On the Boltzmann Distribution Law Leonard K. Nash Harvard University. Cambridge, MA 02138 The standard derivation of the Boltzmann Distribution Law (BDL) is crisp and compelling hut accessihleonly to those familiar with Lagrangean multipliers (I). Efforts to reduce the mathematical sophistication of this derivation usually result in analyses (2) that are quite clumsy. And, though attractive in its naivetA, the most conspicuous alternative mode of derivation is even more clumsy and tedious (3).We wish to show how, while retaining its essential quality, the standard derivation can be made accessible to students who command no more than the hare rudiments of the calculus. Multiplying without Lagrange Let N independent distinguishable units be distributed the number of over a set of uuantum states:. let n, svmholize . units in an n-th quantum state with energy r,. We seek an analytical description of the distrihution that imparts a maximum value to the (microstate number) W defined by the equation:
-
Of course W has no well-defined maximum if the number of units ( N x v , ) and their total energy ( E ZG,~,,) are freely variable. Hence, we seek to maximize W for an isolated system, where the restrictions d N = 0 = d E impose the constraints Ed% =0
(2.4
with the summations extending over all permissible quantum states. Holding in reserve these essential constraints, we rewrite eqn. (1)in logarithmic form, as: In W = In N! - xlnv,!. For the distribution in which W reaches an extremum (easily shown to he a maximum) the criterion d W = 0 entails also that d i n W = dWIW = 0. Hence, with N a constant, differentiation yields as the condition for a maximum: If one prefers to eschew use of Stirling's approximation, the differential may still he attained easily as suggested by Wyatt ( 4 ) ,by writing for large 7,: d i n vn!/dvn N A In vn!/Avn = In v,,! - In(?,, - I)! = In v,,. Hence, the crucial requirement is What is the analytical condition both necessary and sufficient to enforce a zero value for this key summation? The perturbations dq, are not in general equal to zero individually. As the only condition sufficient to enforce the validity of eqn. (3) one might then pardonably suppose that, for each 7, This condition would indeed he not only sufficient but also necessary if the dq, terms were totally independent of each other. But we find no such independence in an isolated assembly, where dh' = 0 = dE. Its sufficiency notwithstanding, the above condition thus fails the test of necessity in an assembly subject to the constraints (2a) and (2h). Consider then the possibility that, for each v,,,
824
Journal of Chemical Education
where a is a constant. Including as a special case our earlier conjecture that in vn = 0, the weaker condition In 11, = a is wholly sufficient to ensure the validity of eqn. (3), since now E l n q,dq, = a E d q n = 0
where the last eoualitv is enforced bv constraint (2a). . . But though the sufficiencybf the condition In 7, = a is thus evident. even vet this is not a condition necessarv for validitv of eqn.(3). F& we now easily see a still weakeicondition chat fully guarantees such validity. Consider the proposition where, like a, in the isolated assembly @ represents some one constant.' Returnine" to the summation in eon. (3). we now find
where the final equality is enforced because the two immediately preceding summations are reduced to zero by the constraints (2a) and (Zh), respectively. Thus demonstrably sufficient, eqn. (4) is a t last a condition also unequivocally necessary for the validity of e m . (3). There can be no weaker condition fur surh vdidhy silriply heriluse condition r $ 1 fully exploits constraints rln) and ('Lh),and beyond these no further constraints characterize isolated systems. The rationale of Lagrangean multipliers now being obvious, one may if one likes comment on the mechanics of their application. But in eqn. (4) we already have established the essential guarantor of eqn. (3), which in turn guarantees d In W = 0. Hence, eqn. (4) is the sought-for analytical description of the distribution in which W attains its maximum value. Writing eqn. (4) for any two quantum levels (i and n), we eliminate a by subtraction, and thus attain T o obtain the simplest form of the BDL, we may take i to be the ground state, with population qi = and an assigned energy ri = ro 10.The last equation then reduces to ?.Iqo = e d C -
(5)
and any of several simple methods suffices to demonstrate that @ = llkT. Expanding without Taylor We often wish to identify the equilibrium state of an isolated system with the configuration described by the BDL. This will be permissible only if the number of microstates ( Wo) associated with the BDL configuration vastly exceeds the number of microstates (W) associated with similar configurations. That Wo >> W is easily shown in the special case of units that have either iust two nossihle auantum states or quantum states with unitorm energy spacing. However, the only demonstration r.51 that is h t h runpic. and g t , n ~ r n lis also
+
' Thouah it mav amear less "natural" than ihe formulation a f l f . the relalion a - j t i ' l s here preferred because. dltwnately, it yieids'i conven ently posilive value for me important parameter .j.
exceedingly tedious and, due to sloppy approximation, yields a result in error by an exponential factor of 2. We now show how this (corrected) analysis can be much abbreviated and freed from all dependence on Stirling's approximation. In an isolated assembly where is the number of units in the n-th quantum state of the BDL configuration, let the corresponding population in any other configuration be 7, 6,. Here 6, is a (positive or negative) number that may he large compared with zero or one provided only that it remain negligible in comparison with an 7, assumed very large. We then call on eqn. (1)to write for the two configurations
+
-wo =
W
nw
N - ncq, +a,)!- N!/II(q, + 6,)! nq,! 7" ! (7, + 6,)(1, + 6, - 1 )...(b + lhn! =n %!
.
+
The polynomial, with 6, terms ranging from (7, 6,) to (vn I), has here been replaced by a product of 6, terms all 6,,/2). The adequacy of this having the same value, (7" approximation is well assured by our initial stipulation that lb,, > Replacing the logarithm of a continuing product by the continuing sum of the corresponding logarithmic terms, we proceed to
+
+
= X 6 , In 7,
1
A Stirling Equivalent
Let 0 symbolize the total number of microstates associated with all configurations of an isolated assembly. In one standard development of statistical mechanics, a fundamental step is the replacement of in Cl by In Wo. Apart from purely verbal arguments, the only simple justification of this replacement merely cites special numerical cases of units that have at most three quantum states evenly spaced in energy ( 6 ) .The following (purely algebraic) treatment generalizes to assemblies of units with an unlimited number of quantum states having a uniform energy spacing Ar = hu. If as required by eqn. (5) all energies are measured from the ground-state energy ro = 0,then r, = nhu. With 7, now referring specifically to the population imposed by the BDL, we may write for the total number (N) of units in a given assembly
- -N.
= xqoe-8% = qore-Blhv)"
where x e-ph". With x < 1, the summation extending from n = 0 to n is easilv shown exoressihle in closed form. yielding N = qol(1
--
...
which result the cognoscente will obtain by a Taylor-series expansion. The first summation, the analog of the differential function x l n qndqn,is easily shown to be zero for any isolated system, in which all shifts of configuration must proceed with AN = E(s. + 6,) - Xs. = 2 6 , = 0 AE = X(7. + 6 A - Eq.r. = Z6.r. = 0 With 7, referring to the BDL configuration described by eqn. (41, it then follows that = 2cin(c2 - or,)=
- x)
(8)
1
With 6,
