1= - W

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(n;)

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This is the same definition as (3.52) except that we have replaced the infinitesimal element d 3r d 3p by a finite cell of volume w. Unfortunately this ensemble average is difficult to calculate. So we shall adopt a somewhat different approach, which will yield the same result for a sufficiently large system. It is clear that if the state of the gas is given, then f is uniquely determined; but if f is given, the state of the gas is not uniquely determined. For example, interchanging the positions of two molecules in the gas leads to a new state of the gas, and hence moves the representative in r space; but that does not change the distribution function. Thus a given distribution function f corresponds not to a point, but to a volume in r space, which we call the volume occupied by f. We shall assume that the equilibrium distribution function is the most probable distribution function, i.e., that which occupies the largest volume in r space. The procedure is then as follows:

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(a) Choose an arbitrary distribution function by choosing an arbitrary set

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of allowed occupations numbers. Calculate the volumes it occupies by counting the number of systems in the ensemble that have these occupation numbers. (b) Vary the distribution function to maximize the volume. Let uS denote by Q {n;} the volume in r space occupied by the distribution function corresponding to the occupation numbers {n;}. It is proportional to the number of ways of distributing N distinguishable molecules among K cells so that there are n; of them in the ith cell (i = 1,2, ... , K). Therefore

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Q {n;}

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n 1 !n 2 !n 3 ! .,. n K !

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g~l g;2

... g;K

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(4.38)

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where g; is a number that we will put equal to unity at the end of the calculation but that is introduced here for mathematical convenience. Taking the logarithm

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THERMODYNAMICS AND KINETIC THEORY

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of (4.38) we obtain log 0 {n i }

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log N! -

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log nil +

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L n i log g; + constant

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Now assume that each n; is a very large integer, so we can use Stirling's approximation, log n;! =:: n; log n; - 1. We then have log 0 {n;} = N log N -

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L n; log nil

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L n i log g; + constant

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(4.39)

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To find the equilibrium distribution we vary the set of integers {n;} subject to the conditions (4.35) and (4.36) until log 0 attains a maximum. Let {ii;} denote the set of occupation numbers that maximizes log O. By the well-known method Lagrange multipliers we have

~[logO{n;}] -~(a;~ln;+,8;~It:;ni)

L [- (log n; + 1) + log g; ;=1

(ni=iiJ

(4.40)

where a, ,8 are Lagrange's multipliers. Now the n; can be considered independent of one another. Substituting (4.39) into (4.40) we obtain

a - ,8d ~n;

Since ~n; are independent variations, we obtain the equilibrium condition by setting the summand equal to zero: log ii; = -1 + log g; - a - ,8t:; (4.41 ) ii = g.e-a-{l<,-l

The most probable distribution function is, by (4.37) and (4.41),

/; =

Ce-{l<,

(4.42)

where C is a constant. The determination of the constants C and ,8 proceeds in the same way as for (4.13). Writing /; == f(p;), we see that f(p) is the MaxwellBoltzmann distribution (4.23) for Po = o. To show that (4.41) actually corresponds to a maximum of log 0 { n i} we calculate the second variation. It is easily shown that the second variation of the quantity on the left side of (4.40), for

-(~nJ2 < 0 ni

We have obtained the Maxwell-Boltzmann distribution as the most probable distribution, in the sense that among all the systems satisfying the macroscopic conditions the Maxwell-Boltzmann distribution is the distribution common to the largest number of them. The question remains: What fraction of these systems have the Maxwell-Boltzmann distributions In other words, how probable is the most probable distribution The probability for the occurrence of any set of