RECENT STATISTICAL THEORIES 683 



of compartments into which the volume V is divided, by Ni the 

 number of particles in the 7'th compartment. A distribution is 

 described by stating all the numbers Ni, Ni- ■ • , Nj, • • ■ N,,,. 



The basis of the classical statistics is the fact that if the particles 

 have identities — if each of them is labelled by a distinctive letter, 

 for instance — there are different ways of arranging them in the same 

 distribution. One starts with any arrangement compatible with the 

 prescribed "populations" Ni, No, ■••N^, and obtains all the other 

 arrangements by interchanging particles ad libitum among the com- 

 partments, respecting only the condition that each of these shall 

 always have as many as it had at first. The total number of distinct 

 arrangements, the number of permutations of the combination Ni, No, 

 • • -Nm, is by a well-known theorem: "* 



W = ^ (7) 



NrlNol-'-NJ. ^^ 



This number has its minimum value of unity for a distribution in 

 which all the particles are crowded into one compartment, which 

 would be the most non-uniform conceivable; and its maximum value 

 for the uniform distribution, as I now proceed to show.'' 



Let us use the logarithm of W instead of W itself. If W has a 

 maximum for any distribution so also will its logarithm, which is 

 easier to handle, and will presently be chosen as the representation 

 of entropy. We have: 



log W = log NI - E log Nil. (8) 



Now we introduce Stirling's approximation for the factorial of a 

 large number — by far the greatest and the most frequently invoked 



* Imagine yourself stationed beside a set of m baskets and an urn filled with N 

 lettered but otherwise indistinguishable balls, which are to be lifted out at random 

 and dropped into the baskets under the following rules of the game: the first A'^i 

 which come to your hand are to be dropped into basket 1, the next N2 to come to 

 your hand are to go into basket 2, and so on to the end. Having acted accordingly, 

 you note down the assortments of balls in the various baskets, and repeat the process 

 ad infinitum. Now there are A'^! different orders in which the balls may come out 

 of the urn. When the inspection of the baskets after two drawings reveals different 

 results, the orders must certainly have been different. But two different orders 

 need not reveal two different results to the inspection. Take any order, to start 

 with; then there are {Q — \) = {N^lNol ••• N,n\ — 1) others which yield the same 

 result. For there are Ni ! orders in which the earliest Ni balls emerge might come 

 out, without any of them losing its place among the first iVi; there are N^l orders 

 in which the next iV2 might come, without any losing its place in the second basket; 

 and so forth. Each of the NI orders then is but one among Q altogether which 

 lead to the same result; so that there are only Nl/Q different results. 



* What will actually be shown is that for the uniform distribution the function 

 W is stationary; that it is maximum (not minimum) seems fairly obvious from the 

 physics of the case, and can be proved. 



