Statistical Thermodynamics

1 1 Statistical ThermodynamicsProfessor Dmitry GaraninStatistical physicsMay 17, 2021I. PREFACES tatistical physics considers systems of a large number of entities (particles) such as atoms, molecules, spins, etc. Forthese system it is impossible and even does not make sense to study the full microscopic dynamics. The only relevantinformation is, say, how many atoms have a particular energy, then one can calculate the observable thermodynamicvalues. That is, one has to know the distribution function of the particles over energies that defines the macroscopicproperties. This gives the namestatistical physicsand defines the scope of this approach outlined above can be used both at and off equilibrium. The branch of physics studying non-equilibrium situations is calledphysical kinetics. In this course, we study only Statistical physics at equilibrium. Itturns out that at equilibrium the energy distribution function has an explicit general form and the only problem isto calculate the observables.

2 The termstatistical mechanicsmeans the same as Statistical physics. One can call itstatistical thermodynamicsas formalism of Statistical Thermodynamics can be developed for both classical and quantum systems. The resultingenergy distribution and calculating observables is simpler in the classical case. However, the very formulation of themethod is more transparent within the quantum mechanical formalism. In addition, the absolute value of the entropy,including its correct value atT 0,can only be obtained in the quantum case. To avoid double work, we will consideronly quantum Statistical Thermodynamics in this course, limiting ourselves to systems without interaction. The moregeneral quantum results will recover their classical forms in the classical MICROSTATES AND MACROSTATESFrom quantum mechanics follows that the states of the system do not change continuously (like in classical physics)but are quantized.

3 There is a huge number of discrete quantum statesiwith corresponding energy values ibeing themain parameter characterizing these states. In the absence of interaction, each particle has its own set of quantumstates which it can occupy. For identical particles these sets of states are identical. One can think of boxesiintowhich particles are placed,Niparticles in theith box. The particles can be distributed over the boxes in a number ofdifferent ways, corresponding to differentmicrostates, in which the stateiof each particle is specified. The informationcontained in the microstates is excessive, and the only meaningful information is provided by the numbersNithatdefine the distribution of particles over their quantum states. These numbersNispecify what in Statistical physics iscalledmacrostate. If these numbers are known, the energy and other quantities of the system can be found. It shouldbe noted that also the Statistical macrostate contains more information than the macroscopic physical quantities thatfollow from it, as a distribution contains more information than an average over macrostatek, specified by the numbersNi, can be realized by a numberwkof microstates, the so-calledthermodynamic probability.

4 The latter is typically a large number, unlike the usual probability that changes between 0and 1. Redistributing the particles over the statesiwhile keeping the same values ofNigenerates different microstateswithin the same macrostate. The basic assumption of Statistical mechanics is the equidistrubution over is, each microstate within a macrostate is equally probable for occupation. Macrostates having a largerwkaremore likely to be realized. As we will see, for large systems, thermodynamic probabilities of different macrostates varyin a wide range and there is the state with the largest value ofwthat wins over all other the number of quantum statesiis finite, the total number of microstates can be written as kwk,(1)2the sum rule for thermodynamic probabilities. For an isolated system the number of particlesNand the energyUare conserved, thus the numbersNisatisfy the constraints iNi=N,(2) iNi i=U(3)that limit the variety of the allowed TWO-STATE PARTICLES (COIN TOSSING)A tossed coin can land in two positions: Head up or tail up.

5 Considering the coin as a particle, one can say thatthis particle has two quantum states, 1 corresponding to the head and 2 corresponding to the tail. IfNcoins aretossed, this can be considered as a system ofNparticles with two quantum states each. The microstates of the systemare specified by the states occupied by each coin. As each coin has 2 states, there are total = 2N(4)microstates. The macrostates of this system are defined by the numbers of particles in each state, numbers satisfy the constraint condition (2), ,N1+N2= one can take, say,N1as the numberklabeling macrostates. The number of microstates in one macrostate (that is, the number of different microstates thatbelong to the same macrostate) is given by the binomial distributionwN1=N!N1!(N N1)!=(NN1).(5)This formula can be derived as follows. We have to pickN1particles to be in the state 1, all others will be in the state2. How many ways are there to do this?

6 The first 0 particle can be picked inNways, the second one can be pickedinN 1 ways since we can choose ofN 1 particles only. The third 1 particle can be picked inN 2 differentways etc. Thus one obtains the number of different ways to pick the particles isN (N 1) (N 2) .. (N N1+ 1) =N!(N N1)!,(6)where the factorial is defined byN! N (N 1) .. 2 1,0! = 1.(7)The recurrent definition of the factorial isN! =N(N 1)!,0! = 1.(8)The expression in Eq. (6) is not yet the thermodynamical probabilitywN1because it contains multiple counting ofthe same microstates. The realizations, in whichN1 1 particles are picked in different orders, have been counted asdifferent microstates, whereas they are the same microstate. To correct for the multiple counting, one has to divideby the number of permutationsN1! of theN1particles that yields Eq. (5). One can check that the condition (1) issatisfied,N N1=0wN1=N N1=0N!N1!(N N1)!= 2N.

7 (9)The thermodynamic probabilitywN1has a maximum atN1=N/2,half of the coins head and half of the coins macrostate is the most probable state. Indeed, as for an individual coin the probabilities to land head up andtail up are both equal to , this is what we expect. For largeNthe maximum ofwN1onN1becomes prove thatN1=N/2 is the maximum ofwN1,one can rewrite Eq. (5) in terms of the new variablep=N1 N/2aswN1=N!(N/2 +p)!(N/2 p)!.(10)3 FIG. 1: The binomial distribution for an ensemble ofNtwo-state systems becomes narrow and peaked atp N1 N/2 = can see thatwpis symmetric aroundN1=N/2, ,p= Eq. (8), one obtainswN/2 1wN/2=(N/2)!(N/2)!(N/2 + 1)!(N/2 1)!=N/2N/2 + 1<1,(11)one can see thatN1=N/2 is indeed the maximum binomial distribution is shown in Fig. 1 for threedifferent valus ofN. As the argument, the variablep/pmax (N1 N/2)/(N/2) is used so that one can put all thedata on the same plot. One can see that in the limit of largeNthe binomial distribution becomes narrow and centeredatp= 0, that is,N1=N/2.

8 This practically means that if the coin is tossed many times, significant deviations fromthe 50:50 relation between the numbers of heads and tails will be extremely STIRLING FORMULA AND THERMODYNAMIC PROBABILITY AT LARGENA nalysis of expressions with large factorials is simplified by the Stirling formulaN! = 2 N(Ne)N.(12)In many important cases, the prefactor 2 Nis irrelevant, as we will see below. With the Stirling formula substituted,Eq. (10) becomeswN1 = 2 N(N/e)N 2 (N/2 +p) [(N/2 +p)/e]N/2+p 2 (N/2 p) [(N/2 p)/e]N/2 p= 2 N1 1 (2pN)2NN(N/2 +p)N/2+p(N/2 p)N/2 p=wN/2 1 (2pN)2(1 +2pN)N/2+p(1 2pN)N/2 p,(13)wherewN/2 = 2 N2N(14)4is the maximal value of the thermodynamic probability. Eq. (13) can be expanded for|p| boththe bases and the exponents, one has to be careful and expand the logarithm ofwN1rather thanwN1itself. Thesquare root term in Eq. (13) can be discarded as it gives a negligible contribution of orderp2 obtainslnwN1 =lnwN/2 (N2+p)ln(1 +2pN) (N2 p)ln(1 2pN) =lnwN/2 (N2+p)[2pN 12(2pN)2] (N2 p)[ 2pN 12(2pN)2] =lnwN/2 p 2p2N+p2N+p 2p2N+p2N= lnwN/2 2p2N(15)and thuswN1 =wN/2exp( 2p2N).

9 (16)One can see thatwN1becomes very small if|p|& Nthat for largeNdoes not violate the applicability condition|p| N. That is,wN1is small in the main part of the interval 0 N1 Nand is sharpy peaked nearN1= MANY-STATE PARTICLESThe results obtained in Sec. III for two-state particles can be generalized forn-state particles. We are looking forthe number of ways to distributeNparticles overnboxes so that there areNiparticles inith box. That is, we lookfor the number of microstates in the macrostate described by the result is given byw=N!N1!N2!..Nn!=N! ni=1Ni!.(17)This formula can be obtained by using Eq. (5) successively. The number of ways to putN1particles in box 1 and theotherN N1in other boxes is given by Eq. (5). Then the number of ways to putN2particles in box 2 is given bya similar formula withN N N1(there are onlyN N1particles afterN1particles have been put in box 1) andN1 numbers of ways should multiply.

10 Then one considers box 3 etc. until the last resultingnumber of microstates isw=N!N1!(N N1)! (N N1)!N2!(N N1 N2)! (N N1 N2)!N3!(N N1 N2 N3)! .. (Nn 2+Nn 1+Nn)!Nn 2! (Nn 1+Nn)! (Nn 1+Nn)!Nn 1!Nn! Nn!Nn!0!.(18)In this formula, all numerators except for the first one and all second terms in the denominators cancel each other, sothat Eq. (17) THERMODYNAMIC PROBABILITY AND ENTROPYWe have seen that different macrostateskcan be realized by largely varying numbers of microstateswk. For largesystems,N 1, the difference between different thermodynamic probabilitieswkis tremendous, and there is a sharpmaximum ofwkat some value ofk=kmax. The main postulate of Statistical physics is that in measurements on largesystems, only the most probable macrostate, satisfying the constraints of Eqs. (2) and (3), makes a contribution. Forinstance, a macrostate of an ideal gas with all molecules in one half of the container is much less probable than themacrostate with the molecules equally distributed over the whole container.