122 BELL SYSTEM TECHNICAL JOURNAL 



of the statement. The student may convince himself of this by applying 

 the second-degree StirHng approximation (equation 2)^. 



1 have called this both a helpful and a troublesome coincidence. It may 

 be deemed a helpful one, because the expression for the total number of in- 

 ventories is easier to derive and easier to remember than the expression for 

 the number of inventories belonging to the most probable distribution. If 

 therefore one has good ground for believing (as here is the case) that the 

 logarithms of the two are approximately equal, one may serenely remember 

 and use In Wtot instead of In IFmax • The troublesome feature is, that some 

 expositors speak of In Wtot throughout and never allude to In Tl^max , thus 

 confusing the student to an extent which (if my experience is typical) may 

 well be serious. I shall later dwell on the fact that In W for any distribution 

 is regarded as a measure of the entropy of that distribution, and In IFmax 

 therefore as a measure of the entropy of the most probable distribution. 

 Some people imply that In Wtot is the true measure of the entropy of the gas, 

 instead of being an approximation to it. They commit no numerical error 

 in so doing, but they blot out the most remarkable quality of the Boltzmann 

 statistics, to wit, the clear distinction which it makes between the most prob- 

 able distribution and those of lesser probability. This mistake is more 

 commonly made in treating the newer statistics. Here I am not so sure 

 that it is a mistake, but I think so. 



Meanings of the Word "State" 



The word "state", which turns up continually in this essay, is one of those 

 words of which a proper definition is hardly less than a full description of the 

 theory which employs it. When the theory changes so also does the meaning 

 of the word. In the welter of statistical theories, the word "state" has 

 several different meanings. In thermodynamics also it has more than one 

 meaning, but one is preeminent. 



Thermodynamics usually concerns itself with gases (not to speak of li- 

 quids and solids) which are in what I earlier called a "uniform" state: uniform 

 density, uniform pressure, uniform temperature. For a gas of a single kind 

 ("kind" being a word which it is the business of chemistry to define) it is a 

 fact of experience that any two of these three variables suffice to define the 

 third and also all of the other variables which thermodynamics cares about. 

 Of these others there are two in particular which I mention at this point, the 



2 Actually if one goes from the "most probable state" Ni = const. = N/M to the 

 "next most probable" in which one ball is taken out of one of the baskets and put into 

 another, the change in W is in the ratio of {N/M) to {N/M) + 1, which is practically 

 no change at all when N/M is so high as is commonly taken. This shows that the state- 

 ment could not be true if it were made about the numbers Umax and Wtot rather than 

 about the logarithms thereof. It certainly looks as though the statement could not be 

 true even when made of the logarithms, but this is evidently one of the cases where 

 "intuition" is a fallible guide. 



