Page:Elementary Principles in Statistical Mechanics (1902).djvu/114

the potential energy, either alone or with quantities which are constant in the integration.

We may often avoid the inconvenience occasioned by formulae becoming illusory on account of discontinuities in the values of $V_{q}$ as function of $\epsilon _{q}$ by substituting for the given discontinuous function a continuous function which is practically equivalent to the given function for the purposes of the evaluations desired. It only requires infinitesimal changes of potential energy to destroy the finite extensions-in-configuration of constant potential energy which are the cause of the difficulty.

In the case of an ensemble of systems canonically distributed in configuration, when $V_{q}$ is, or may be regarded as, a continuous function of $\epsilon _{q}$ (within the limits considered), the probability that the potential energy of an unspecified system lies between the limits $\epsilon _{q}'$ and $\epsilon _{q}''$ is given by the integral

$\int _{\epsilon _{q}'}^{\epsilon _{q}''}e^{{\frac {\psi _{q}-\epsilon _{q}}{\Theta }}+\phi _{q}}d\epsilon _{q},$

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where $\psi$ may be determined by the condition that the value of the integral is unity, when the limits include all possible values of $\epsilon _{q}$. In the same case, the average value in the ensemble of any function of the potential energy is given by the equation

${\overline {u}}=\int _{V_{q}=0}^{\epsilon _{q}=\infty }ue^{{\frac {\psi -\epsilon _{q}}{\Theta }}+\phi _{q}}d\epsilon _{q}.$

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When $V_{q}$ is not a continuous function of $\epsilon _{q}$, we may write $dV_{q}$ for $e^{\phi _{q}}d\epsilon _{q}$ in these formulae.

In like manner also, for any given configuration, let us denote by $V_{p}$ the extension-in-velocity below a certain limit of kinetic energy specified by $\epsilon _{p}$. That is, let

$V_{p}=\int \ldots \int \Delta _{p}^{\frac {1}{2}}dp_{1}\ldots dp_{n},$

(276)