《热力学与统计物理》课程教学资源（书籍文献）A Modern Course in Statistical Physics - 2016 - Linda E. Reichl，4th revised and updated edition.pdf_P31-P35

2.9 Problems 23 This formula can be simplified if we use Stirling’s approximation. Then the entropy takes the form S = 5 2 NkB + NkB ln [ V N (4πmE 3h2N )3∕2 ] . (2.50) Equation (2.50) is called the Sackur–Tetrode equation and gives the entropy of an ideal gas of indistinguishable particles. We can obtain a relation between the energy and temperature of an ideal gas from the Sackur–Tetrode equation using the thermodynamic equation 1 T = ( 𝜕S 𝜕E ) V,N = 3 2 NkB E , (2.51) where the last term was obtained by taking the derivative of Eq. (2.50). Therefore, E = (3∕2)NkBT. The pressure of the ideal gas can be obtained from the entropy using another thermodynamic relation P T = ( 𝜕S 𝜕V ) E,N = NkB V , (2.52) where, again, the last term was obtained by taking the derivative of Eq. (2.50). Thus, we obtain PV = NkBT, which is the equation of state of an ideal gas. 2.9 Problems Problem 2.1 A bus has nine seats facing forward and eight seats facing backward. In how many ways can seven passengers be seated if two refuse to ride facing forward and three refuse to ride facing backward? Problem 2.2 Find the number of ways in which eight persons can be assigned to two rooms (A and B) if each room must have at least three persons in it. Problem 2.3 Find the number of permutations of the letters in the word, MONOTONOUS. In how many ways are four O’s together? In how many ways are (only) 3 O’s together? Problem 2.4 In how many ways can five red balls, four blue balls, and four white balls be placed in a row so that the balls at the ends of the row are the same color? Problem 2.5 Various six digit numbers can be formed by permuting the digits 666655. All arrangements are equally likely. Given that a number is even, what is the probability that two fives are together? Problem 2.6 Fifteen boys go hiking. Five get lost, eight get sunburned, and six return home without problems. (a) What is the probability that a sunburned boy got lost? (b) What is the probability that a lost boy got sunburned?

2Complexityand EntropyProblem2.7Adeck ofcards contains52cards,dividedequallyamongfour suits,Spades (S), Clubs (C), Diamonds (D), and Hearts (H). Each suit has 13 cards whicharedesignated:(2,3,4,5,6,7,8,9,10,J,Q,K,A).Assumethatthe deck isalwayswell shuffled so it is equally likely to receive any card in the deck, when a card isdealt.(a)Ifadealthandconsists offivecards,howmanydifferent hands can onebe dealt (assume the cards in the hand can be received in any order)? (b) If thegame is poker,what is the probability ofbeing dealt a Royal Flush (1o,J,Q,K, andA all ofone suit)? (c) Ifoneisdealta hand with sevencards,and thefirstfourcardsare spades,what is theprobabilityof receiving at least oneadditional spade?Problem2.8Afair six-sided die is thrown N times and the resultofeach throwis recorded. (a) If the die is thrown N = 12 times, what is the probability that oddnumbersoccurthreetimes?IfitisthrownN=120times,whatistheprobabilitythat odd numbers occur30 times?Use the binomial distribution.(b)Computethesamequantitiesas in part (a)but use theGaussian distribution.(Note:Forpart (a)computeyour answers tofour places.)(c)Plotthebinomial and Gaussiandistributions forN=2andN=12.Problem2.9Agasof Nidentical particles is free to move amongM distinguish-ablelattice sites on a lattice with volume V,such that each lattice site can have atmostoneparticleatanytime.Thedensityoflatticesitesisμ=M/V.AssumethatN≤M and thatall configurations of thelattice have the same energy.(a)Com-pute the entropy of the gas. (b) Find the equation of state of the gas. (Note: thepressure ofan ideal gas isan exampleof an entropicforce.)Problem2.10An Einstein solid (in 3D space)has100 lattice sites and300phonons, each with energyho =0.01 eV. (a) What is the entropy of the solid(give a number)? (b) What is the temperature of the solid (give a number)?Problem 2.11 A system consists of N noninteracting,distinguishable two-levelatoms.Each atom can exist in oneoftwo energy states,Eo=Oor E, =.Thenumberofatoms in energylevel,Eo,is ngandthe number ofatoms in energylevel, E,, is ny. The internal energy of this system is U = noEo + n,Er. (a) Computethe multiplicityofmicroscopic states. (b)Compute the entropy ofthis system asafunctionofinternalenergy.(c)Computethetemperatureofthissystem.Underwhat conditions can it benegative? (d)Computethe heat capacityfor a fixednumberofatoms,N.Problem2.12Alattice contains Nnormal lattice sites and Ninterstitial latticesites.The lattice sites are all distinguishable.N identical atoms sit on thelattice,Mon theinterstitial sites,andN-Mon thenormal sites(N》M》1).Ifanatomoccupiesa normal site,it has energyE=0.If an atomoccupies an interstitial site,it has energyE= . Compute the internal energy and heat capacity as a functionoftemperatureforthislatticeProblem2.13Considera latticewithN spin-1 atomswith magneticmomentμ.Each atom can be in one of three spin states, S, = -1,o, +1. Let n_i, no, andn,denotethe respectivenumber of atoms in eachof those spin states.Find the

24 2 Complexity and Entropy Problem 2.7 A deck of cards contains 52 cards, divided equally among four suits, Spades (S), Clubs (C), Diamonds (D), and Hearts (H). Each suit has 13 cards which are designated: (2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A). Assume that the deck is always well shuffled so it is equally likely to receive any card in the deck, when a card is dealt. (a) If a dealt hand consists of five cards, how many different hands can one be dealt (assume the cards in the hand can be received in any order)? (b) If the game is poker, what is the probability of being dealt a Royal Flush (10, J, Q, K, and A all of one suit)? (c) If one is dealt a hand with seven cards, and the first four cards are spades, what is the probability of receiving at least one additional spade? Problem 2.8 A fair six-sided die is thrown N times and the result of each throw is recorded. (a) If the die is thrown N = 12 times, what is the probability that odd numbers occur three times? If it is thrown N = 120 times, what is the probability that odd numbers occur 30 times? Use the binomial distribution. (b) Compute the same quantities as in part (a) but use the Gaussian distribution. (Note: For part (a) compute your answers to four places.) (c) Plot the binomial and Gaussian distributions for N = 2 and N = 12. Problem 2.9 A gas of N identical particles is free to move among M distinguishable lattice sites on a lattice with volume V, such that each lattice site can have at most one particle at any time. The density of lattice sites is μ = M∕V. Assume that N ≪ M and that all configurations of the lattice have the same energy. (a) Compute the entropy of the gas. (b) Find the equation of state of the gas. (Note: the pressure of an ideal gas is an example of an entropic force.) Problem 2.10 An Einstein solid (in 3D space) has 100 lattice sites and 300 phonons, each with energy ℏω = 0.01 eV. (a) What is the entropy of the solid (give a number)? (b) What is the temperature of the solid (give a number)? Problem 2.11 A system consists of N noninteracting, distinguishable two-level atoms. Each atom can exist in one of two energy states, E0 = 0 or E1 = ε. The number of atoms in energy level, E0, is n0 and the number of atoms in energy level, E1, is n1. The internal energy of this system is U = n0E0 + n1E1. (a) Compute the multiplicity of microscopic states. (b) Compute the entropy of this system as a function of internal energy. (c) Compute the temperature of this system. Under what conditions can it be negative? (d) Compute the heat capacity for a fixed number of atoms, N. Problem 2.12 A lattice contains N normal lattice sites and N interstitial lattice sites. The lattice sites are all distinguishable. N identical atoms sit on the lattice, M on the interstitial sites, and N − M on the normal sites (N ≫ M ≫ 1). If an atom occupies a normal site, it has energy E = 0. If an atom occupies an interstitial site, it has energy E = ε. Compute the internal energy and heat capacity as a function of temperature for this lattice. Problem 2.13 Consider a lattice with N spin-1 atoms with magnetic moment μ. Each atom can be in one of three spin states, Sz = −1, 0, +1. Let n−1, n0, and n1 denote the respective number of atoms in each of those spin states. Find the

[273Thermodynamics3.1IntroductionThe science of thermodynamics began with the observation that matter in theaggregate can exist in macroscopic states which are stable and do not change intime.Onceasystemreaches itsequilibrium state,thesystem will remainforeverinthat state unless some external influence acts to change it. This inherent stabilityand reproducibilityof theequilibriumstatescanbe seeneverywhere in the worldaround us.Thermodynamics has been able to describe, with remarkable accuracy, themacroscopicbehaviorofahugevarietyof systemsovertheentirerangeofexperimentally accessible temperatures (10-9 to 106 K). It provides a truly universaltheory of matter in the aggregate. And yet, the entire subject is based on onlyfourlaws [183,220],which maybe stated rather simply asfollows:ZerothLawTwo bodies, each in equilibrium with a third body, are in equilibrium with each other.First LawEnergy is conserved.Second LawHeatflows spontaneouslyfromhightemperaturetolowtempera-ture.Third LawIt is not possible to reach the coldest temperature usinga finite setofreversiblesteps.Even thoughtheselaws soundrathersimple,their implications arevast andgiveusimportant tools for studying the behavior and stability of systems in equilibriumand, in somecases, of systems farfrom equilibrium.The stateofthermodynamicequilibriumcanbespecified completelyintermsofa fewparameters-calledstate variables.State variables emerge from theconser-vation laws governingtheunderlying dynamics of these systems.Statevariablesmay be either extensive or intensive.Extensive variables always change in valuewhen the size (spatial extent and number of degrees of freedom) of the system ischanged, and intensive variables donot.Certain pairs of intensive and extensive state variables occur together becausetheycorrespondtogeneralizedforcesanddisplacementswhichappear inexpres-AModern Course in Statistical Physics,4.Edition.Linda E.Reichl.@ 2016 WILEY-VCH Verlag GmbH & Co. KGaA.Published 2016 by WILEY-VCH Verlag GmbH& Co. KGaA

27 3 Thermodynamics 3.1 Introduction The science of thermodynamics began with the observation that matter in the aggregate can exist in macroscopic states which are stable and do not change in time. Once a system reaches its equilibrium state, the system will remain forever in that state unless some external influence acts to change it. This inherent stability and reproducibility of the equilibrium states can be seen everywhere in the world around us. Thermodynamics has been able to describe, with remarkable accuracy, the macroscopic behavior of a huge variety of systems over the entire range of experimentally accessible temperatures (10−9 to 106 K). It provides a truly universal theory of matter in the aggregate. And yet, the entire subject is based on only four laws [183, 220], which may be stated rather simply as follows: Zeroth Law Two bodies, each in equilibrium with a third body, are in equilibrium with each other. First Law Energy is conserved. Second Law Heat flows spontaneously from high temperature to low temperature. Third Law It is not possible to reach the coldest temperature using a finite set of reversible steps. Even though these laws sound rather simple, their implications are vast and give us important tools for studying the behavior and stability of systems in equilibrium and, in some cases, of systems far from equilibrium. The state of thermodynamic equilibrium can be specified completely in terms of a few parameters – called state variables. State variables emerge from the conservation laws governing the underlying dynamics of these systems. State variables may be either extensive or intensive. Extensive variables always change in value when the size (spatial extent and number of degrees of freedom) of the system is changed, and intensive variables do not. Certain pairs of intensive and extensive state variables occur together because they correspond to generalized forces and displacements which appear in expresA Modern Course in Statistical Physics, 4. Edition. Linda E. Reichl. © 2016WILEY-VCH Verlag GmbH & Co.KGaA. Published 2016 byWILEY-VCH Verlag GmbH & Co.KGaA

3Thermodynamicssions forthermodynamic work.Examples of such extensive and intensive pairsinclude, respectively, volume, V, and pressure, P; magnetization, M, and mag-netic field, H; length, L, and tension,J;area, A,and surface tension, o; electricpolarization, P, and electric field, E. The relation of these state variables to un-derlying conservation laws is direct in some cases.For example, the existenceofmagnetization is related to the conservation of the internal angular momen-tum of atoms and electricpolarization isrelated totheconservation of electriccharge.The pair of state variables related to heat content of a thermodynamic systemare thetemperature,T,whichis intensive,and the entropy,S, whichis extensive.There is also a pairof state variables associated with"chemicalproperties ofa sys-tem.Theyare the number of particles N (number of moles n)which is extensive,and the chemical potential per particle (permole),μ,which is intensive.If there ismore than one type of particle in the system, then there will be a particle numberormolenumber and chemical potential associated with eachtype of particle.Otherstatevariablesusedtodescribethethermodynamicbehaviorofasystemare thevarious response functions,such as heat capacity, C; compressibility,xmagnetic susceptibility,xandvarious thermodynamic potentials,suchasthein-ternal energy, U; enthalpy,H; Helmholtz free energy,A; Gibbs free energy, G; andthe grand potential, Q2.We shallbecome acquainted with these state variables insubsequent sections.Ifwechangethethermodynamic stateof our system,the amount by which thestate variables change must be independent ofthe path taken.If this werenot so,the statevariables would contain information about the history ofthe system.It isprecisely this property of state variables which makes them so useful in studyingchanges in the equilibrium state of various systems.Mathematically,changes instatevariables correspond to exact diferentials.Themathematics ofexact differ-entials is reviewed in Appendix B.It is useful to distinguish between three types of thermodynamic systems. Anisolated system is one which is surrounded by an insulating wall,so that no heatormatter can be exchanged withthesurroundingmedium.Aclosed systemisone which is surrounded bya conducting wall that allows heat to be exchangedwiththesurrounding medium,but not matter.An open system is one which al-lowsbothheatandmatterexchangewiththesurrounding medium.Iftheinsulating/conducting wall canmove,thenmechanical work can beexchanged with thesurrounding medium.It is possibleto changefrom one equilibrium state to anotherSuchchanges canoccur reversiblyorirreversibly.Areversiblechangeisoneforwhichthe systemalways remains infinitesimally close to the thermodynamic equilibrium that is,isperformed quasi-statically.Such changescan always bereversed and the systembrought back to its original thermodynamic state without causing any changes inthethermodynamic state ofthe universe. For each step ofa reversible process,thestate variables have a well-defined meaning.An irreversible or spontaneous change from one equilibrium state to another isone in which the system does not stay infinitesimallyclose to equilibrium during

28 3 Thermodynamics sions for thermodynamic work. Examples of such extensive and intensive pairs include, respectively, volume, V, and pressure, P; magnetization, M, and magnetic field, H; length, L, and tension, J; area, A, and surface tension, σ; electric polarization, P, and electric field, E. The relation of these state variables to underlying conservation laws is direct in some cases. For example, the existence of magnetization is related to the conservation of the internal angular momentum of atoms and electric polarization is related to the conservation of electric charge. The pair of state variables related to heat content of a thermodynamic system are the temperature, T, which is intensive, and the entropy, S, which is extensive. There is also a pair of state variables associated with “chemical” properties of a system. They are the number of particles N (number of moles 𝔫) which is extensive, and the chemical potential per particle (per mole), μ, which is intensive. If there is more than one type of particle in the system, then there will be a particle number or mole number and chemical potential associated with each type of particle. Other state variables used to describe the thermodynamic behavior of a system are the various response functions, such as heat capacity, C; compressibility, κ; magnetic susceptibility, χ; and various thermodynamic potentials, such as the internal energy, U; enthalpy, H; Helmholtz free energy, A; Gibbs free energy, G; and the grand potential, Ω. We shall become acquainted with these state variables in subsequent sections. If we change the thermodynamic state of our system, the amount by which the state variables change must be independent of the path taken. If this were not so, the state variables would contain information about the history of the system. It is precisely this property of state variables which makes them so useful in studying changes in the equilibrium state of various systems. Mathematically, changes in state variables correspond to exact differentials. The mathematics of exact differentials is reviewed in Appendix B. It is useful to distinguish between three types of thermodynamic systems. An isolated system is one which is surrounded by an insulating wall, so that no heat or matter can be exchanged with the surrounding medium. A closed system is one which is surrounded by a conducting wall that allows heat to be exchanged with the surrounding medium, but not matter. An open system is one which allows both heat and matter exchange with the surrounding medium. If the insulating/conducting wall can move, then mechanical work can be exchanged with the surrounding medium. It is possible to change from one equilibrium state to another. Such changes can occur reversibly or irreversibly. A reversible change is one for which the system always remains infinitesimally close to the thermodynamic equilibrium – that is, is performed quasi-statically. Such changes can always be reversed and the system brought back to its original thermodynamic state without causing any changes in the thermodynamic state of the universe. For each step of a reversible process, the state variables have a well-defined meaning. An irreversible or spontaneous change from one equilibrium state to another is one in which the system does not stay infinitesimally close to equilibrium during