike the scalar temperature the vectorial force has a direction a complication that we have ignored The pressure of a gas piston provides a more apt analogy The mechanical equilibrium between several such bodies is also transitive and the latter could have been used as a starting point for deducing the existence of a mechanical force 1 From a purely mathematical perspective it is not necessarily possible to solve an arbitrary constraint condition for C1 However the requirement that the constraint describes real physical parameters clearly implies that we can obtain C1 as a function of the remaining parameters As an example let us consider the following three systems A a wire of length L with tension F B a paramagnet of magnetization M in a magnetic field B and C a gas of volume V at pressure P Observations indicate that when these systems are in equilibrium the following constraints are satisfied between their coordinates P aV2V bL L0cF KL L0 0 P aV2VbM dB 0 17 The two conditions can be organized into three empirical temperature functions as Θ P aV2V b c FL L0 K d BM 18 Note that the zeroth law severely constrains the form of the constraint equation describing the equilibrium between two bodies Any arbitrary function cannot necessarily be organized into an equality of two empirical temperature functions The constraints used in the above example were in fact chosen to reproduce three wellknown equations of state that will be encountered and discussed later in this book In their more familiar forms they are written as P aV2V b NkB T van der Waals gas M N μB2 B3kB T Curie paramagnet F K DTL L0 Hookes law for rubber 19 Note that we have employed the symbol for Kelvin temperature T in place of the more general empirical temperature Θ This concrete temperature scale can be constructed using the properties of the ideal gas The ideal gas temperature scale while the zeroth law merely states the presence of isotherms to set up a practical temperature scale at this stage a reference system is necessary The ideal gas occupies an important place in thermodynamics and provides the necessary reference Empirical observations indicate that the product of pressure and volume is constant along the isotherms of any gas that is sufficiently dilute The ideal gas refers to this dilute limit of Fig 13 The triple point of ice water and steam occurs at a unique point in the PT phase diagram real gases and the ideal gas temperature is proportional to the product The constant of proportionality is determined by reference to the temperature of the triple point of the icewatergas system which was set to 27316 degrees kelvin K by the 10th General Conference on Weights and Measures in 1954 Using a dilute gas ie as P 0 as thermometer the temperature of a system can be obtained from TK 27316 limP0PVsystemlimP0PVicewatergas 13 The first law In dealing with simple mechanical systems conservation of energy is an important principle For example the location of a particle can be changed in a potential by external work which is then related to a change in its potential energy Observations indicate that a similar principle operates at the level of macroscopic bodies provided that the system is properly insulated that is when the only sources of energy are of mechanical origin We shall use the following formulation of these observations The amount of work required to change the state of an otherwise adiabatically isolated system depends only on the initial and final states and not on the means by which the work is performed or on the intermediate stages through which the system passes For a particle moving in a potential the required work can be used to construct a potential energy landscape Similarly for the thermodynamic system we can construct another state function the internal energy EX Up to a constant EX can be obtained from the amount of work ΔW needed for an adiabatic transformation from an initial state Xi to a final state Xf using ΔW EXf EXi Another set of observations indicate that once the adiabatic constraint is removed the amount of work is no longer equal to the change in the internal energy The difference ΔQ ΔE ΔW is defined as the heat intake of the system from its surroundings Clearly in such transformations ΔQ and ΔW are not separately functions of state in that they depend on external factors such as the means of applying work and not only on the final states To emphasize this for a differential transformation we write dQ dE dW where dE Σi i E dXi can be obtained by differentiation while dQ and dW generally cannot Also note that our convention is such that the signs of work and heat indicate the energy added to the system and not vice versa The first law of thermodynamics thus states that to change the state of a system we need a fixed amount of energy which can be in the form of mechanical work or heat This can also be regarded as a way of defining and quantifying the exchanged heat A quasistatic transformation is one that is performed sufficiently slowly so that the system is always in equilibrium Thus at any stage of the process the thermodynamic coordinates of the system exist and can in principle be computed For such transformations the work done on the system equal in magnitude but opposite in sign to the work done by the system can be related to changes in these coordinates As a mechanical example consider the stretching of a spring or rubber band To construct the potential energy of the system as a function of its length L we can pull the spring sufficiently slowly so that at each stage the external force is matched by the internal force F from the spring For such a quasistatic process the change in the potential energy of the spring is F dL If the spring is pulled abruptly some of the external work is converted into kinetic energy and eventually lost as the spring comes to rest Generalizing from the above example one can typically divide the state functions X into a set of generalized displacements x and their conjugate generalized forces J such that for an infinitesimal quasistatic transformation dW Σi Ji dxi I denote force by the symbol J rather than F to reserve the latter for the free energy I hope the reader is not confused with currents sometimes also denoted by J which rarely appear in this book Table 11 Generalized forces and displacements System Force Displacement Wire tension F length L Film surface tension S area A Fluid pressure P volume V Magnet magnetic field H magnetization M Dielectric electric field E polarization P Chemical reaction chemical potential μ particle number N Table 11 provides some common examples of such conjugate coordinates Note that pressure P is by convention calculated from the force exerted by the system on the walls as opposed to the force on a spring which is exerted in the opposite direction This is the origin of the negative sign that accompanies hydrostatic work The displacements are usually extensive quantities that is proportional to system size while the forces are intensive and independent of size The latter are indicators of equilibrium for example the pressure is uniform in a gas in equilibrium in the absence of external potentials and equal for two equilibrated gases in contact As discussed in connection with the zeroth law temperature plays a similar role when heat exchanges are involved Is there a corresponding displacement and if so what is it This question will be answered in the following sections The ideal gas we noted in connection with the zeroth law that the equation of state of the ideal gas takes a particularly simple form PV T The internal energy of the ideal gas also takes a very simple form as observed for example by Joules free expansion experiment measurements indicate that if an ideal gas expands adiabatically but not necessarily quasistatically from a volume Vi to Vf the initial and final temperatures are the same As the transformation is adiabatic ΔQ 0 and there is no external work done on the system ΔW 0 the internal energy of the gas is unchanged Since the pressure and volume of the gas change in the process but its temperature does not we conclude that the internal energy depends only on temperature that is EVT ET This property of the ideal gas is in fact a consequence of the form of its equation of state as will be proved in one of the problems at the end of this chapter Fig 15 A gas initially confined in the left chamber is allowed to expand rapidly to both chambers Response functions are the usual method for characterizing the macroscopic behavior of a system They are experimentally measured from the changes of thermodynamic coordinates with external probes Some common response functions are as follows Heat capacities are obtained from the change in temperature upon addition of heat to the system Since heat is not a function of state the path by which it is supplied must also be specified For example for a gas we can calculate the heat capacities at constant volume or pressure denoted by CV dQdTV and CP dQdTP respectively The latter is larger since some of the heat is used up in the work done in changes of volume CV dQdTV dE dWdT V dE PdVdTV dEdTV CP dQdTP dE dWdT P dE PdVdT P dEdTP PdVdTP Force constants measure the infinitesimal ratio of displacement to force and are generalizations of the spring constant Examples include the isothermal compressibility of a gas kappaT dVdP T V and the susceptibility of a magnet chiT dMdBT V From the equation of state of an ideal gas PV T we obtain kappaT 1P Thermal responses probe the change in the thermodynamic coordinates with temperature For example the expansivity of a gas is given by alphaP dVdTP V which equals 1T for the ideal gas Since the internal energy of an ideal gas depends only on its temperature dEdT V dEdT P dEdT and Eq 114 simplifies to CP CV P dVdTP PV alphaP PVT N kB The last equality follows from extensivity for a given amount of ideal gas the constant PVT is proportional to N the number of particles in the gas the ratio is Boltzmanns constant with a value of kB 14 1023 JK1 14 The second law The practical impetus for development of the science of thermodynamics in the nineteenth century was the advent of heat engines The increasing reliance on machines to do work during the industrial revolution required better understanding of the principles underlying conversion of heat to work It is quite interesting to see how such practical considerations as the efficiency of engines can lead to abstract ideas like entropy An idealized heat engine works by taking in a certain amount of heat QH from a heat source for example a coal fire converting a portion of it to work W and dumping the remaining heat QC into a heat sink eg atmosphere The efficiency of the engine is calculated from η WQH QH QCQH 1 An idealized refrigerator is like an engine running backward that is using work W to extract heat QC from a cold system and dumping heat QH at a higher temperature We can similarly define a figure of merit for the performance of a refrigerator as ω QCW QC QH QC Source QH Engine QC Sink W Exhaust QH W Refrigerator QC Icebox Fig 16 The idealized engine and refrigerator The first law rules out socalled perpetual motion machines of the first kind that is engines that produce work without consuming any energy However the conservation of energy is not violated by an engine that produces work by converting water to ice Such a perpetual motion machine of the second kind would certainly solve the worlds energy problems but is ruled out by the second law of thermodynamics The observation that the natural direction for the flow of heat is from hotter to colder bodies is the essence of the second law of thermodynamics There is a number of different formulations of the second law such as the following two statements Kelvins statement No process is possible whose sole result is the complete conversion of heat into work Clausiuss statement No process is possible whose sole result is the transfer of heat from a colder to a hotter body A perfect engine is ruled out by the first statement a perfect refrigerator by the second Since we shall use both statements we first show that they are equivalent Proof of the equivalence of the Kelvin and Clausius statements proceeds by showing that if one is violated so is the other a Let us assume that there is a machine that violates Clausiuss statement by taking heat Q from a cooler region to a hotter one Now consider an engine operating between these two regions taking heat QH from the hotter one and dumping QC at the colder sink The combined system takes QH Q from the hot source produces work equal to QH QC and dumps QC Q at the cold sink If we adjust the engine output such that QC Q the net result is a 100 efficient engine in violation of Kelvins statement Fig 17 A machine violating Clausiuss statement C can be connected to an engine resulting in a combined device K that violates Kelvins statement hot Q C Q cold Engine W QH QC Q hot QH QC K W b Alternatively assume a machine that violates Kelvins law by taking heat Q and converting it completely to work The work output of this machine can be used to run a refrigerator with the net outcome of transferring heat from a colder to a hotter body in violation of Clausiuss statement Fig 18 A machine violating Kelvins statement can be connected to a refrigerator resulting in violation of Clausiuss statement hot Q K W QH Refrigerator QC hot QH Q C QC cold 15 Carnot engines A Carnot engine is any engine that is reversible runs in a cycle with all of its heat exchanges taking place at a source temperature TH and a sink temperature TC Fig 19 A Carnot engine operates between temperatures TH and TC with no other heat exchanges TH QH Carnot engine W QC TC A reversible process is one that can be run backward in time by simply reversing its inputs and outputs It is the thermodynamic equivalent of frictionless motion in mechanics Since time reversibility implies equilibrium a reversible transformation must be quasistatic but the reverse is not necessarily true eg if there is energy dissipation due to friction An engine that runs in a cycle returns to its original internal state at the end of the process The distinguishing characteristic of the Carnot engine is that heat exchanges with the surroundings are carried out only at two temperatures The zeroth law allows us to select two isotherms at temperatures TH and TC for these heat exchanges To complete the Carnot cycle we have to connect these isotherms by reversible adiabatic paths in the coordinate space Since heat is not a function of state we dont know how to construct such paths in general Fortunately we have sufficient information at this point to construct a Carnot engine using an ideal gas as its internal working substance For the purpose of demonstration let us compute the adiabatic curves for a monatomic ideal gas with an internal energy E32 N kBT 32 PV Along a quasistatic path dQ dE dW d 32 PV PdV 52 PdV 32 V dP 118 The adiabatic condition dQ 0 then implies a path dPP 53 dVV 0 PVγ constant 119 with γ 53 Fig 110 The Carnot cycle for an ideal gas with isothermal and adiabatic paths indicated by solid and dashed lines respectively The adiabatic curves are clearly distinct from the isotherms and we can select two such curves to intersect our isotherms thereby completing a Carnot cycle The assumption of E T is not necessary and in one of the problems provided at the end of this chapter you will construct adiabatics for any ET In fact a similar construction is possible for any twoparameter system with EJx Carnots theorem No engine operating between two reservoirs at temperatures TH and TC is more efficient than a Carnot engine operating between them Since a Carnot engine is reversible it can be run backward as a refrigerator Use the nonCarnot engine to run the Carnot engine backward Let us denote the heat exchanges of the nonCarnot and Carnot engines by QH QC and QH QC respectively The net effect of the two engines is to transfer heat equal to QH QH QC QC from TH to TC According to Clausiuss statement the quantity of transferred heat cannot be negative that is QH QH Since the same quantity of work W is involved in this process we conclude that WQH WQH ηCarnot ηnonCarnot 120 Fig 111 A generic engine is used to run a Carnot engine in reverse Corollary All reversible Carnot engines have the same universal efficiency ηTH TC since each can be used to run any other one backward The thermodynamic temperature scale as shown earlier it is at least theoretically possible to construct a Carnot engine using an ideal gas or any other twoparameter system as working substance We now find that independent of the material used and design and construction all such cyclic and reversible engines have the same maximum theoretical efficiency Since this maximum efficiency is only dependent on the two temperatures it can be used to construct a temperature scale Such a temperature scale has the attractive property Fig 112 Two Carnot engines connected in series are equivalent to a third of being independent of the properties of any material eg the ideal gas To construct such a scale we first obtain a constraint on the form of ηTH TC Consider two Carnot engines running in series one between temperatures T1 and T2 and the other between T2 and T3 T1 T2 T3 Denote the heat exchanges and work outputs of the two engines by Q1 Q2 W12 and Q2 Q3 W23 respectively Note that the heat dumped by the first engine is taken in by the second so that the combined effect is another Carnot engine since each component is reversible with heat exchanges Q1 Q3 and work output W13 W12 W23 Using the universal efficiency the three heat exchanges are related by Q2 Q1 W12 Q11 ηT1 T2 Q3 Q2 W23 Q21 ηT2 T3 Q11 ηT1 T21 ηT2 T3 Q3 Q1 W13 Q11 ηT1 T3 Comparison of the final two expressions yields 1 ηT1 T3 1 ηT1 T21 ηT2 T3 121 This property implies that 1 ηT1 T2 can be written as a ratio of the form fT2fT1 which by convention is set to T2T1 that is 1 ηT1 T2 Q2Q1 T2T1 ηTH TC TH TCTH 122 Equation 122 defines temperature up to a constant of proportionality which is again set by choosing the triple point of water ice and steam to 27316 K So far we have used the symbols Θ and T interchangeably In fact by running a Carnot cycle for a perfect gas it can be proved see problems that the ideal gas and thermodynamic temperature scales are equivalent Clearly the thermodynamic scale is not useful from a practical standpoint its advantage is conceptual in that it is independent of the properties of any substance All thermodynamic temperatures are positive since according to Eq 122 the heat extracted from a temperature T is proportional to it If a negative temperature existed an engine operating between it and a positive temperature would extract heat from both reservoirs and convert the sum total to work in violation of Kelvins statement of the second law 16 Entropy To construct finally the state function that is conjugate to temperature we appeal to the following theorem Clausiuss theorem For any cyclic transformation reversible or not dQT 0 where dQ is the heat increment supplied to the system at temperature T Subdivide the cycle into a series of infinitesimal transformations in which the system receives energy in the form of heat dQ and work dW The system need not be in equilibrium at each interval Direct all the heat exchanges Fig 113 The heat exchanges of the system are directed to a Carnot engine with a reservoir at T₀ of the system to one port of a Carnot engine which has another reservoir at a fixed temperature T₀ There can be more than one Carnot engine as long as they all have one end connected to T₀ Since the sign of dQ is not specified the Carnot engine must operate a series of infinitesimal cycles in either direction To deliver heat dQ to the system at some stage the engine has to extract heat dQR from the fixed reservoir If the heat is delivered to a part of the system that is locally at a temperature T then according to Eq 122 dQR T₀ dQT 123 After the cycle is completed the system and the Carnot engine return to their original states The net effect of the combined process is extracting heat QR dQR from the reservoir and converting it to external work W The work W QR is the sum total of the work elements done by the Carnot engine and the work performed by the system in the complete cycle By Kelvins statement of the second law QR W 0 that is T₀ dQT 0 dQT 0 124 since T₀ 0 Note that T in Eq 124 refers to the temperature of the whole system only for quasistatic processes in which it can be uniquely defined throughout the cycle Otherwise it is just a local temperature say at a boundary of the system at which the Carnot engine deposits the element of heat Consequences of Clausiuss theorem 1 For a reversible cycle dQrevT 0 since by running the cycle in the opposite direction dQrev dQrev and by the above theorem dQrevT is both nonnegative and nonpositive hence zero This result implies that the integral of dQrevT between any two points A and B is independent of path since for two paths 1 and 2 AB dQrev1T₁ BA dQrev2T₂ 0 AB dQrev1T₁ AB dQrev2T₂ 125 2 Using Eq 125 we can construct yet another function of state the entropy S Since the integral is independent of path and only depends on the two endpoints we can set SB SA AB dQrevT 126 Fig 114 a A reversible cycle b Two reversible paths between A and B c The cycle formed from a generic path between A and B and a reversible one For reversible processes we can now compute the heat from dQrev TdS This allows us to construct adiabatic curves for a general multivariable system from the condition of constant S Note that Eq 126 only defines the entropy up to an overall constant 3 For a reversible hence quasistatic transformation dQ TdS and dW i Ji dxi and the first law implies dE dQ dW TdS i Ji dxi 127 We can see that in this equation S and T appear as conjugate variables with S playing the role of a displacement and T as the corresponding force This identification allows us to make the correspondence between mechanical and thermal exchanges more precise although we should keep in mind that unlike its mechanical analog temperature is always positive While to obtain Eq 127 we had to appeal to reversible transformations it is important to emphasize that it is a relation between functions of state Equation 127 is likely the most fundamental and useful identity in thermodynamics 4 The number of independent variables necessary to describe a thermodynamic system also follows from Eq 127 If there are n methods of doing work on a system represented by n conjugate pairs Ji xi then n 1 independent coordinates are necessary to describe the system We shall ignore possible constraints between the mechanical coordinates For example choosing E xi as coordinates it follows from Eq 127 that SE x 1T and Sxi Exi JiT 128 x and J are shorthand notations for the parameter sets xi and Ji Fig 115 The initially isolated subsystems are allowed to interact resulting in an increase of entropy 5 Consider an irreversible change from A to B Make a complete cycle by returning from B to A along a reversible path Then AB dQT BA dQrevT 0 AB dQT SB SA 129 In differential form this implies that dS dQT for any transformation In particular consider adiabatically isolating a number of subsystems each initially separately in equi librrium As they come to a state of joint equilibrium since the net dQ 0 we must have δS 0 Thus an adiabatic system attains a maximum value of entropy in equilibrium since spontaneous internal changes can only increase S The direction of increasing entropy thus points out the arrow of time and the path to equilibrium The mechanical analog is a point mass placed on a surface with gravity providing a downward force As various constraints are removed the particle will settle down at locations of decreasing height The statement about the increase of entropy is thus no more mysterious than the observation that objects tend to fall down under gravity 17 Approach to equilibrium and thermodynamic potentials The time evolution of systems toward equilibrium is governed by the second law of thermodynamics For example in the previous section we showed that for an adiabatically isolated system entropy must increase in any spontaneous change and reaches a maximum in equilibrium What about outofequilibrium systems that are not adiabatically isolated and may also be subject to external mechanical work It is usually possible to define other thermodynamic potentials that are extremized when the system is in equilibrium Enthalpy is the appropriate function when there is no heat exchange dQ 0 and the system comes to mechanical equilibrium with a constant external force The minimum enthalpy principle merely formulates the observation that stable mechanical equilibrium is obtained by minimizing the net potential energy of the system plus the external agent For example consider a spring of natural extension L₀ and spring constant K subject to the force J mg exerted by a particle of mass m For an extension x L L₀ the internal potential energy of the spring is Ux Kx²2 while there is a change of mgx in the potential energy of the particle Mechanical equilibrium is obtained by minimizing Kx²2 mgx at an extension xeq mgK The spring at any other value of the displacement initially oscillates before coming to rest at xeq due to friction For a more general potential energy Ux the internally generated force Ji dUdx has to be balanced with the external force J at the equilibrium point For any set of displacements x at constant externally applied generalized forces J the work input to the system is dW J δx Equality is achieved for a quasistatic change with J Ji but there is generally some loss of the external work to dissipation Since dQ 0 using the first law δE J δx and δH 0 where H E J x 130 is the enthalpy The variations of H in equilibrium are given by dH dE dJ x TdS J dx x dJ J dx TdS x dJ 131 The equality in Eq 131 and the inequality in Eq 130 are a possible source of confusion Equation 130 refers to variations of H on approaching equilibrium as some parameter that is not a function of state is varied eg the velocity of the particle joined to the spring in the above example By contrast Eq 131 describes a relation between equilibrium coordinates To differentiate the two cases I will denote the former nonequilibrium variations by δ The coordinate set S J is the natural choice for describing the enthalpy and it follows from Eq 131 that xi H Ji SJji 132 Variations of the enthalpy with temperature are related to heat capacities at constant force for example CP dQ dT P dE PdV dT P dE PV dT P dH dT P 133 Note however that a change of variables is necessary to express H in terms of T rather than the more natural variable S Helmholtz free energy is useful for isothermal transformations in the absence of mechanical work dW 0 It is rather similar to enthalpy with T taking the place of J From Clausiuss theorem the heat intake of a system at a constant temperature T satisfies dQ TdS Hence δE dQ dW TdS and δF 0 where F E TS 134 is the Helmholtz free energy Since dF dE dTS TdS J dx SdT TdS SdT J dx 135 the coordinate set T x the quantities kept constant during an isothermal transformation with no work is most suitable for describing the free energy The equilibrium forces and entropy can be obtained from Ji F xi Txji S F T x 136 The internal energy can also be calculated from F using E F TS F T F T x T² FT T x 137 Table 12 Inequalities satisfied by thermodynamic potentials dQ 0 constant T dW 0 δS 0 δF 0 constant J δH 0 δG 0 Gibbs free energy applies to isothermal transformations involving mechanical work at constant external force The natural inequalities for work and heat input into the system are given by dW J δx and dQ TdS Hence δE TdS J δx leading to δG 0 where G E TS J x 138 is the Gibbs free energy Variations of G are given by dG dE dTS dJ x TdS J dx SdT TdS x dJ J dx SdT x dJ 139 and most easily expressed in terms of T J Table 12 summarizes the above results on thermodynamic functions Equations 130 134 and 138 are examples of Legendre transformations used to change variables to the most natural set of coordinates for describing a particular situation So far we have implicitly assumed a constant number of particles in the system In chemical reactions and in equilibrium between two phases the number of particles in a given constituent may change The change in the number of particles necessarily involves changes in the internal energy which is expressed in terms of a chemical work dW µ dN Here N N1 N2 lists the number of particles of each species and µ µ1 µ2 the associated chemical potentials that measure the work necessary to add additional particles to the system Traditionally chemical work is treated differently from mechanical work and is not subtracted from E in the Gibbs free energy of Eq 138 For chemical equilibrium in circumstances that involve no mechanical work the appropriate state function is the grand potential given by G E TS µ N 140 GT µ x is minimized in chemical equilibrium and its variations in general satisfy dG SdT J dx N dµ 141 Example To illustrate the concepts of this section consider N particles of supersaturated steam in a container of volume V at a temperature T How can we describe the approach of steam to an equilibrium mixture with Nw particles in the liquid and Ns particles in the gas phase The fixed coordinates describing this system are V T and N The appropriate thermodynamic function from Table 12 is the Helmholtz free energy FV T N whose variations satisfy dF dE TS SdT PdV µdN 142 Fig 117 Condensation of water from supersaturated steam Before the system reaches equilibrium at a particular value of Nw it goes through a series of nonequilibrium states with smaller amounts of water If the process is sufficiently slow we can construct an outofequilibrium value for F as FV T NNw FwT Nw FsV T N Nw 143 which depends on an additional variable Nw It is assumed that the volume occupied by water is small and irrelevant According to Eq 134 the equilibrium point is obtained by minimizing F with respect to this variable Since δF Fw Nw TV δNw Fs Ns TV δNw 144 and FNTV µ from Eq 142 the equilibrium condition can be obtained by equating the chemical potentials that is from µwV T µsV T Fig 118 The net free energy has a minimum as a function of the amount of water The identity of chemical potentials is the condition for chemical equilibrium Naturally to proceed further we need expressions for µw and µs c More careful observations show that at higher surfactant densities SAT N kB TA Nb2 2aANA2 and TSA A NbN kB where a and b are constants Obtain the expression for SA T and explain qualitatively the origin of the corrections described by a and b d Find an expression for CS CA in terms of EAT EAS S SAT and TSA 3 Temperature scales prove the equivalence of the ideal gas temperature scale Θ and the thermodynamic scale T by performing a Carnot cycle on an ideal gas The ideal gas satisfies PV N kB Θ and its internal energy E is a function of Θ only However you may not assume that E Θ You may wish to proceed as follows a Calculate the heat exchanges QH and QC as a function of ΘH ΘC and the volume expansion factors b Calculate the volume expansion factor in an adiabatic process as a function of Θ c Show that QHQC ΘHΘC 4 Equations of state the equation of state constrains the form of internal energy as in the following examples a Starting from dE TdS PdV show that the equation of state PV N kB T in fact implies that E can only depend on T b What is the most general equation of state consistent with an internal energy that depends only on temperature c Show that for a van der Waals gas CV is a function of temperature alone 5 The ClausiusClapeyron equation describes the variation of boiling point with pressure It is usually derived from the condition that the chemical potentials of the gas and liquid phases are the same at coexistence For an alternative derivation consider a Carnot engine using one mole of water At the source P T the latent heat L is supplied converting water to steam There is a volume increase V associated with this process The pressure is adiabatically decreased to P dP At the sink P dP T dT steam is condensed back to water a Show that the work output of the engine is W VdP OdP2 Hence obtain the ClausiusClapeyron equation dPdTboiling LTV 1 18 Useful mathematical results In the preceding sections we introduced a number of state functions However if there are n ways of doing mechanical work n 1 independent parameters suffice to characterize an equilibrium state There must thus be various constraints and relations between the thermodynamic parameters some of which are explored in this section 1 Extensivity including chemical work variations of the extensive coordinates of the system are related by generalizing Eq 127 dE TdS J dx μ dN 145 For fixed intensive coordinates the extensive quantities are simply proportional to size or to the number of particles This proportionality is expressed mathematically by EλS λx λN λES x N 146 Evaluating the derivative of the above equation with respect to λ at λ 1 results in ESxN S i ExiSxjiN xi α ENαSxNβα Nα ES x N 147 The partial derivatives in the above equation can be identified from Eq 145 as T Ji and μα respectively Substituting these values into Eq 147 leads to E TS J x μ N 148 Combining the variations of Eq 148 with Eq 145 leads to a constraint between the variations of intensive coordinates SdT x dJ N dμ 0 149 known as the GibbsDuhem relation While Eq 148 is sometimes referred to as the fundamental equation of thermodynamics I regard Eq 145 as the more fundamental The reason is that extensivity is an additional assumption and in fact relies on shortrange interactions between constituents of the system It is violated for a large system controlled by gravity such as a galaxy while Eq 145 remains valid Example For a fixed amount of gas variations of the chemical potential along an isotherm can be calculated as follows Since dT 0 the GibbsDuhem relation gives V dP N dμ 0 and dμ VN dP kB T dPP 150 where we have used the ideal gas equation of state PV NkB T Integrating the above equation gives μ μ0 kB T ln PP0 μ0 kB T ln VV0 151 where P0 V0 μ0 refer to the coordinates of a reference point 2 Maxwell relations Combining the mathematical rules of differentiation with thermodynamic relationships leads to several useful results The most important of these are Maxwells relations which follow from the commutative property x y fx y y x f x y of derivatives For example it follows from Eq 145 that ESxN T and ExiSxjiN Ji 152 The joint second derivative of E is then given by ²ESxi ²Exi S TxiS JiSxi 153 Since yx xy1 the above equation can be inverted to give SJixi xiTS 154 Similar identities can be obtained from the variations of other state functions Supposing that we are interested in finding an identity involving SxT We would like to find a state function whose variations include SdT and J dx The correct choice is dF dE TS SdT Jdx Looking at the second derivative of F yields the Maxwell relation SxT JTx 155 To calculate SJT consider dE TS Jx SdT xdJ which leads to the identity SJT xTj 156 There is a variety of mnemonics that are supposed to help you remember and construct Maxwell relations such as magic squares Jacobians etc I personally dont find any of these methods worth learning The most logical approach is to remember the laws of thermodynamics and hence Eq 127 and then to manipulate it so as to find the appropriate derivative using the rules of differentiation Example To obtain μPNT for an ideal gas start with dE TS PV SdT V dP μ dN Clearly μPNT VNTP VN kB TP 157 as in Eq 150 From Eq 127 it also follows that SVEN PT EVSN ESVN 158 where we have used Eq 145 for the final identity The above equation can be rearranged into SVEN ESVN VESN 1 159 which is an illustration of the chain rule of differentiation 3 The Gibbs phase rule In constructing a scale for temperature we used the triple point of steamwaterice in Fig 13 as a reference standard How can we be sure that there is indeed a unique coexistence point and how robust is it The phase diagram in Fig 13 depicts the states of the system in terms of the two intensive parameters P and T Quite generally if there are n ways of performing work on a system that can also change its internal energy by transformations between c chemical constituents the number of independent intensive parameters is n c Of course including thermal exchanges there are n c 1 displacementlike variables in Eq 145 but the intensive variables are constrained by Eq 149 at least one of the parameters characterizing the system must be extensive The system depicted in Fig 13 corresponds to a onecomponent system water with only one means of doing work hydrostatic and is thus described by two independent intensive coordinates for example P T or μ T In a situation such as depicted in Fig 117 where two phases liquid and gas coexist there is an additional constraint on the intensive parameters as the chemical potentials must be equal in the two phases This constraint reduces the number of independent parameters by 1 and the corresponding manifold for coexisting phases in Fig 13 is onedimensional At the triple point where three phases coexist we have to impose another constraint so that all three chemical potentials are equal The Gibbs phase rule states that quite generally if there are p phases in coexistence the dimensionality number of degrees of freedom of the corresponding loci in the space of intensive parameters is f n c 1 p 160 The triple point of pure water thus occurs at a single point f 1 1 1 3 0 in the space of intensive parameters If there are additional constituents for example a concentration of salt in the solution the number of intensive quantities increases and the triple point can drift along a corresponding manifold by intensive state functions TJμ and extensive variables E x N Now imagine that the system is arbitrarily divided into two equal parts and that one part spontaneously transfers some energy to the other part in the form of work or heat The two subsystems A and B initially have the same values for the intensive variables while their extensive coordinates satisfy EA EB E xA xB x and NA NB N After the exchange of energy between the two subsystems the coordinates of A change to EA δE xA δx NA δN and TA δTA JA δJA μA δμA 165 and those of B to EB δE xB δx NB δN and TB δTB JB δJB μB δμB 166 Fig 120 Spontaneous change between two halves of a homogeneous system Note that the overall system is maintained at constant E x and N Since the intensive variables are themselves functions of the extensive coordinates to first order in the variations of E x N we have δTA δTB δT δJA δJB δJ δμA δμB δμ 167 Using Eq 148 the entropy of the system can be written as S SA SB EA TA JA TA xA μA TA NA EB TB JB TB xB μB TB NB 168 Since by assumption we are expanding about the equilibrium point the firstorder changes vanish and to second order δS δSA δSB 2 δ 1TA δEA δ JATA δxA δ μATA δNA 169 We have used Eq 167 to note that the secondorder contribution of B is the same as A Equation 169 can be rearranged to δS 2TA δTA δEA JA δxA μA δNATA δJA δxA δμA δNA 2TA δTA δSA δJA δxA δμA δNA 170 external potential Ux dissipates energy and settles to equilibrium at a minimum value of U The vanishing of the force Ji dUdx is not by itself sufficient to ensure stability as we must check that it occurs at a minimum of the potential energy such that d2Udx2 0 In the presence of an external force J we must minimize the enthalpy H U Jx which amounts to tilting the potential At the new equilibrium point xeqJ we must require d2Hdx2 d2Udx2 0 Thus only the convex portions of the potential Ux are physically accessible With more than one mechanical coordinate the requirement that any change δx results in an increase in energy or enthalpy can be written as Σij 2 Uxi xj δxi δxj 0 161 We can express the above equation in more symmetric form by noting that the corresponding change in forces is given by δJi δUxi Σj 2 Uxi xj δxj 162 Thus Eq 161 is equivalent to Σi δJi δxi 0 163 When dealing with a thermodynamic system we should allow for thermal and chemical inputs to the internal energy of the system Including the corresponding pairs of conjugate coordinates the condition for mechanical stability should generalize to δTδS Σi δJi δxi Σα δμα δNα 0 164 Before examining the consequences of the above condition I shall provide a more standard derivation that is based on the uniformity of an extended thermodynamic body Consider a homogeneous system at equilibrium characterized Fig 119 Possible types of mechanical equilibrium for a particle in a potential The convex portions solid line of the potential can be explored with a finite force J while the concave dashed line portion around the unstable point is not accessible The condition for stable equilibrium is that any change should lead to a decrease in entropy and hence we must have δTδS δJ δx δμ δN 0 171 We have now removed the subscript A as Eq 171 must apply to the whole system as well as to any part of it The above condition was obtained assuming that the overall system is kept at constant E x and N In fact since all coordinates appear symmetrically in this expression the same result is obtained for any other set of constraints For example variations in δT and δx with δN 0 lead to δS STx δT SxiT δxi δJi JiTx δT JixjT δxj 172 Substituting these variations into Eq 171 leads to STx δT2 JixjT δxi δxj 0 173 Note that the cross terms proportional to δTδxi cancel due to the Maxwell relation in Eq 156 Equation 173 is a quadratic form and must be positive for all choices of δT and δx The resulting constraints on the coefficients are independent of how the system was initially partitioned into subsystems A and B and represent the conditions for stable equilibrium If only δT is nonzero Eq 171 requires STx 0 implying a positive heat capacity since Cx dQdTx T STx 0 174 If only one of the δxi in Eq 171 is nonzero the corresponding response function xiJiTxji must be positive However a more general requirement exists since all δx values may be chosen nonzero The general requirement is that the matrix of coefficients JixjT must be positive definite A matrix is positive definite if all of its eigenvalues are positive It is necessary but not sufficient that all the diagonal elements of such a matrix the inverse response functions be positive leading to further constraints between the response functions Including chemical work for a gas the appropriate matrix is PVTN PNTV μVTN μNTV 175 In addition to the positivity of the response functions κTN V1 VPTN and NμTV the determinant of the matrix must be positive requiring PNTV μVTN PVTN μNTV 0 176 Another interesting consequence of Eq 171 pertains to the critical point of a gas where PV Tc N 0 Assuming that the critical isotherm can be analytically expanded as δPT Tc PV Tc N δV 12 ²PV² Tc N δV² 16 ³PV³ Tc N δV³ 177 the stability condition δPδV 0 implies that ²PV² Tc N must be zero and the third derivative negative if the first derivative vanishes This condition is used to obtain the critical point of the gas from approximations to the isotherms as we shall do in a later chapter for the van der Waals equation In fact it is usually not justified to make a Taylor expansion around the critical point as in Eq 177 although the constraint δPδV 0 remains applicable Fig 121 Stability condition at criticality illustrated for van der Waals isotherms 110 The third law Differences in entropy between two states can be computed using the second law from ΔS dQrevT Lowtemperature experiments indicate that ΔSX T vanishes as T goes to zero for any set of coordinates X This observation is independent of the other laws of thermodynamics leading to the formulation of a third law by Nernst which states The entropy of all systems at zero absolute temperature is a universal constant that can be taken to be zero The above statement actually implies that lim T0 SX T 0 178 which is a stronger requirement than the vanishing of the differences ΔSX T This extended condition has been tested for metastable phases of a substance Certain materials such as sulfur or phosphine can exist in a number of rather similar crystalline structures allotropes Of course at a given temperature only one of these structures is truly stable Let us imagine that as the hightemperature equilibrium phase A is cooled slowly it makes a transition at a temperature T to phase B releasing latent heat L Under more rapid cooling conditions the transition is avoided and phase A persists in metastable equilibrium The entropies in the two phases can be calculated by measuring the heat capacities CAT and CBT Starting from T 0 the entropy at a temperature slightly above T can be computed along the two possible paths as ST ϵ SA0 ₀T dT CATT SB0 ₀T dT CBTT LT 179 By such measurements we can indeed verify that SA0 SB0 0 Fig 122 Heat capacity measurements on allotropes of the same material Consequences of the third law 1 Since ST 0 X 0 for all coordinates X lim T0 SX T 0 180 2 Heat capacities must vanish as T 0 since ST X S0 X ₀T dT CXTT 181 and the integral diverges as T 0 unless lim T0 CXT 0 182 3 Thermal expansivities also vanish as T 0 since αJ 1x xT J 1x SJ T 183 The second equality follows from the Maxwell relation in Eq 156 The vanishing of the latter is guaranteed by Eq 180 4 It is impossible to cool any system to absolute zero temperature in a finite number of steps For example we can cool a gas by an adiabatic reduction in pressure Since the curves of S versus T for different pressures must join at T 0 successive steps involve progressively smaller changes in S and in T on approaching zero temperature Alternatively the unattainability of zero temperatures implies that ST 0 P is independent of P This is a weaker statement of the third law which also implies the equality of zero temperature entropy for different substances Fig 123 An infinite number of steps is required to cool a gas to T 0 by a series of isothermal decompressions In the following chapters we shall attempt to justify the laws of thermodynamics from a microscopic point of view The first law is clearly a reflection of the conservation of energy which also operates at the microscopic level The zeroth and second laws suggest an irreversible approach to equilibrium a concept that has no analog at the particulate level It is justified as reflecting the collective tendencies of large numbers of degrees of freedom In statistical mechanics the entropy per particle is calculated as SN kB lngNN where gN is the degeneracy of the states number of configurations with the same energy of a system of N particles The third law of thermodynamics thus requires that lim N lngNN 0 at T 0 limiting the possible number of ground states for a manybody system The above condition does not hold within the framework of classical statistical mechanics as there are examples of both noninteracting such as an ideal gas and interacting the frustrated spins in a triangular antiferromagnet systems with a large number of degenerate ground states and a finite zerotemperature entropy However classical mechanics is inapplicable at very low temperatures and energies where quantum effects become important The third law is then equivalent to a restriction on the degeneracy of ground states of a quantum mechanical system³ While this can be proved for a noninteracting ³ For any spin system with rotational symmetry such as a ferromagnet there are of course many possible ground states related by rotations However the number of such states does not grow with the number of spins N thus such degeneracy does not affect the absence of a thermodynamic entropy at zero temperature system of identical particles as we shall demonstrate in the final chapter there is no general proof of its validity with interactions Unfortunately the onset of quantum effects and other possible origins of the breaking of classical degeneracy is systemspecific Hence it is not a priori clear how low the temperature must be before the predictions of the third law can be observed Another deficiency of the law is its inapplicability to glassy phases Glasses result from the freezing of supercooled liquids into configurations with extremely slow dynamics While not truly equilibrium phases and hence subject to all the laws of thermodynamics they are effectively so due to the slowness of the dynamics A possible test of the applicability of the third law to glasses is discussed in the problems Problems for chapter 1 1 Surface tension thermodynamic properties of the interface between two phases are described by a state function called the surface tension S It is defined in terms of the work required to increase the surface area by an amount dA through dW S dA a By considering the work done against surface tension in an infinitesimal change in radius show that the pressure inside a spherical drop of water of radius R is larger than outside pressure by 2SR What is the air pressure inside a soap bubble of radius R b A water droplet condenses on a solid surface There are three surface tensions involved Saw Ssw and Ssa where a s and w refer to air solid and water respectively Calculate the angle of contact and find the condition for the appearance of a water film complete wetting c In the realm of large bodies gravity is the dominant force while at small distances surface tension effects are all important At room temperature the surface tension of water is So 7 102 N m1 Estimate the typical length scale that separates large and small behaviors Give a couple of examples for where this length scale is important 2 Surfactants surfactant molecules such as those in soap or shampoo prefer to spread on the airwater surface rather than dissolve in water To see this float a hair on the surface of water and gently touch the water in its vicinity with a piece of soap This is also why a piece of soap can power a toy paper boat a The airwater surface tension So assumed to be temperatureindependent is reduced roughly by N kB TA where N is the number of surfactant particles and A is the area Explain this result qualitatively b Place a drop of water on a clean surface Observe what happens to the airwater surface contact angle as you gently touch the droplet surface with a small piece of soap and explain the observation c More careful observations show that at higher surfactant densities SAT N kB TA Nb2 2aANA2 and TSA A NbN kB where a and b are constants Obtain the expression for SA T and explain qualitatively the origin of the corrections described by a and b d Find an expression for CS CA in terms of EAT EAS S SAT and TSA 3 Temperature scales prove the equivalence of the ideal gas temperature scale Θ and the thermodynamic scale T by performing a Carnot cycle on an ideal gas The ideal gas satisfies PV N kB Θ and its internal energy E is a function of Θ only However you may not assume that E Θ You may wish to proceed as follows a Calculate the heat exchanges QH and QC as a function of ΘH ΘC and the volume expansion factors b Calculate the volume expansion factor in an adiabatic process as a function of Θ c Show that QHQC ΘHΘC 4 Equations of state the equation of state constrains the form of internal energy as in the following examples a Starting from dE TdS PdV show that the equation of state PV N kB T in fact implies that E can only depend on T b What is the most general equation of state consistent with an internal energy that depends only on temperature c Show that for a van der Waals gas CV is a function of temperature alone 5 The ClausiusClapeyron equation describes the variation of boiling point with pressure It is usually derived from the condition that the chemical potentials of the gas and liquid phases are the same at coexistence For an alternative derivation consider a Carnot engine using one mole of water At the source P T the latent heat L is supplied converting water to steam There is a volume increase V associated with this process The pressure is adiabatically decreased to P dP At the sink P dP T dT steam is condensed back to water a Show that the work output of the engine is W VdP OdP2 Hence obtain the ClausiusClapeyron equation dPdTboiling LTV b What is wrong with the following argument The heat QH supplied at the source to convert one mole of water to steam is LT At the sink LT dT is supplied to condense one mole of steam to water The difference dT dLdT must equal the work W VdP equal to LdTT from Eq 1 Hence dLdT LT implying that L is proportional to T c Assume that L is approximately temperatureindependent and that the volume change is dominated by the volume of steam treated as an ideal gas that is V NkB TP Integrate Eq 1 to obtain PT d A hurricane works somewhat like the engine described above Water evaporates at the warm surface of the ocean steam rises up in the atmosphere and condenses to water at the higher and cooler altitudes The Coriolis force converts the upward suction of the air to spiral motion Using ice and boiling water you can create a little storm in a tea cup Typical values of warm ocean surface and high altitude temperatures are 80F and 120F respectively The warm water surface layer must be at least 200 feet thick to provide sufficient water vapor as the hurricane needs to condense about 90 million tons of water vapor per hour to maintain itself Estimate the maximum possible efficiency and power output of such a hurricane The latent heat of vaporization of water is about 23 106 J kg1 e Due to gravity atmospheric pressure Ph drops with the height h By balancing the forces acting on a slab of air behaving like a perfect gas of thickness dh show that Ph P0 expmghkT where m is the average mass of a molecule in air f Use the above results to estimate the boiling temperature of water on top of Mount Everest h 9 km The latent heat of vaporization of water is about 23 106 J kg1 6 Glass liquid quartz if cooled slowly crystallizes at a temperature Tm and releases latent heat L Under more rapid cooling conditions the liquid is supercooled and becomes glassy a As both phases of quartz are almost incompressible there is no work input and changes in internal energy satisfy dE TdS μdN Use the extensivity condition to obtain the expression for μ in terms of E T S and N b The heat capacity of crystalline quartz is approximately CX αT3 while that of glassy quartz is roughly CG βT where α and β are constants Assuming that the third law of thermodynamics applies to both crystalline and glass phases calculate the entropies of the two phases at temperatures T Tm c At zero temperature the local bonding structure is similar in glass and crystalline quartz so that they have approximately the same internal energy E0 Calculate the internal energies of both phases at temperatures T Tm d Use the condition of thermal equilibrium between two phases to compute the equilibrium melting temperature Tm in terms of α and β e Compute the latent heat L in terms of α and β f Is the result in the previous part correct If not which of the steps leading to it is most likely to be incorrect 7 Filament for an elastic filament it is found that at a finite range in temperature a displacement x requires a force J ax bT cTx where a b and c are constants Furthermore its heat capacity at constant displacement is proportional to temperature that is Cx AxT a Use an appropriate Maxwell relation to calculate Sx T b Show that A has to in fact be independent of x that is dAdx 0 c Give the expression for ST x assuming S0 0 S0 d Calculate the heat capacity at constant tension that is CJ T ST J as a function of T and J 8 Hard core gas a gas obeys the equation of state PV Nb NkB T and has a heat capacity CV independent of temperature N is kept fixed in the following a Find the Maxwell relation involving SV TN b By calculating dET V show that E is a function of T and N only c Show that γ CPCV 1 NkBCV independent of T and V d By writing an expression for EP V or otherwise show that an adiabatic change satisfies the equation PV Nbγ constant 9 Superconducting transition many metals become superconductors at low temperatures T and magnetic fields B The heat capacities of the two phases at zero magnetic field are approximately given by CsT VαT3 in the superconducting phase CnT V βT3 γT in the normal phase where V is the volume and α β γ are constants There is no appreciable change in volume at this transition and mechanical work can be ignored throughout this problem a Calculate the entropies SsT and SnT of the two phases at zero field using the third law of thermodynamics b Experiments indicate that there is no latent heat L 0 for the transition between the normal and superconducting phases at zero field Use this information to obtain the transition temperature Tc as a function of α β and γ c At zero temperature the electrons in the superconductor form bound Cooper pairs As a result the internal energy of the superconductor is reduced by an amount VΔ that is EnT 0 E0 and EsT 0 E0 VΔ for the metal and superconductor respectively Calculate the internal energies of both phases at finite temperatures d By comparing the Gibbs free energies or chemical potentials in the two phases obtain an expression for the energy gap Δ in terms of α β and γ e In the presence of a magnetic field B inclusion of magnetic work results in dE TdS BdM μdN where M is the magnetization The superconducting phase is a perfect diamagnet expelling the magnetic field from its interior such that Ms VB4π in appropriate units The normal metal can be regarded as approximately nonmagnetic with Mn 0 Use this information in conjunction with previous results to show that the superconducting phase becomes normal for magnetic fields larger than BcT B0 1 T2 Tc2 giving an expression for B0 10 Photon gas Carnot cycle the aim of this problem is to obtain the blackbody radiation relation ET V VT4 starting from the equation of state by performing an infinitesimal Carnot cycle on the photon gas figure a Express the work done W in the above cycle in terms of dV and dP b Express the heat absorbed Q in expanding the gas along an isotherm in terms of P dV and an appropriate derivative of ET V c Using the efficiency of the Carnot cycle relate the above expressions for W and Q to T and dT d Observations indicate that the pressure of the photon gas is given by P AT4 where A π2 kB4 45 ħc3 is a constant Use this information to obtain ET V assuming E0 V 0 e Find the relation describing the adiabatic paths in the above cycle 11 Irreversible processes a Consider two substances initially at temperatures T10 and T20 coming to equilibrium at a final temperature Tf through heat exchange By relating the direction of heat flow to the temperature difference show that the change in the total entropy which can be written as ΔS ΔS1 ΔS2 T10Tf dQ1 T1 T20Tf dQ2 T2 T1 T2 T1 T2 dQ must be positive This is an example of the more general condition that in a closed system equilibrium is characterized by the maximum value of entropy S b Now consider a gas with adjustable volume V and diathermal walls embedded in a heat bath of constant temperature T and fixed pressure P The change in the entropy of the bath is given by ΔSbath ΔQbath T ΔQgas T 1T ΔEgas PΔVgas By considering the change in entropy of the combined system establish that the equilibrium of a gas at fixed T and P is characterized by the minimum of the Gibbs free energy G E PV TS 12 The Solar System originated from a dilute gas of particles sufficiently separated from other such clouds to be regarded as an isolated system Under the action of gravity the particles coalesced to form the Sun and planets a The motion and organization of planets is much more ordered than the original dust cloud Why does this not violate the second law of thermodynamics b The nuclear processes of the Sun convert protons to heavier elements such as carbon Does this further organization lead to a reduction in entropy c The evolution of life and intelligence requires even further levels of organization How is this achieved on Earth without violating the second law 2 Probability 21 General definitions The laws of thermodynamics are based on observations of macroscopic bodies and encapsulate their thermal properties On the other hand matter is composed of atoms and molecules whose motions are governed by more fundamental laws classical or quantum mechanics It should be possible in principle to derive the behavior of a macroscopic body from the knowledge of its components This is the problem addressed by kinetic theory in the following chapter Actually describing the full dynamics of the enormous number of particles involved is quite a daunting task As we shall demonstrate for discussing equilibrium properties of a macroscopic system full knowledge of the behavior of its constituent particles is not necessary All that is required is the likelihood that the particles are in a particular microscopic state Statistical mechanics is thus an inherently probabilistic description of the system and familiarity with manipulations of probabilities is an important prerequisite The purpose of this chapter is to review some important results in the theory of probability and to introduce the notations that will be used in the following chapters The entity under investigation is a random variable x which has a set of possible outcomes S x₁ x₂ The outcomes may be discrete as in the case of a coin toss Scoins head tail or a dice throw Sdice 1 2 3 4 5 6 or continuous as for the velocity of a particle in a gas Sv vx vy vz or the energy of an electron in a metal at zero temperature Sε 0 ε εF An event is any subset of outcomes E S and is assigned a probability pE for example pdice1 16 or pdice1 3 13 From an axiomatic point of view the probabilities must satisfy the following conditions i Positivity pE 0 that is all probabilities must be real and nonnegative ii Additivity pA or B pA pB if A and B are disconnected events iii Normalization pS 1 that is the random variable must have some outcome in S From a practical point of view we would like to know how to assign probability values to various outcomes There are two possible approaches 1 Objective probabilities are obtained experimentally from the relative frequency of the occurrence of an outcome in many tests of the random variable If the random process is repeated N times and the event A occurs NA times then pA lim N NA N For example a series of N 100 200 300 throws of a dice may result in N₁ 19 30 48 occurrences of 1 The ratios 019 015 016 provide an increasingly more reliable estimate of the probability pdice1 2 Subjective probabilities provide a theoretical estimate based on the uncertainties related to lack of precise knowledge of outcomes For example the assessment pdice1 16 is based on the knowledge that there are six possible outcomes to a dice throw and that in the absence of any prior reason to believe that the dice is biased all six are equally likely All assignments of probability in statistical mechanics are subjectively based The consequences of such subjective assignments of probability have to be checked against measurements and they may need to be modified as more information about the outcomes becomes available 22 One random variable As the properties of a discrete random variable are rather well known here we focus on continuous random variables which are more relevant to our purposes Consider a random variable x whose outcomes are real numbers that is Sx x The cumulative probability function CPF Px is the probability of an outcome with any value less than x that is Px probE x Px must be a monotonically increasing function of x with P 0 and P 1 Fig 21 A typical cumulative probability function The probability density function PDF is defined by px dPxdx Hence px dx probE x x dx As a probability density it is positive and normalized such that probS dx px 1 21 Note that since px is a probability density it has dimensions of x¹ and changes its value if the units measuring x are modified Unlike Px the PDF has no upper bound that is 0 px and may contain divergences as long as they are integrable The expectation value of any function Fx of the random variable is Fx dx px Fx 22 Fig 22 A typical probability density function Fig 23 Obtaining the PDF for the function Fx The function Fx is itself a random variable with an associated PDF of pFf df probFx f f df There may be multiple solutions xi to the equation Fx f and pFf df Σi pxi dxi pFf Σi pxi dxdFxxi 23 The factors of dxdF are the Jacobians associated with the change of variables from x to F For example consider px λ expλx2 and the function Fx x2 There are two solutions to Fx f located at x f with corresponding Jacobians f122 Hence PFf λ2 expλf 12f 12f λ expλf 2f for f 0 and pFf 0 for f 0 Note that pFf has an integrable divergence at f 0 Fig 24 Probability density functions for x and F x2 Moments of the PDF are expectation values for powers of the random variable The nth moment is mn xn dx px xn The characteristic function is the generator of moments of the distribution It is simply the Fourier transform of the PDF defined by pk eikx dx px eikx The PDF can be recovered from the characteristic function through the inverse Fourier transform px 12π dk pk eikx Moments of the distribution are obtained by expanding pk in powers of k pk n0 ikn n xn n0 ikn n xn Moments of the PDF around any point x₀ can also be generated by expanding eikx₀ pk eikxx₀ n0 ikn n xx₀n The cumulant generating function is the logarithm of the characteristic function Its expansion generates the cumulants of the distribution defined through ln pk n1 ikn n xnc Relations between moments and cumulants can be obtained by expanding the logarithm of pk in Eq 27 and using ln1 ε n1 1n1 εn n The first four cumulants are called the mean variance skewness and curtosis or kurtosis of the distribution respectively and are obtained from the moments as xc x x2c x2 x2 x3c x3 3x2 x 2 x3 x4c x4 4 x3 x 3 x22 12 x2 x2 6 x4 The cumulants provide a useful and compact way of describing a PDF An important theorem allows easy computation of moments in terms of the cumulants represent the nth cumulant graphically as a connected cluster of n points The mth moment is then obtained by summing all possible subdivisions of m points into groupings of smaller connected or disconnected clusters The contribution of each subdivision to the sum is the product of the connected cumulants that it represents Using this result the first four moments are computed graphically x x2 x3 3 x4 4 3 6 The corresponding algebraic expressions are x xc x2 x2c xc2 x3 x3c 3 x2c xc xc3 x4 x4c 4 x3c xc 3 x2c2 6 x2c xc2 xc4 This theorem which is the starting point for various diagrammatic computations in statistical mechanics and field theory is easily proved by equating the expressions in Eqs 27 and 29 for pk m0 ikm m xm exp n1 ikn n xnc n pn ikn pn pn xnc npn Matching the powers of ikm on the two sides of the above expression leads to xm pn m n 1 pn npn xncpn The sum is restricted such that n pn m and leads to the graphical interpretation given above as numerical factor is simply the number of ways of breaking m points into pn clusters of n points 23 Some important probability distributions The properties of three commonly encountered probability distributions are examined in this section 1 The normal Gaussian distribution describes a continuous real random variable x with px 1 2πσ2 expx λ2 2σ2 The corresponding characteristic function also has a Gaussian form pk dx 1 2πσ2 expx λ2 2σ2 i k x expi k λ k2 σ2 2 Cumulants of the distribution can be identified from ln pk i k λ k2 σ2 2 using Eq 29 as xc λ x2c σ2 x3c x4c 0 The normal distribution is thus completely specified by its two first cumulants This makes the computation of moments using the cluster expansion Eqs 212 particularly simple and x λ x2 σ2 λ2 x3 3σ2 λ λ3 x4 3σ4 6σ2 λ2 λ4 The normal distribution serves as the starting point for most perturbative computations in field theory The vanishing of higher cumulants implies that all graphical computations involve only products of onepoint and twopoint known as propagators clusters 2 The binomial distribution consider a random variable with two outcomes A and B eg a coin toss of relative probabilities pA and pB 1 pA The probability that in N trials the event A occurs exactly NA times eg 5 heads in 12 coin tosses is given by the binomial distribution pNNA N choose NA pANA pBN NA The prefactor N choose NA NNAN NA is just the coefficient obtained in the binomial expansion of pA pBN and gives the number of possible orderings of NA events A and NB N NA events B The characteristic function for this discrete distribution is given by tilde pNk eikNA sumNA0N NNAN NA pANA pBN NA eikNA pA eik pBN The resulting cumulant generating function is ln tilde pNk N ln pA eik pB N ln tilde p1k where ln tilde p1k is the cumulant generating function for a single trial Hence the cumulants after N trials are simply N times the cumulants in a single trial In each trial the allowed values of NA are 0 and 1 with respective probabilities pB and pA leading to NAell pA for all ell After N trials the first two cumulants are NAc N pA NA2c N pA pA2 N pA pB A measure of fluctuations around the mean is provided by the standard deviation which is the square root of the variance While the mean of the binomial distribution scales as N its standard deviation only grows as sqrtN Hence the relative uncertainty becomes smaller for large N The binomial distribution is straightforwardly generalized to a multinomial distribution when the several outcomes A B M occur with probabilities pA pB pM The probability of finding outcomes NA NB NM in a total of N NA NB NM trials is pNNA NB NM NNA NB NM pANA pBNB pMNM 3 The Poisson distribution the classical example of a Poisson process is radioactive decay Observing a piece of radioactive material over a time interval T shows that a The probability of one and only one event decay in the interval t t dt is proportional to dt as dt 0 b The probabilities of events at different intervals are independent of each other Fig 26 Subdividing the time interval into small segments of size dt The probability of observing exactly M decays in the interval T is given by the Poisson distribution It is obtained as a limit of the binomial distribution by subdividing the interval into N Tdt 1 segments of size dt In each segment an event occurs with probability p alpha dt and there is no event with probability q 1 alpha dt As the probability of more than one event in dt is too small to consider the process is equivalent to a binomial one Using Eq 221 the characteristic function is given by tilde pk p eik qn limdt 0 1 alpha dt eik 1Tdt exp alpha eik 1 T The Poisson PDF is obtained from the inverse Fourier transform in Eq 26 as px integralinftyinfty dk2pi exp alpha eik 1 T i k x ealpha T integralinftyinfty dk2pi eik x sumM0infty alpha TMM eik M using the power series for the exponential The integral over k is integralinftyinfty dk2pi ei k xM deltaxM leading to palpha Tx sumM0infty ealpha T alpha TMM deltaxM This answer clearly realizes that the only possible values of x are integers M The probability of M events is thus palpha TM ealpha T alpha TM M The cumulants of the distribution are obtained from the expansion ln tilde palpha Tk alpha T eik 1 alpha T sumn1infty ikn n Mnc alpha T All cumulants have the same value and the moments are obtained from Eqs 212 as M alpha T M2 alpha T2 alpha T M3 alpha T3 3 alpha T2 alpha T Example Assuming that stars are randomly distributed in the Galaxy clearly unjustified with a density n what is the probability that the nearest star is at a distance R Since the probability of finding a star in a small volume dV is n dV and they are assumed to be independent the number of stars in a volume V is described by a Poisson process as in Eq 228 with alpha n The probability pR of encountering the first star at a distance R is the product of the probabilities pnV0 of finding zero stars in the volume V 4 pi R3 3 around the origin and pndV1 of finding one star in the shell of volume dV 4 pi R2 dR at a distance R Both pnV0 and pndV1 can be calculated from Eq 228 and pR dR pnV0 pndV1 e4 pi R3 n 3 e4 pi R2 n dR 4 pi R2 n dR pR 4 pi R2 n exp4 pi R3 n 3 24 Many random variables With more than one random variable the set of outcomes is an Ndimensional space Sx infty x1 x2 xN infty For example describing the location and velocity of a gas particle requires six coordinates The joint PDF px is the probability density of an outcome in a volume element dNx producti1N dxi around the point x x1 x2 xN The joint PDF is normalized such that pxS integral dN x px 1 If and only if the N random variables are independent the joint PDF is the product of individual PDFs px producti1N pixi The unconditional PDF describes the behavior of a subset of random variables independent of the values of the others For example if we are interested only in the location of a gas particle an unconditional PDF can be constructed by integrating over all velocities at a given location pvecx integral d3 vecv pvecx vecv more generally px1 xm integral productim1N dxi px1 xN The conditional PDF describes the behavior of a subset of random variables for specified values of the others For example the PDF for the velocity of a particle at a particular location x denoted by pvx is proportional to the joint PDF pvx px vN The constant of proportionality obtained by normalizing pvx is N d³vpx v px 235 the unconditional PDF for a particle at x In general the unconditional PDFs are obtained from Bayes theorem as px₁ xₘxₘ₁ xₙ px₁ xₙpxₘ₁ xₙ 236 Note that if the random variables are independent the unconditional PDF is equal to the conditional PDF The expectation value of a function Fx is obtained as before from Fx dⁿxpxFx 237 The joint characteristic function is obtained from the Ndimensional Fourier transformation of the joint PDF pk expij1 to N kⱼxⱼ 238 The joint moments and joint cumulants are generated by pk and ln pk respectively as x₁ⁿ¹x₂ⁿ² xₙⁿᴺ ik₁ⁿ¹ik₂ⁿ² ikₙⁿᴺ pk 0 x₁ⁿ¹ x₂ⁿ² xₙⁿᴺ ik₁ⁿ¹ik₂ⁿ² ikₙⁿᴺ ln pk 0 239 The previously described graphical relation between joint moments all clusters of labeled points and joint cumulant connected clusters is still applicable For example from x₁x₂ x₁²x₂ 2 we obtain x₁x₂ x₁cx₂c x₁ x₂c and x₁²x₂ x₁c²x₂c x₁²cx₂c 2x₁ x₂cx₁c x₁² x₂c 240 The connected correlation xα xβc is zero if xα and xβ are independent random variables The joint Gaussian distribution is the generalization of Eq 215 to N random variables as px 12πⁿ detC exp12ₘₙC¹ₘₙxₘ λₘxₙ λₙ 241 where C is a symmetric matrix and C¹ is its inverse The simplest way to get the normalization factor is to make a linear transformation from the variables yⱼ xⱼ λⱼ using the unitary matrix that diagonalizes C This reduces the normalization to that of the product of N Gaussians whose variances are determined by the eigenvalues of C The product of the eigenvalues is the determinant detC This also indicates that the matrix C must be positive definite The corresponding joint characteristic function is obtained by similar manipulations and is given by pk expikₘλₘ 12Cₘₙkₘkₙ 242 where the summation convention implicit summation over a repeated index is used The joint cumulants of the Gaussian are then obtained from ln pk as xₘc λₘ xₘ xₙc Cₘₙ 243 with all higher cumulants equal to zero In the special case of λₘ 0 all odd moments of the distribution are zero while the general rules for relating moments to cumulants indicate that any even moment is obtained by summing over all ways of grouping the involved random variables into pairs for example xαxβxγxδ CαβCγδ CαγCβδ CαδCβγ 244 In the context of field theories this result is referred to as Wicks theorem 25 Sums of random variables and the central limit theorem Consider the sum X i1 to N xᵢ where xᵢ are random variables with a joint PDF of px The PDF for X is pₓx dᴺxpxδx xᵢ i1 to N1 dxᵢ px₁ xₙ₁ x x₁ xₙ₁ 245 and the corresponding characteristic function using Eq 238 is given by pₓk expikj1 to N xⱼ pk₁ k₂ kₙ k 246 Cumulants of the sum are obtained by expanding ln pₓk ln pk₁ k₂ kₙ k iki1 to N xᵢc ik²2 i₁ i₂ 1 to N xᵢ₁xᵢ₂c 247 as Xc i1 to N xᵢc X²c i j 1 to N xᵢxⱼc 248 If the random variables are independent px pᵢxᵢ and pₓk pᵢk the cross cumulants in Eq 248 vanish and the nth cumulant of X is simply the sum of the individual cumulants Xⁿc i1 to N xᵢⁿc When all the N random variables are independently taken from the same distribution px this implies Xⁿc Nxⁿc generalizing the result obtained previously for the binomial distribution For large values of N the average value of the sum is proportional to N while fluctuations around the mean as measured by the standard deviation grow only as N The random variable y X NxcN has zero mean and cumulants that scale as yⁿc N¹ⁿ² As N only the second cumulant survives and the PDF for y converges to the normal distribution limN py i1 to N xᵢ NxcN 12πx²c expy²2x²c 249 Note that the Gaussian distribution is the only distribution with only first and second cumulants The convergence of the PDF for the sum of many random variables to a normal distribution is an essential result in the context of statistical mechanics where such sums are frequently encountered The central limit theorem states a more general form of this result it is not necessary for the random variables to be independent as the condition i₁iₘxᵢ₁ xᵢₘc ONᵐ² is sufficient for the validity of Eq 249 Note that the above discussion implicitly assumes that the cumulants of the individual random variables appearing in Eq 248 are finite What happens if this is not the case that is when the variables are taken from a very wide PDF The sum may still converge to a socalled Levy distribution Consider a sum of N independent identically distributed random variables with the mean set to zero for convenience The variance does not exist if the individual PDF falls off slowly at large values as pᵢx p₁x 1x¹α with 0 α 2 α 0 is required to make sure that the distribution is normalizable while for α 2 the variance is finite The behavior of p₁x at large x determines the behavior of p₁k at small k and simple power counting indicates that the expansion of p₁k is singular starting with kᵅ Based on this argument we conclude that ln pₓk N ln p₁k Nakᵅ higher order terms 250 As before we can define a rescaled variable y X N1 α to get rid of the N dependence of the leading term in the above equation resulting in limN pyk akalpha The higherorder terms appearing in Eq 250 scale with negative powers of N and vanish as N The simplest example of a Levy distribution is obtained for α1 and corresponds to py a π y² a²This is the Cauchy distribution discussed in the problems section For other values of α the distribution does not have a simple closed form but can be written as the asymptotic series pαy 1 π sumn1 1n1 sinn π 2 α Γ1 nα n an y1 n α Such distributions describe phenomena with large rare events characterized here by a tail that falls off slowly as pα y y1α 26 Rules for large numbers To describe equilibrium properties of macroscopic bodies statistical mechanics has to deal with the very large number N of microscopic degrees of freedom Actually taking the thermodynamic limit of N leads to a number of simplifications some of which are described in this section There are typically three types of N dependence encountered in the thermodynamic limit a Intensive quantities such as temperature T and generalized forces for example pressure P and magnetic field B are independent of N that is O N⁰ b Extensive quantities such as energy E entropy S and generalized displacements for example volume V and magnetization M are proportional to N that is O N¹ c Exponential dependence that is O exp Nφ is encountered in enumerating discrete microstates or computing available volumes in phase space Other asymptotic dependencies are certainly not ruled out a priori For example the Coulomb energy of N ions at fixed density scales as Q² R N53 Such dependencies are rarely encountered in everyday physics The Coulomb interaction of ions is quickly screened by counterions resulting in an extensive overall energy This is not the case in astrophysical problems since the gravitational energy is not screened For example the entropy of a black hole is proportional to the square of its mass In statistical mechanics we frequently encounter sums or integrals of exponential variables Performing such sums in the thermodynamic limit is considerably simplified due to the following results 1 Summation of exponential quantities consider the sum S sumi1N εi where each term is positive with an exponential dependence on N that is 0 εi O expNφi and the number of terms N is proportional to some power of N Fig 27 A sum over N exponentially large quantities is dominated by the largest term Such a sum can be approximated by its largest term εmax in the following sense Since for each term in the sum 0 εi εmax εmax S N εmax An intensive quantity can be constructed from ln S N which is bounded by ln εmax N ln S N ln εmax N ln N N For N Np the ratio ln N N vanishes in the large N limit and limN ln S N ln εmax N φmax 2 Saddle point integration similarly an integral of the form I dx exp Nφx can be approximated by the maximum value of the integrand obtained at a point xmax that maximizes the exponent φx Expanding the exponent around this point gives I dx exp N φxmax 12 φxmaxx xmax² Note that at the maximum the first derivative φxmax is zero while the second derivative φxmax is negative Terminating the series at the quadratic order results in I eNφxmax dx exp N2 φxmaxx xmax² sqrt2π N φxmax eNφxmax where the range of integration has been extended to The latter is justified since the integrand is negligibly small outside the neighborhood of xmax Fig 28 Saddle point evaluation of an exponential integral There are two types of correction to the above result Firstly there are higherorder terms in the expansion of φx around xmax These corrections can be looked at perturbatively and lead to a series in powers of 1 N Secondly there may be additional local maxima for the function A maximum at xmax leads to a similar Gaussian integral that can be added to Eq 260 Clearly such contributions are smaller by O expNφxmax φxmax Since all these corrections vanish in the thermodynamic limit limN ln I N limN φxmax 1 2N ln N φxmax 2π O 1N² φxmax The saddle point method for evaluating integrals is the extension of the above result to more general integrands and integration paths in the complex plane The appropriate extremum in the complex plane is a saddle point The simplified version presented above is sufficient for our needs Stirlings approximation for N at large N can be obtained by saddle point integration In order to get an integral representation of N start with the result ₀ dxeαx 1 α Repeated differentiation of both sides of the above equation with respect to α leads to ₀ dx xN eαx N αN1 Although the above derivation only applies to integer N it is possible to define by analytical continuation a function ΓN1 N 0 dx xN ex 264 for all N While the integral in Eq 264 is not exactly in the form of Eq 258 it can still be evaluated by a similar method The integrand can be written as expNϕx with ϕx ln x xN The exponent has a maximum at xmax N with ϕxmax ln N 1 and ϕxmax 1N2 Expanding the integrand in Eq 264 around this point yields N dx expN ln N N 12N x N2 NN eN 2πN 265 where the integral is evaluated by extending its limits to Stirlings formula is obtained by taking the logarithm of Eq 265 as ln N N ln N N 12 ln2πN O1N 266 27 Information entropy and estimation Information consider a random variable with a discrete set of outcomes S xi occurring with probabilities pi for i 1 M In the context of information theory there is a precise meaning to the information content of a probability distribution Let us construct a message from N independent outcomes of the random variable Since there are M possibilities for each character in this message it has an apparent information content of N ln2 M bits that is this many binary bits of information have to be transmitted to convey the message precisely On the other hand the probabilities pi limit the types of messages that are likely For example if p2 p1 it is very unlikely to construct a message with more x1 than x2 In particular in the limit of large N we expect the message to contain roughly Ni Npi occurrences of each symbol The number of typical messages thus corresponds to the number of ways of rearranging the Ni occurrences of xi and is given by the multinomial coefficient g N i1M Ni 267 This is much smaller than the total number of messages Mn To specify one out of g possible sequences requires ln2 g N i1M pi ln2 pi for N 268 bits of information The last result is obtained by applying Stirlings approximation for ln N It can also be obtained by noting that 1 i pi N Ni N i1M piNi Ni g i1M piNpi 269 where the sum has been replaced by its largest term as justified in the previous section Shannons theorem proves more rigorously that the minimum number of bits necessary to ensure that the percentage of errors in N trials vanishes in the N limit is ln2 g For any nonuniform distribution this is less than the N ln2 M bits needed in the absence of any information on relative probabilities The difference per trial is thus attributed to the information content of the probability distribution and is given by Ipi ln2 M i1M pi ln2 pi 270 Entropy Eq 267 is encountered frequently in statistical mechanics in the context of mixing M distinct components its natural logarithm is related to the entropy of mixing More generally we can define an entropy for any probability distribution as S i1M pi ln pi ln pi 271 The above entropy takes a minimum value of zero for the delta function distribution pi δij and a maximum value of ln M for the uniform distribution pi 1M S is thus a measure of dispersity disorder of the distribution and does not depend on the values of the random variables xi A onetoone mapping to fi Fxi leaves the entropy unchanged while a manytoone mapping makes the distribution more ordered and decreases S For example if the two values x1 and x2 are mapped onto the same f the change in entropy is ΔSx1 x2 f p1 ln p1p1 p2 p2 ln p2p1 p2 0 272 Estimation the entropy S can also be used to quantify subjective estimates of probabilities In the absence of any information the best unbiased estimate is that all M outcomes are equally likely This is the distribution of maximum entropy If additional information is available the unbiased estimate is obtained by maximizing the entropy subject to the constraints imposed by this information For example if it is known that Fx f we can maximize Sα β pi i pi ln pi α i pi 1 β i pi Fxi f 273 where the Lagrange multipliers α and β are introduced to impose the constraints of normalization and Fx f respectively The result of the optimization is a distribution pi expβFxi where the value of β is fixed by the constraint This process can be generalized to an arbitrary number of conditions It is easy to see that if the first n 2k moments and hence n cumulants of a distribution are specified the unbiased estimate is the exponential of an nthorder polynomial In analogy with Eq 271 we can define an entropy for a continuous random variable Sx x as S dx px ln px ln px 274 There are however problems with this definition as for example S is not invariant under a onetoone mapping After a change of variable to f Fx the entropy is changed by Fx As discussed in the following chapter canonically conjugate pairs offer a suitable choice of coordinates in classical statistical mechanics since the Jacobian of a canonical transformation is unity The ambiguities are also removed if the continuous variable is discretized This happens quite naturally in quantum statistical mechanics where it is usually possible to work with a discrete ladder of states The appropriate volume for discretization of phase space is then set by Plancks constant ℏ Problems for chapter 2 1 Characteristic functions calculate the characteristic function the mean and the variance of the following probability density functions a Uniform px 12a for a x a and px 0 otherwise b Laplace px 12a expxa c Cauchy px aπx2a2 The following two probability density functions are defined for x 0 Compute only the mean and variance for each d Rayleigh px xa2 expx22a2 e Maxwell px 2π x2a3 expx22a2 2 Directed random walk the motion of a particle in three dimensions is a series of independent steps of length ℓ Each step makes an angle θ with the z axis with a probability density pθ 2 cos2θ2π while the angle φ is uniformly distributed between 0 and 2π Note that the solid angle factor of sin θ is already included in the definition of pθ which is correctly normalized to unity The particle walker starts at the origin and makes a large number of steps N a Calculate the expectation values z x y z2 x2 and y2 and the covariances xy xz and yz b Use the central limit theorem to estimate the probability density px y z for the particle to end up at the point x y z 3 Tchebycheff inequality consider any probability density px for x with mean λ and variance σ2 Show that the total probability of outcomes that are more than nσ away from λ is less than 1n2 that is xλnσ dxpx 1n2 Hint Start with the integral defining σ2 and break it up into parts corresponding to xλ nσ and xλ nσ 4 Optimal selection in many specialized populations there is little variability among the members Is this a natural consequence of optimal selection a Let rα be n random numbers each independently chosen from a probability density pr with r 0 1 Calculate the probability density pnx for the largest value of this set that is for x maxr1 rn b If each rα is uniformly distributed between 0 and 1 calculate the mean and variance of x as a function of n and comment on their behavior at large n 5 Benfords law describes the observed probabilities of the first digit in a great variety of data sets such as stock prices Rather counterintuitively the digits 1 through 9 occur with probabilities 0301 0176 0125 0097 0079 0067 0058 0051 0046 respectively The key observation is that this distribution is invariant under a change of scale for example if the stock prices were converted from dollars to Persian rials Find a formula that fits the above probabilities on the basis of this observation 6 Information consider the velocity of a gas particle in one dimension v a Find the unbiased probability density p1v subject only to the constraint that the average speed is c that is v c b Now find the probability density p2v given only the constraint of average kinetic energy mv22 mc22 c Which of the above statements provides more information on the velocity Quantify the difference in information in terms of I2 I1 ln p2 ln p1 ln 2 7 Dice a dice is loaded such that 6 occurs twice as often as 1 a Calculate the unbiased probabilities for the six faces of the dice b What is the information content in bits of the above statement regarding the dice 8 Random matrices as a model for energy levels of complex nuclei Wigner considered N N symmetric matrices whose elements are random Let us assume that each element Mij for i j is an independent random variable taken from the probability density function pMij 12a for a Mij a and pMij 0 otherwise a Calculate the characteristic function for each element Mij b Calculate the characteristic function for the trace of the matrix T tr M i Mii c What does the central limit theorem imply about the probability density function of the trace at large N d For large N each eigenvalue λα α 1 2 N of the matrix M is distributed according to a probability density function pλ 2πλ02 sqrt1 λ2λ02 for λ0 λ λ0 and pλ 0 otherwise known as the Wigner semicircle rule Find the variance of λ Hint Changing variables to λ λ0 sin θ simplifies the integrals e If in the previous result we have λ02 4Na23 can the eigenvalues be independent of each other 9 Random deposition a mirror is plated by evaporating a gold electrode in vacuum by passing an electric current The gold atoms fly off in all directions and a portion of them sticks to the glass or to other gold atoms already on the glass plate Assume that each column of deposited atoms is independent of neighboring columns and that the average deposition rate is d layers per second a What is the probability of m atoms deposited at a site after a time t What fraction of the glass is not covered by any gold atoms b What is the variance in the thickness 10 Diode the current I across a diode is related to the applied voltage V via I I0 expeVkT 1 The diode is subject to a random potential V of zero mean and variance σ2 which is Gaussian distributed Find the probability density pI for the current I flowing through the diode Find the most probable value for I the mean value of I and indicate them on a sketch of pI 11 Mutual information consider random variables x and y distributed according to a joint probability px y The mutual information between the two variables is defined by Mx y xy px y ln px y pxx pyy where px and py denote the unconditional probabilities for x and y a Relate Mx y to the entropies Sx y Sx and Sy obtained from the corresponding probabilities b Calculate the mutual information for the joint Gaussian form px y expax22 by22 cxy 12 Semiflexible polymer in two dimensions configurations of a model polymer can be described by either a set of vectors ti of length a in two dimensions for i 1 N or alternatively by the angles φi between successive vectors as indicated in the figure below The polymer is at a temperature T and subject to an energy H κ i1N1 ti ti1 κa2 i1N1 cos φi where κ is related to the bending rigidity such that the probability of any configuration is proportional to expHkB T a Show that tm tn expnm ξ and obtain an expression for the persistence length lp aξ You can leave the answer as the ratio of simple integrals b Consider the endtoend distance R as illustrated in the figure Obtain an expression for R2 in the limit of N 1 c Find the probability pR in the limit of N 1 d If the ends of the polymer are pulled apart by a force F the probabilities for polymer configurations are modified by the Boltzmann weight exp FR kB T By expanding this weight or otherwise show that R K1 F OF3 and give an expression for the Hookian constant K in terms of quantities calculated before 3 Kinetic theory of gases 31 General definitions Kinetic theory studies the macroscopic properties of large numbers of particles starting from their classical equations of motion Thermodynamics describes the equilibrium behavior of macroscopic objects in terms of concepts such as work heat and entropy The phenomenological laws of thermodynamics tell us how these quantities are constrained as a system approaches its equilibrium At the microscopic level we know that these systems are composed of particles atoms molecules whose interactions and dynamics are reasonably well understood in terms of more fundamental theories If these microscopic descriptions are complete we should be able to account for the macroscopic behavior that is derive the laws governing the macroscopic state functions in equilibrium Kinetic theory attempts to achieve this objective In particular we shall try to answer the following questions 1 How can we define equilibrium for a system of moving particles 2 Do all systems naturally evolve towards an equilibrium state 3 What is the time evolution of a system that is not quite in equilibrium The simplest system to study the veritable workhorse of thermodynamics is the dilute nearly ideal gas A typical volume of gas contains of the order of 1023 particles and in kinetic theory we try to deduce the macroscopic properties of the gas from the time evolution of the set of atomic coordinates At any time t the microstate of a system of N particles is described by specifying the positions qit and momenta pit of all particles The microstate thus corresponds to a point μt in the 6Ndimensional phase space Γ Πi1N qi pi The time evolution of this point is governed by the canonical equations dqidt Hpi dpidt Hqi where the Hamiltonian H p q describes the total energy in terms of the set of coordinates q q1 q2 qN and momenta p p1 p2 pN The microscopic equations of motion have time reversal symmetry that is if all the momenta are suddenly reversed p p at t 0 the particles retrace their previous trajectory qt qt This follows from the invariance of H under the transformation Tp q p q Fig 31 The phase space density is proportional to the number of representative points in an infinitesimal volume As formulated within thermodynamics the macrostate M of an ideal gas in equilibrium is described by a small number of state functions such as E T P and N The space of macrostates is considerably smaller than the phase space spanned by microstates Therefore there must be a very large number of microstates μ corresponding to the same macrostate M This manytoone correspondence suggests the introduction of a statistical ensemble of microstates Consider N copies of a particular macrostate each described by a different representative point μt in the phase space Γ Let dN p q t equal the number of representative points in an infinitesimal volume dΓ Πi1N d3 pi d3 qi around the point p q A phase space density ρp q t is then defined by ρp q t dΓ limN dNp q t N This quantity can be compared with the objective probability introduced in the previous section Clearly dΓ ρ 1 and ρ is a properly normalized probability density function in phase space To compute macroscopic values for various functions Op q we shall use the ensemble averages O dΓ ρp q t Op q When the exact microstate μ is specified the system is said to be in a pure state On the other hand when our knowledge of the system is probabilistic in the sense of its being taken from an ensemble with density ρΓ it is said to belong to a mixed state It is difficult to describe equilibrium in the context of a pure state since μt is constantly changing in time according to Eqs 31 Equilibrium is more conveniently described for mixed states by examining the time evolution of the phase space density 77 Superfluid He4 199 26 atm P T 23 atm 218 K 52 K He 4 II He 4 I Fig 712 Schematic phase diagram of He4 with a phase transition between normal and superfluid forms of the liquid interaction makes He a universal wetting agent As the He atoms have a stronger attraction for practically all other molecules they easily spread over the surface of any substance Due to its light mass the He atom undergoes large zeropoint fluctuations at T 0 These fluctuations are sufficient to melt a solid phase and thus He remains in a liquid state at ordinary pressures Pressures of over 25 atmospheres are required to sufficiently localize the He atoms to result in a solid phase at T 0 A remarkable property of the quantum liquid at zero temperature is that unlike its classical counterpart it has zero entropy 4He Pump vapor out Fig 713 Evaporative cooling of helium by pumping out vapor The lighter isotope of He3 has three nucleons and obeys fermi statistics The liquid phase is very well described by a fermi gas of the type discussed in Section 75 The interactions between atoms do not significantly change the noninteracting picture described earlier By contrast the heavier isotope of He4 is a boson Helium can be cooled down by a process of evaporation liquid helium is placed in an isolated container in equilibrium with the gas phase at a finite vapor density As the helium gas is pumped out part of the liquid evaporates to take its place The evaporation process is accompanied by the release of latent heat which cools the liquid The boiling liquid is quite active and turbulent just as a boiling pot of water However when the liquid is cooled down to below 22 K it suddenly becomes quiescent and the turbulence disappears The liquids on the two sides of this phase transition are usually referred to as HeI and HeII 200 Ideal quantum gases Fig 714 Pushing superfluid helium through a superleak can be achieved for P1 P2 and leads to cooling of the intake chamber Packed powder P2 P1 T2 T1 HeII has unusual hydrodynamic properties It flows through the finest cap illaries without any resistance Consider an experiment that pushes HeII from one container to another through a small tube packed with powder For ordi nary fluids a finite pressure difference between the containers proportional to viscosity is necessary to maintain the flow HeII flows even in the limit of zero pressure difference and acts as if it has zero viscosity For this reason it is referred to as a superfluid The superflow is accompanied by heating of the container that loses HeII and cooling of the container that accepts it the mechanocaloric effect Conversely HeII acts to remove temperature differ ences by flowing from hot regions This is demonstrated dramatically by the fountain effect in which the superfluid spontaneously moves up a tube from a heated container Fig 715 Helium spurts out of the resistively heated chamber in the fountain effect In some other circumstances HeII behaves as a viscous fluid A classical method for measuring the viscosity of a liquid is via torsional oscillators a collection of closely spaced disks connected to a shaft is immersed in the fluid and made to oscillate The period of oscillations is proportional to the moment of inertia which is modified by the quantity of fluid that is dragged by the oscillator This experiment was performed on HeII by Andronikashvilli who indeed found a finite viscous drag Furthermore the changes in the frequency of the oscillator with temperature indicate that the quantity of fluid that is dragged by the oscillator starts to decrease below the transition temperature The measured normal density vanishes as T 0 approximately as T 4 In 1938 Fritz London suggested that a good starting hypothesis is that the transition to the superfluid state is related to the BoseEinstein condensation This hypothesis can account for a number of observations Index adiabatic 6 11 30 219 adiabatic walls 1 adsorption 123 268 allotropes 26 Andronikashvilli 201 assumption of equal a priori equilibrium probabilities 100 174 assumption of molecular chaos 75 average occupation number 188 Bayes theorem 44 BBGKY hierarchy 64 Benfords law 54 232 binary alloy 156 binomial distribution 41 black body 293 33 blackbody radiation 168 black hole 177 292 Boltzmann equation 66 72 Boltzmanns constant 8 bose gas degenerate 195 bose occupation number 195 BoseEinstein condensation 196 206 307 317 boson 183 187 203 303 Brillouin zone 163 167 canonical ensemble 111 175 Carnot cycle 30 217 222 Carnot engine 10 11 30 Carnots theorem 11 Casimir force 169 Cauchy distribution 48 53 227 central limit theorem 46 characteristic function 38 53 226 chemical potential 18 119 140 192 Clausiuss theorem 13 ClausiusClapeyron 30 197 205 312 220 closed systems 1 cluster expansion 131 coexistence 140 197 collision 69 collision terms 65 commutation relation 171 compressibility 8 139 156 197 condensation 141 146 153 conditional PDF 44 connected clusters 134 conservation laws 79 83 conserved quantities 62 76 80 contact line 214 continuous transition 149 convexity 145 Coulomb system 90 149 151 153 178 243 271 281 299 critical exponents 149 critical point 146 147 155 cumulant expansion 127 cumulant generating function 39 cumulants 39 127 134 cumulative probability function 36 curtosis 39 cycle 10 Debye model 165 Debye screening 66 90 150 244 273 Debye temperature 167 DebyeHückel 271 149 degeneracy 189 degenerate quantum limit 195 density matrix 173 177 185 295 density of states 167 diagrammatic representation 129 132 diathermic walls 1 diatomic gas 123 159 266 Dietericis equation 151 279 Dirac fermions 207 211 318 directed random walk 53 229 disconnected clusters 129 discontinuous transition 149 dispersion relation 167 169 displacement 7 displacements 6 efficiency 9 12 13 effusion 97 Einstein model 165 empirical temperature 78 energy gap 105 engine 8 10 engine Carnot 10 ensemble 59 173 ensemble average 173 enthalpy 16 117 entropy 13 14 26 entropy of mixing 52 entropy Boltzmann 79 entropy microcanonical 100 110 entropy Shannon 52 equation of state 3 30 146 219 equations of motion 58 equilibrium 1 2 174 equipartition 122 265 ergodic 63 estimation 52 exchange 183 excluded volume 136 138 expansivity 8 expectation value 37 44 exponential 48 extensive 7 20 48 108 extensivity 20 129 134 fermi energy 191 fermi gas degenerate 191 fermi occupation number 191 318 Index 319 fermi sea 191 206 314 fermi temperature 194 fermion 183 187 203 205 303 312 first law 5 6 102 Fock space 175 force 7 forces 6 fountain effect 201 Fourier equation 87 fractal 123 269 free energy Gibbs 18 free energy Helmholtz 17 fugacity 134 Gaussian distribution 40 Gibbs canonical ensemble 116 Gibbs free energy 18 117 Gibbs inequality 145 Gibbs paradox 108 182 Gibbs partition function 116 152 285 Gibbs phase rule 22 GibbsDuhem relation 20 glass 31 223 grand canonical ensemble 119 140 175 grand partition function 119 133 188 305 grand potential 18 120 134 Htheorem 72 Hamiltonian 58 hard core 130 131 136 138 142 hard rods 154 hard sphere 71 122 263 harmonic oscillator 121 164 168 177 258 294 heat 6 102 heat capacity 8 25 105 113 159 166 168 194 198 205 311 heat flux 82 87 helium 199 204 209 311 Helmholtz free energy 17 112 Hermitian 174 171 Hilbert space 171 175 182 hurricane 31 223 hydrodynamic equations 80 hydrodynamics first order 85 hydrodynamics zeroth order 83 94 hydrophilic 215 hydrophobic 215 ideal gas 4 7 106 115 117 120 204 305 ideal gas free expansion 7 ideal gas quantum 182 189 204 305 identical particles 109 111 182 186 information 51 54 73 76 79 233 intensive 7 20 48 internal energy 5 irreversibility 74 irreversible 15 Ising 118 isobaric 117 isotherm 3 4 140 isothermal 17 joint characteristic function 44 joint cumulants 44 joint Gaussian distribution 46 joint moments 44 joint PDF 43 kinetic theory 58 kurtosis 39 Lagrange multipliers 53 Langmuir isotherm 124 Laplace distribution 53 226 latent heat 27 31 197 200 224 lattice 162 law of corresponding states 146 Levy distribution 47 linked clusters 132 Liouvilles theorem 60 100 174 liquid 142 146 local density 80 local equilibrium 76 80 83 local expectation value 80 local velocity 81 lognormal distribution 232 longitudinal 166 209 Lorentz gas 92 250 macrostate 59 173 magnetization 118 Manning transition 153 Maxwell construction 140 Maxwell distribution 53 228 Maxwell relations 21 mean 39 meanfield theory 141 mean free time 67 mechanocaloric effect 201 microcanonical ensemble 99 174 microstate 58 171 mixed state 59 173 mixing entropy 108 182 moments 38 127 134 mutual information 56 negative temperature 104 Nernst 26 normal distribution 40 normal modes 158 163 168 176 288 occupation number 165 205 312 occupation numbers 103 184 188 onedimensional gas 152 283 oneparticle density 63 oneparticle irreducible 130 135 open systems 1 optimal selection 54 231 parity 183 partition function 112 175 186 Pauli paramagnetism 204 307 periodic boundary conditions 167 168 175 permutation 182 perpetual motion machine 9 phase diagram 146 phase separation 140141 phase space 58 111 171 175 186 phase space density 59 60 173 phase transition 142 144 phonons 167 203 209 photons 168 Planck 169 Poisson bracket 61 172 Poisson distribution 42 PoissonBoltzmann equation 90 244 polarization 166 168 polyatomic molecules 157 polyelectrolyte 153 polymer 123 125 153 269 pressure 78 138 pressure tensor 82 87 probability 35 probability density function 37 pure state 59 173 quantum degenerate limit 191 quantum observable 171 quasistatic 6 320 Index random matrix 55 random variable 35 36 Rayleigh distribution 53 227 refrigerator 9 reservoir 111 175 response function 25 response functions 8 reversible 10 14 ring diagrams 149 271 rotational modes 161 rotons 203 saddle point integration 49 second law 8 74 102 second law Clausius 9 second law Kelvin 9 second virial coefficient 136 137 151 187 276 Shannon 52 single collision time 86 skewness 39 solid angle 106 solid heat capacity 162 solid vibrations 162 Sommerfeld expansion 191 sound 84 203 209 sound speed 166 spin 118 124 125 179 189 stability 22 103 139 146 standard deviation 41 state functions 1 statistical mechanics classical 99 statistical mechanics quantum 157 StefanBoltzmann law 293 170 Stirlings approximation 50 Stoner ferromagnetism 205 314 streaming terms 65 67 superconductor 32 superfluid 199 surface tension 7 29 155 surfactant 29 123 155 215 268 susceptibility 8 119 124 195 204 307 Tchebycheff inequality 54 230 temperature empirical 2 temperature ideal gas 4 30 217 temperature Kelvin 4 30 217 temperature thermodynamic 12 thermal conductivity 87 93 255 thermal equilibrium 1 thermal wavelength 176 187 thermodynamic coordinates 1 thermodynamic limit 114 144 thermodynamics 1 third law 26 31 224 time reversal 61 62 70 74 torsional oscillator 202 translational symmetry 163 transverse 166 168 209 triple point 5 22 twoparticle density 63 68 twobody interaction 64 ultraviolet catastrophe 168 unbiased estimate 52 unconditional PDF 43 uniform density approximation 155 uniform density assumption 142 uniform distribution 53 226 universality 149 van der Waals equation 136 137 142 van der Waals interaction 130 136 137 199 variance 39 variational approximation 144 vibrational modes 159 virial coefficient 131 135 virial coefficients 151 276 virial expansion 131 135 149 271 virial theorem 179 301 viscosity 87 94 Vlasov equation 67 89 90 241 243 wavelength 166 wavevector 166 168 163 wetting 200 Wicks theorem 46 86 93 255 Wigner 55 work 5 102 Youngs equation 214 zeroth law 2 101