Statistical Thermodynamics

5We have seen in thermodynamics that an isolated system, initially in a non-equilibrium state, evolves to the equilibrium state characterized by the maximal entropy. On this way one comes to the idea that entropy S and thermodynamicprobability w should be related, one being a monotonic function of the other. The form of this function can be foundif one notices that entropy is additive while the thermodynamic probability is miltiplicative. If a system consists oftwo subsystems that weakly interact with each other (that is almost always the case as the intersubsystem interactionis limited to the small region near the interface between them), then S S1 S2 and w w1 w2 . If one chooses (L.Boltzmann)S kB ln w,(19)then S S1 S2 and w w1 w2 are in accord since ln (w1 w2 ) ln w1 ln w2 . In Eq. (19) kB is the Boltzmannconstant,kB 1.38 10 23 J K 1 .(20)It will be shown that the statistically defined entropy above coincides with the thermodynamic entropy at equilibrium.On the other hand, statistical entropy is well defined for nonequilibrium states as well, where as the thermodynamicentropy is usually undefined off equilibrium.VII.BOLTZMANN DISTRIBUTION AND CONNECTION WITH THERMODYNAMICSIn this section we obtain the distribution of particles over energy levels i as the most probable macrostate bymaximizing its thermodynamic probability w. We label quantum states by the index i and use Eq. (17). The taskis to find the maximum of w with respect to all Ni that satisfy the constraints (2) and (3). Practically it is moreconvenient to maximize ln w than w itself. Using the method of Lagrange multipliers, one searches for the maximumof the target functionXXΦ(N1 , N2 , · · · , Nn ) ln w αNi βεi Ni .(21)iiHere α and β are Lagrange multipliers with (arbitrary) signs chosen anticipating the final result. The maximumsatisfies Φ 0, Nii 1, 2, . . . .(22)As we are interested in the behavior of macroscopic systems with Ni 1, the factorials can be simplified with thehelp of Eq. (12) that takes the form (23)ln N ! N ln N N ln 2πN .Neglecting the relatively small last term in this expression, one obtainsXXXXΦ Nj ln Nj Nj αNj βεj Nj . ln N ! jjj(24)jThe first term here is a constant and it won’t contribute to the derivatives Φ/ Ni . In the latter, the only contributioncomes from the terms with j i in the above expression. One obtains the equations Φ ln Ni α βεi 0 Ni(25)Ni eα βεi ,(26)that yieldthe Boltzmann distribution with yet undefined Lagrange multipliers α and β. The latter can be found from Eqs. (2)and (3) in terms of N and U. Summing Eq. (26) over i, one obtainsN eα Z,α ln (N/Z) ,(27)

6whereZ Xe βεi(28)iis the so-called partition function (German Zustandssumme) that plays a major role in statistical physics. Then,eliminating α from Eq. (26) yieldsNi N βεi,eZ(29)the Boltzmann distribution. After that for the internal energy U one obtainsU Xiεi Ni N X βεiN Zεi e Z iZ β(30)orU N ln Z. β(31)This formula implicitly defines the Lagrange multiplier β as a function of U.The statistical entropy, Eq. (19), within the Stirling approximation becomes!XS kB ln w kB N ln N Ni ln Ni .(32)iInserting here Eq. (26) and α from Eq. (27), one obtainsXS N ln N Ni (α βεi ) N ln N αN βU N ln Z βU.kBiThe statistical entropy depends only on the parameter β. Its differential is given by ln ZdUdSdβ N U βdβ βdU.dS dβ βdβ(33)(34)Comparing this with the thermodynamic relation dU T dS at V 0, one identifiesβ 1.kB T(35)Now, with the help of dT /dβ kB T 2 , one can represent the internal energy, Eq. (31), via the derivative withrespect to T asU N kB T 2 ln Z. T(36)U.T(37)Eq. (33) becomesS N kB ln Z From here one obtains the statistical formula for the free energyF U T S N kB T ln Z.(38)One can see that the partition function Z contains the complete information of the system’s thermodynamics sinceother quantities such as pressure P follow from F. In particular, one can check F/ T S.

7VIII.QUANTUM STATES AND ENERGY LEVELSA.Stationary Schrödinger equationIn the formalism of quantum mechanics, quantized states and their energies E are the solutions of the eigenvalueproblem for a matrix or for a differential operator. In the latter case the problem is formulated as the so-calledstationary Schrödinger equationĤΨ EΨ,(39)where Ψ Ψ (r) is the complex function called wave function. The physical interpretation of the wave function is that2 Ψ (r) gives the probability for a particle to be found near the space point r. As above, the number of measurementsdN of the toral N measurements in which the particle is found in the elementary volume d3 r dxdydz around r isgiven by2dN N Ψ (r) d3 r.(40)The wave function satisfies the normalization condition 2d3 r Ψ (r) .1 (41)The operator Ĥ in Eq. (39) is the so-called Hamilton operator or Hamiltonian. For one particle, it is the sum ofkinetic and potential energiesĤ p̂2 U (r),2m(42)where the classical momentum p is replaced by the operatorp̂ i . r(43)The Schrödinger equation can be formulated both for single particles and the systems of particles. In this course wewill restrict ourselves to single particles. In this case the notation ε will be used for single-particle energy levels insteadof E. One can see that Eq. (39) is a second-order linear differential equation. It is an ordinary differential equationin one dimension and partial differential equation in two and more dimensions. If there is a potential energy, this isa linear differential equation with variale coefficients that can be solved analytically only in special cases. In the caseU 0 this is a linear differential equation with constant coefficients that is easy to solve analytically.An important component of the quantum formalism is boundary conditions for the wave function. In particular, fora particle inside a box with rigid walls the boundary condition is Ψ 0 at the walls, so that Ψ (r) joins smoothly with2the value Ψ (r) 0 everywhere outside the box. In this case it is also guaranteed that Ψ (r) is integrable and Ψ canbe normalized according to Eq. (41). It turns out that the solution of Eq. (39) that satisfies the boundary conditionsexists only for a discrete set of E values that are called eigenvalues. The corresponding Ψ are calles eigenfunctions,and all together is called eigenstates. Eigenvalue problems, both for matrices and differential operators, were known inmathematics before the advent of quantum mechanics. The creators of quantum mechanics, mainly Schrödinger andHeisenberg, realised that this mathematical formalism can accurately describe quantization of energy levels observed inexperiments. Whereas Schrödinger formlated his famous Schrödinger equation, Heisenberg made a major contributioninto description of quantum systems with matrices.B.Energy levels of a particle in a boxAs an illustration, consider a particle in a one-dimensional rigid box, 0 x L. In this case the momentum becomesp̂ i ddx(44)and Eq. (39) takes the form 2 d 2Ψ(x) εΨ(x)2m dx2(45)

8and can be represented asd2Ψ(x) k 2 Ψ(x) 0,dx2k2 2mε. 2(46)The solution of this equation satisfying the boundary conditions Ψ(0) Ψ(L) 0 has the formΨν (x) A sin (kν x) ,kν πν,Lν 1, 2, 3, . . .(47)where eigenstates are labeled by the index ν. The constant A following from Eq. (41) is A eigenvalues are given byεν p2/L. The energy 2 kν2π 2 2 ν 2. 2m2mL2(48)One can see the the energy ε is quandratic in the momentum p k (de Broglie relation), as it should be, butthe energy levels are discrete because of the quantization. For very large boxes, L , the energy levels becomequasicontinuous. The lowest-energy level with ν 1 is called the ground state.For a three-dimensional box with sides Lx , Ly , and Lz one has to solve the Schrödinger equation 2 dd2d2 2 Ψ(x) εΨ(x)(49)2m dx2dy 2dz 2with similar boundary conditions. The solution factorizes and has the form Ψν x ,ν y ,ν z (x, y, z) A sin (kν x x) sin kν y x sin (kν z x) ,kα πνα,Lαν α 1, 2, 3, . . . ,(50)where α x, y, z. The energy levels are 2 kx2 ky2 kz2 2 k 2ε 2m2m (51)and parametrized by the three quantum numbers ν α . The ground state is (ν x , ν y , ν z ) (1, 1, 1). One can order thestates in increasing ε and number them by the index j, the ground state being j 1. If Lx Ly Lz L, thenεν x ,ν y ,ν z π 2 2ν 2x ν 2y ν 2z22mL(52)and the same value of εj can be realized for different sets of ν x , ν y , and ν z , for instance, (1,5,12), (5,12,1), etc. Thenumber of different sets of (ν x , ν y , ν z ) having the same εj is called degeneracy and is denoted as gj . States with ν x ν y ν z have gj 1 and they are called non-degenerate. If only two of the numbers ν x , ν y , and ν z coincide, thedegeneracy is gj 3. If all numbers are different, gj 3! 6. If one sums over the energy levels parametrized by j,one has to multiply the summands by the corresponding degeneracies.C.Density of statesFor systems of a large size and thus very finely quantized states, one can define the density of states ρ(ε) as thenumber of energy levels dn in the interval dε, that is,dn ρ(ε)dε.(53)It is convenient to start calculation of ρ(ε) by introducing the number of states dn in the “elementary volume”dν x dν y dν z , considering ν x , ν y , and ν z as continuous. The result obviously isdn dν x dν y dν z ,(54)that is, the corresponding density of states is 1. Now one can rewrite the same number of states in terms of the wavevector k using Eq. (51)dn Vdkx dky dkz ,π3(55)

9where V Lx Ly Lz is the volume of the box. After that one can go over to the number of states within the shell dk,as was done for the distribution function of the molecules over velocities. Taking into account that kx , ky , and kz areall positive, this shell is not a complete spherical shell but 1/8 of it. Thusdn V 4π 2k dk.π3 8(56)The last step is to change the variable from k to ε using Eq. (51). Withrr2m dk1 2m 12m2 k ε,k 2 ε, 2dε2 2 ε(57)one obtains Eq. (53) withρ(ε) IX. V(2π)22m 2 3/2 ε.(58)STATISTICAL THERMODYNAMICS OF THE IDEAL GASIn this section we demonstrate how the results for the ideal gas, previously obtained within the molecular theory,follow from the more general framework of statistical physics in the preceding section. Consider an ideal gas in asufficiently large container. In this case the energy levels of the system, see Eq. (51), are quantized so finely that onecan introduce the density of states ρ (ε) defined by Eq. (53). The number of particles in quantum states within theenergy interval dε is the product of the number of particles in one state N (ε) and the number of states dnε in thisenergy interval. With N (ε) given by Eq. (29) one obtainsN βεeρ (ε) dε.ZdNε N (ε)dnε (59)For finely quantized levels one can replace summation in Eq. (28) and similar formulas by integration asX . dερ (ε) . . .(60)iThus for the partition function (28) one has Z dερ (ε) e βε .(61)For quantum particles in a rigid box with the help of Eq. (58) one obtainsZ V(2π)22m 2 3/2 dε εe βε 0 V(2π)22m 2 3/2 π2β 3/2 VmkB T2π 2 3/2.(62)Using the classical relation ε mv 2 /2 and thus dε mvdv, one can obtain the formula for the number of particles inthe speed interval dvdNv N f (v)dv,(63)where the speed distribution function f (v) is given by 3/2 3/2 r12π 2V1 βε2mmeρ (ε) mv v mv e βεf (v) 22ZV mkB T 2(2π) 3/2 mmv 22 4πv exp .2πkB T2kB T(64)This result coincides with the Maxwell distribution function obtained earlier from the molecular theory of gases. ThePlank’s constant that links to quantum mechanics disappeared from the final result.

10The internal energy of the ideal gas is its kinetic energyU Nε̄,(65)ε̄ mv̄ 2 /2 being the average kinetic energy of an atom. The latter can be calculated with the help of the speeddistribution function above, as was done in the molecular theory of gases. The result has the formε̄ fkB T,2(66)where in our case f 3 corresponding to three translational degrees of freedom. The same result can be obtainedfrom Eq. (31)" 3/2 # ln Z3 3 13mU N N Nln Vln β N N kB T.(67) β β2π 2 β2 β2 β2After that the known result for the heat capacity CV ( U/ T )V follows.The pressure P is defined by the thermodynamic formula FP . V T(68)With the help of Eqs. (38) and (62) one obtainsP N kB T ln VN kB T ln Z N kB T V VV(69)that amounts to the equation of state of the ideal gas P V N kB T.X.STATISTICAL THERMODYNAMICS OF HARMONIC OSCILLATORSConsider an ensemble of N identical harmonic oscillators, each of them described by the HamiltonianĤ p̂2kx2 .2m2(70)Here the momentum operator p̂ is given by Eq. (44) and k is the spring constant. This theoretical model can describe,for instance, vibrational degrees of freedom of diatomic molecules. In this case x is the elongation of the chemicalbond between the two atoms, relative to the equilibrium bond length. The stationary Schrödinger equation (39) fora harmonic oscillator becomes 2 d 2kx2 Ψ(x) εΨ(x).(71)2m dx22The boundary conditions for this equation require that Ψ( ) 0 and the integral in Eq. (41) converges. In contrastto Eq. (45), this is a linear differential equation with a variable coefficient. Such differential equations in general donot have solutions in terms of known functions. In some cases the solution can be expressed through special functionssuch as hypergeometrical, Bessel functions, etc. Solution of Eq. (71) can be found but this task belongs to quantummechanics courses. The main result that we need here is that the energy eigenvalues of the harmonic oscillator havethe form 1εν ω 0 ν ,ν 0, 1, 2, . . . ,(72)2pwhere ω 0 k/m is the frequency of oscillations. The energy level with ν 0 is the ground state. The ground-stateenergy ε0 ω 0 /2 is not zero, as would be the case for a classical oscillator. This quantum ground-state energy iscalled zero-point energy. It is irrelevant in the calculation of the heat capacity of an ensemble of harmonic oscillators.The partition function of a harmonic oscillator isZ Xν 0e βεν e β ω0 /2 Xν 0e β ω0 ν e β ω0 /212 β ω /2 , β ω β ω/20001 esinh (β ω 0 /2)e e(73)

11where the result for the geometrical progression1 x x2 x3 . . . (1 x) 1,x 1(74)was used. The hyperbolic functions are defined byex e x2cosh(x)1coth(x) .sinh(x)tanh(x)ex e x,2sinh(x)tanh(x) ,cosh(x)cosh(x) sinh(x) (75)We will be using the formulassinh(x)0 cosh(x),cosh(x)0 sinh(x)(76)andsinh(x) x, x 1,ex /2, x 1tanh(x) x, x 11, x 1.(77)The internal (average) energy of the ensemble of oscillators is given by Eq. (31). This yields ln sinh (β ω 0 /2)N sinh (β ω 0 /2) ω 0 ln ZU N N Ncoth β βsinh (β ω 0 /2) β2 β ω 02 (78)orU N ω 0coth2 ω 02kB T .(79)Another way of writing the internal energy is U N ω 01eβ ω0 1 12 ,(80)where the constant term with 1/2 is the zero-point energy. The limiting low- and high-temperature cases of thisformula are N ω 0 /2, kB T ω 0U (81) N kB T, kB T ω 0 .In the low-temperature case, almost all oscillators are in their ground states, since e β ω0 1. Thus the main termthat contributes into the partition function in Eq. (73) is ν 0. Correspondingly, U is the sum of ground-stateenergies of all oscillators.At high temperatures, the previously obtained classical result is reproduced. The Planck’s constant that links toquantum mechanics disappears. In this case, e β ω0 is only slightly smaller than one, so that very many different ncontribute to Z in Eq. (73). In this case one can replace summation by integration, as was done above for the particlein a potential box. In that case one also obtained the classical results.One can see that the crossover between the two limiting cases corresponds to kB T ω 0 . In the high-temperaturelimit kB T ω 0 , many low-lying energy levels are populated. The top populated level ν can be estimates fromkB T ω 0 ν , so thatν kB T1. ω 0Probability to find an oscillator in the states with ν ν is exponentially small.The heat capacity is defined by 2 dU ω 01 ω 0 ω 0 /(2kB T )C N N kB.dT22kB T 2sinh [ ω 0 /(2kB T )]sinh2 [ ω 0 /(2kB T )](82)(83)This formula has limiting cases(C N kB ω 0kB T 2 0exp k ω, kB T ω 0TBN kB ,kB T ω 0 .(84)

12FIG. 2: Heat capacity of harmonic oscillators.One can see that at high temperatures the heat capacity can be written asC fN kB ,2(85)where the effective number of degrees of freedom for an oscillator is f 2. The explanation of the additional factor2 is that the oscillator has not only the kinetic, but also the potential energy, and the average values of these twoenergies are the same. Thus the total amount of energy in a vibrational degree of freedom doubles with respect to thetranslational and rotational degrees of freedom.At low temperatures the vibrational heat capacity above becomes exponentially small. One says that vibrationaldegrees of freedom are getting frozen out at low temperatures.The heat capacity of an ensemble of harmonic oscillators in the whole temperature range, Eq. (83), is plotted inFig. 2.The average quantum number of an oscillator is given byn hνi 1 X βεννe.Z ν 0(86)Using Eq. (72), one can calculate this sum as followsn 1 X 11 X 1 βεν1 1 XZ1 1 Z1U1 ν e βεν e εν e βεν .Z ν 0 2Z ν 0 2 ω 0 Z ν 02Z ω 0 Z β2N ω 02(87)Finally with the help of Eq. (80), one findsn 11. eβ ω0 1exp [ ω 0 /(kB T )] 1(88)This is the Bose-Einstein distribution that will be considered later. The meaning of it is the following. Consideringthe oscillator as a “box” or “mode”, one can ask what is the average number of quanta (that is, “particles” orquasiparticles) in this box at a given temperature. The latter is given by the formula above.Substituting Eq. (88) into Eq. (80), one obtains a nice formula 1(89)U N ω 0 n 2

13resembling Eq. (72). At low temperatures, kB T ω 0 , the average quantum number n becomes exponentially small.This means that the oscillator is predominantly in its ground state, ν 0. At high temperatures, kB T ω 0 , Eq.(88) yieldsn kB T ω 0(90)that has the same order of magnitude as the top populated quantum level number ν given by Eq. (82).The density of states ρ (ε) for the harmonic oscillator can be easily found. The number of levels dnν in the intervaldν isdnν dν(91)(that is, there is only one level in the interval dν 1). Then, changing the variables with the help of Eq. (72), onefinds the number of levels dnε in the interval dε as 1dνdεdnε dε ρ (ε) dε,(92)dε dεdνwhereρ (ε) 1. ω 0(93)To conclude this section, let us reproduce the classical high-temperature results by a simpler method. At hightem

to calculate the observables. The term statistical mechanics means the same as statistical physics. One can call it statistical thermodynamics as well. The formalism of statistical thermodynamics can be developed for both classical and quantum systems. The resulting energy distribution and calculating observables is simpler in the classical case.