Page:Elementary Principles in Statistical Mechanics (1902).djvu/134

The impossibility of a canonical distribution occurs when the equation

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

(361)

fails to determine a finite value for $\psi$. Evidently the equation cannot make $\psi$ an infinite positive quantity, the impossibility therefore occurs when the equation makes $\psi =-\infty$. Now we get easily from (191)

$d{\frac {\psi }{\Theta }}=-{\frac {\overline {\epsilon }}{\Theta ^{2}}}\,d\Theta .$

If the canonical distribution is possible for any values of $\Theta$, we can apply this equation so long as the canonical distribution is possible. The equation shows that as $\Theta$ is increased (without becoming infinite) $-\psi$ cannot become infinite unless ${\overline {\epsilon }}$ simultaneously becomes infinite, and that as $\Theta$ is decreased (without becoming zero) $-\psi$ cannot become infinite unless simultaneously ${\overline {\epsilon }}$ becomes an infinite negative quantity. The corresponding cases in thermodynamics would be bodies which could absorb or give out an infinite amount of heat without passing certain limits of temperature, when no external work is done in the positive or negative sense. Such infinite values present no analytical difficulties, and do not contradict the general laws of mechanics or of thermodynamics, but they are quite foreign to our ordinary experience of nature. In excluding such cases (which are certainly not entirely devoid of interest) we do not exclude any which are analogous to any actual cases in thermodynamics.

We assume then that for any finite value of $\Theta$ the second member of (361) has a finite value.

When this condition is fulfilled, the second member of (359) will vanish for $u=e^{-\phi }V$. For, if we set $\Theta '=2\Theta$,

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