J. W. Lorimer
Universitv of Western Ontario London, Ontorio Conodo
I
I
Elementary Statistical Mechanics Without Lagrange Multipliers
I n the elementary deduction of the three statistical mechanical distribution laws (MaxwellBoltzmann, Bose-Einstein, and Fermi-Dirac), two purely mathematical complications arise. The first involves a simplified expression for the logarithm of a factorial, for which Stirling's approximation is usually invoked. Wyatt ( I ) has shown that an approximation to the differential of a factorial may be obtained simply, and that this approximation is sufficient for the deduction of the distribution laws. If desired, the usual first approximation to the complete Stirling theorem may be obt,ainedby integration of the differential. The second complication involves finding the conditions for a maximum, subject to two conditions of constraint, and the Lagrange method of undertermined multipliers (2) is usually used for this purpose. At first sight, the Lagrange method appears to be a highly arbitrary procedure, "the validity of which is not obvious" (3). The alternative "direct" method given
below avoids the arbitrary feature of Lagrange multipliers. At the same time, i t provides complete justifice tion for the Lagrange method as it is usually presented. Application of this direct method in the deduction of the Maxwell-Boltzmann distribution law will serve as an example, hut it should be noted that the method is equally useful in the deduction of the other distribution laws, and, in fact, in any other problems involving constrained maxima or minima. The probability that a system contains n, independent, indistinguishable molecules with energy el, nzwith energy et,. . . n, with energy r, is (4) :
subject to the conditions that the total number of molecules,
Volume 43, Number 1, January 1966
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and the total energy,
are given constant values. The problem is to find the conditions under which W is a maximum. I n other words, arbitrary variations in W (or In W for convenience), N and E with respect to the ni must all be zero: 6 In
W
=
-
In (ni/N)Sni = 0
(4)
i = l
and
Then eliminate an,-, from equations (10) and (11):
I n eqn. (12), the r - 2 values of 6nl are all independent, so that the bracketed term must he zero for i = 1,2,. . . r - 2, giving, after rearrangement, In eqn. (4), Wyatt's approximation to the differential of a factorial (1) has been used. The usual (Lagrange) method of finding the appropriate set of ni's is to multiply eqn. (5) by the constant a, eqn. (6) by the constant 8, and subtract the resulting equations from eqn. (4) to give:
The right-hand side of eqn. (13) contains only the variables n,_] and n,. In other words, eqn. (13) states that ln (n&)/(w
- e,)
= -P
(14)
where p is a constant, for any of the r - 1 values of i. Rearrangement of (14) gives: I n eqn. (7), only r - 2 of the variations are independent, hecause of equations (5) and (6). For this reason, it is necessary to choose a and p such that the term in brackets is zero for two values of i, say the rth and (r - 1)th values. With a and 0 now determined, the remaining r - 2 values of Fni are independent, and the term in brackets must be zero for each value of i, giving: In (ni/N)
+ a + Pei = 0
(8)
the Maxwell-Boltzmann distribution law. The interpretation of or as the logarithm of the partition function and of p as l/kT follows in the usual manner (4). Unfortunately, the logical steps in the Lagrange method which lead from eqn. (7) to eqn. (8) are frequently ignored. I n fact, they appear to be redundant, for if all the 6 n ~in eqn. (7) are considered to be independent, the same result is obtained. For this reason, the student with little mathematical sophistication can find this aspect of the Lagrange method somewhat mysterious. An alternative to the Lagrange method is to solve eons. (4)-(6) ~, ~, directlv. To do this. i t is sufficient to note that two of the"r variations 6n; can be eliminated from eqn. (4) by use of eqns. (5) and (6). First eliminate 6n, from the pairs (4) and (6) and (5) and (6), to obtain:
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Journal of Chemical Education
where a' is a second constant introduced for the same reason as (3. Except for the constant In N, eqn. (15) is identical to eqn. (24, and holds for all r values of i. The appearance of the constants a' and p comes as no surprise, for these constants are direct expressions of the existence of the two conditions of constraint (2) and (3), and thus characterize the state of the system. The method given above has been used in some texts (5, 6) to solve problems in which a maximum snbject to one constraint is sought. However, ahen two constraints are present, both texts referred to fall back on the Lagrange method. The "direct" method is essentially the Lagrange method in reverse, and may be extended if desired to cases in which any number of conditions of constraint are present. Literature Cited (1) W Y AP.~A. H., J. CHEM. EDUC., 39,27 (1962). (2) PAGE,L., "An Introduction to Theoretical Physics," 3rd ed., D. Van Nostrand Co., Inc., New York, 1952, pp. 329-30. (3) WALL,F. T., "Chemical Thermodynamics,'' 2nd. ed., W. H. Freeman and Co., San Francisco, 1965, p. 231. (4) Ibid., chaps. 11,13. ( 5 ) Ibid., pp. 221t30. ( 6 ) CRAWFORD, F. H., "Heat, Thermodynamics, and Statistical Physics," Harcourt, Brace and World, Inc., New York, 1963, pp. 464-5.
