are divided into sets such that the states belonging

Transcription

1 Gibbs' Canonical in Ensemble Statistical By Takuzo Distribution Mechanics. Law SAKAI (Read Feb. 17,1940) 1. librium The distribution state is obtained law for an assembly various methods. of molecules There are in an equiboltzmann's method, Darwin-Fowler's method, method to use Gibbs' canonical ensemble method to use Gibbs' gr ensemble. The use of Stirling's formula for factorials in Boltzmann's method(1) is avoided in Darwin-Fowler's method use of orem of steepest descent in function ory. The latter method is elegant but its formal proof is not simple. The use of Stirling's formula is also avoided in method of canonical ensemble, but it is believed that it is only applicable to Boltzmann's statistics will be too complicated in new statistics. Thus Pauli used Gibbs' gr ensemble to deduce distribution law in new statistics.(2) But system composed of localized molecules can not be treated on this way, as re appears a difficulty concerning convergency of a series. We shall show here a very simple method to use canonical ensemble, which is applicable to all statistics. 2. We shall call an assembly of molecules a system. Consider a system which is composed of similar molecules. The whole quantum states of a molecule are divided into sets such that states belonging to a set are indistinguishable from one anor on macroscopic observations. Let be number of quantum states belonging to -th set, energy of which being. Take a configuration of system, in which N1, N2, N3,... molecules are distributed into 1, 2, 3,... states respectively. Let us first consider gaseous systems. The number of states of system representing above configuration is evidently given (1) The use of Stirling's formula is criticized R. H. " Fbwler s btatistical Mechanics " (1936). some authors. (2) W. Pauli : Zeits. f. Phys., 41 (1927), 81. R. C. Tolman tistical Mechanics" (1938) 510. : See, for example, " The Principles of Sta-

2 200 whore Takuzo SASAI. N is total number of molecules [vol. 22 ; superscript N in G(N) means that G(N) refers to system which contaius N molecules. These G`' states of system are indistinguishable from one anor on macroscopic observations, so y have equal a priori probability w(n)(n1, N2,... ). Thus G(N) is statistical weight of macroscopic configuration under consideration. If we take all permissible configurations, we must have each of N1,N2,... takes 0, 1, 2,... under condition (2). The average number of molecules belonging to is given whore respect to 3. In ensemble. given superscript N in N(N) system containing case of rmal The probability that means that average is taken with N molecules. equilibrium, we take Gibbs' canonical system will have energy E(N) is (N) is free energy of system =kt, k being Boltzmann's constant T absolute temperature. Take Einstein-Bose' case for sake of illustration. The distribution law is deduced as follows. From (4) (5), (1) In gaseous Systems we can not recognize individuality of each of moleules. Hence we divide usual statistical weight N! II( N /N 1) N1 cas is well-known. This is a special case of new statistics at high temperatures, as one can see from fact. that (1 b) (1c) are reduced to (1a) if we assume >N, an obvious assumption at high temperatures.

3 1940] Gibbs' Canonical Ensemble Distribution Law. 201 if mutual interactions of molecules is small. Now Writing. On using relations (3) (4) for system with N-1 molecules, When total number N of molecules is very large compared with unity, we may put Then solving (7) with respect to N, This is usual distribution law for Einstein-Bose' gases. Similarly in case of Fermi-Dirac's statistics, so that distribution law is In case of Boltzmann's statistics, (1) As NA will be proportional to N, error induced this assumption is of order of 1/N compared with unity.

4 202 Takuzo SAKAI. [Vol. 22 at once, without assumption (9). 4. What is physical meaning of relation (6)? In difference (N) - (N-1), temperature T volume V must be kept constant. In particular, volume is involved in (cf. (17)), which we have kept constant in above calculation. Hence if Nis verylarge.fromrmodynamics weknow that( / N)r,r is equivalent to chemical potential per molecule. Thus from (6) we see that is chemical potential per molecule. Furr we know, from rmodynamics, that rmodynamic potential is equal to chemical potential per molecule multiplied total number of molecules. Hence we can set, When V definition p is of free pressure. N energy, The are canonical variable, this ensemble expression is characterized represents general gr ensemble.(1) When molecule has internal motion, is of form = +uj i is kinetic energy of motion of centre of gravity uj energy of internal motion. In case of gaseous systems, we have gj is statistical weight of j-th internal motion, m mass of molecule h Planck's constant. The total energy is of form & represents that part (1) E. A. Guggenheim : of Journ. energy Chem. Phys. which is contributed 7 (1939) 103. from

5 1940] Gibbs' Canonical Ensemble Distribution Law. 203 motion of centre of gravity Ei that which is contributed from internal motion of molecules. The equation of states is obtained calculating pressure exerted molecules on wall of vessel well-known elementary method. Evidently Analytically we can proceed as follows. From (15), On or h, on using (17) distribution law, r=0,1, -1 mean Boltzmann's, Einstein-Bose' Fermi-Dirac's statistics respectively. Differentiating above equation with respect to V, Eliminating ( / V)T from this equation (20), Similarly on using relation S=-( / T)v, S=(E- )T is entropy, Hence pv=2/3 Ek+TV const. When T is very large tends Ek to 3/2 NkT, empirically pv tends to NkT. Hence constant in equation of state above obtained must vanish equation (19). The expression for entropy becomes

6 204 Takuzo SAKAI. [Vol. 22 Using relation (21), this takes following form: or on using expression for N. In Boltz- mann's method last form (23) for entropy is directly obtained from assumption S= k Jog G, on iusng Stirling's formula for factorials. 5. In case of localized molecules (for example, a system of rotators, each of which has its centre of gravity at a specified position, or a system of oscillators, each of which has its centre of force at a specified position), we can enumerate each of m. The statistical weight is of form Then following same method as before, at once, without assumption (9). is weight of state with energy, does not contain V as a factor as in (17). As we assume that molecules are localized mutual interaction is negligible, re appears no term like pv corresponding to external work in rmodynamic potential. Hence The expression as usual. 6. The The us consider Take for entropy canonical problem Einstein-Bose' becomes ensemble is characterized of fluctuation fluctuation N case for is treated of N, which on a similar is defined Fake of illustration. Then way. Let

7 1940] Gibbs' Canonical Ensemble Distribution Law. 205 Therefore Substituting this into (27), On a similar way, in case of Fermi-Dirac's statistics, for gaseous systems obeying Boltzmann's statistics. On or h for systems with localized molecules. The fluctuation of total energy is given Taking Einstein-Bose' case, wee get easily on way similar to that used in obtaining (20). Here according as =µ or µ. Hence or C

8 206 Takuzo SAKAI [vol. 22 Therefore In case in for gasous of Fermi-Dirac's case of systems Boltzmann's statistics, statistics for systems with localized molecules. All of above results are same with those obtained or methods. 7. In foregoing articles we have explained validity of present method in case of equilibrium states. But formulae (3) (4) are not restricted only to equilibrium states. The essential idea is that macroscopically indistinguishable states have a equal a priori probability(1) that statistical weight is given our G's when we can classify molecular states. The case of non-equilibrium states will be considered at anor occasion. 8. Finally a few remarks will be mails about fluctuation, since statistical mechanics is applied to problems of nuclei of atoms. The fluctuation is generally small when number of particles is very large. But in problems of nuclei, number of protons or neutrons is not necessarily large. If we assume that nuclei are gaseous systems (which will not be a good model but in frequently assumed) particles obey Fermi-Dirac's statistics, we can not directly apply results above obtained, because we have assumed that N is sufficiently large compared with unity. This assumption is also made in DarwinFowler's method or in Boltzmann's method. In case of gr (1) J. v. Neuumann: "Mamatische Grundlagen der Quantcnmechanik" (1932) 219. R. C. Tolman ; l.e. T. Sakai : Pros. Phys. Math. Soc. Japan, 19 (1937), 172, in which Pauli's hypose of rom a priori phases Is discussed.

9 1940] Gibb's Canonical Ensemble Distribution Law. 207 ensemble we must take average number N in place of N in above formula. Here gr ensemble is an ensemble which consists of systems with different total number N. In this case, N can be a small number. In this sense, we shall be able to apply above formulae to nuclear problem, although n re come in or difficulties implied in conception of gr ensemble, namely interchange of rmal energy or of particles in nuclei with surroundings. But anyhow putting aside se detailed considerations for present, following results at low temperatures, assuming structureless particles: 0 is Fermi energy quantities of order of / 30 are neglected compared with unity. The calculation is carried out Sommerfeld's method in electron ory of metals. The fluctuation of total number is given if we take gr ensemble. At low temperatures, neglecting 3 30 compared with unity. Hence fluctuation of total energy is small when << even if N is not large, but fluctuation of total number of particles is small only when N is large. If we take =1/5, n N must be greater than about 190 in order that N N may be smaller than 1/5. At high temperatures problem is same as in Boltzmann's statistics. Therefore fluctuation is small only when total number of particles is large. Considered from se facts, gaseous models in nuclear problems are crude approximations. Faculty of Science, Tokyo Imperial University. (Received Feb.19,1940)