Page:Elementary Principles in Statistical Mechanics (1902).djvu/124

CHAPTER IX.

THE FUNCTION ϕ AND THE CANONICAL DISTRIBUTION.

In this chapter we shall return to the consideration of the canonical distribution, in order to investigate those properties which are especially related to the function of the energy which we have denoted by $\phi$.

If we denote by $N$, as usual, the total number of systems in the ensemble,

$Ne^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon$

will represent the number having energies between the limits $\epsilon$ and $\epsilon +d\epsilon$. The expression

$Ne^{{\frac {\psi -\epsilon }{\Theta }}+\phi }$

(317)

represents what may be called the density-in-energy. This vanishes for $\epsilon =\infty$, for otherwise the necessary equation

$\int \limits _{V=0}^{\epsilon =\infty }e^{{\frac {\psi -\epsilon }{\Theta }}+\phi }\,d\epsilon =1$

(318)

could not be fulfilled. For the same reason the density-in-energy will vanish for $\epsilon =-\infty$, if that is a possible value of the energy. Generally, however, the least possible value of the energy will be a finite value, for which, if $n>2$, $e^{\phi }$ will vanish,^[1] and therefore the density-in-energy. Now the density-in-energy is necessarily positive, and since it vanishes for extreme values of the energy if $n>2$, it must have a maximum in such cases, in which the energy may be said to have

1. See page 96.