A Ensemble statistics

In ensemble statistics, it is necessary to compute the number of complexions of the ensemble for a given distribution of the X member systems among the possible states of a system. A distribution is defined by the set of integers { N _i} = ( N ₁, N ₂, , N _j ) such that N _j is the number of member systems in the jth quantum state. The number of complexions is just the number of ways of realizing this distribution.

Since the members of the ensemble are macroscopic systems, they are distinguishable from one another. Therefore, we need to count the number of ways of arranging X distinguishable systems such that N ₁ are in state 1, N ₂ are in state 2, , N _j are in state j, and so on, with the condition that the total number of systems is

Clearly, this is equivalent to putting X marbles in boxes such that N ₁ are in the first box, N ₂ are in the second box, and so on, without regard to the order of the arrangement of marbles in a particular box. Call this number W.

First, let us show that the number of ways of putting the marbles in the boxes in such a way that the ordering of the marbles is taken into account is X!. For this case,...