Page:Elementary Principles in Statistical Mechanics (1902).djvu/121

That is, ${\displaystyle V_{q}}$ is the same function of ${\displaystyle \epsilon +q}$, as ${\displaystyle {\frac {1}{(2\pi )^{\tfrac {n}{2}}}}{\frac {d^{\tfrac {n}{2}}V}{d\epsilon ^{\tfrac {n}{2}}}}}$ of ${\displaystyle \epsilon }$.

When ${\displaystyle n}$ is large, approximate formulae will be more available. It will be sufficient to indicate the method proposed, without precise discussion of the limits of its applicability or of the degree of its approximation. For the value of ${\displaystyle e^{\phi }}$ corresponding to any given ${\displaystyle \epsilon }$, we have

${\displaystyle e^{\phi }=\int \limits _{V_{q}=0}^{\epsilon _{q}=\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{q}=\int \limits _{0}^{\epsilon }e^{\phi _{p}+\phi _{q}}\,d\epsilon _{p},}$

(308)

where the variables are connected by the equation (300). The maximum value of ${\displaystyle \phi _{p}+\phi _{q}}$ is therefore characterized by the equation

${\displaystyle {\frac {d\phi _{p}}{d\epsilon _{p}}}={\frac {d\phi _{q}}{d\epsilon _{q}}}.}$

(309)

The values of ${\displaystyle \epsilon _{p}}$ and ${\displaystyle \epsilon _{q}}$ determined by this maximum we shall distinguish by accents, and mark the corresponding values of functions of ${\displaystyle \epsilon _{p}}$ and ${\displaystyle \epsilon _{q}}$ in the same way. Now we have by Taylor's theorem

${\displaystyle \phi _{p}=\phi _{p}'+{\bigg (}{\frac {d\phi _{p}}{d\epsilon _{p}}}{\bigg )}'(\epsilon _{p}-\epsilon _{p}')+{\bigg (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\bigg )}'{\frac {(\epsilon _{p}-\epsilon _{p}')^{2}}{2}}+\mathrm {etc.} }$

(310)

${\displaystyle \phi _{q}=\phi _{q}'+{\bigg (}{\frac {d\phi _{q}}{d\epsilon _{q}}}{\bigg )}'(\epsilon _{q}-\epsilon _{q}')+{\bigg (}{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\bigg )}'{\frac {(\epsilon _{q}-\epsilon _{q}')}{2}}+\mathrm {etc.} }$

(311)

If the approximation is sufficient without going beyond the quadratic terms, since by (300)

${\displaystyle \epsilon _{p}-\epsilon _{p}'=-(\epsilon _{q}-\epsilon _{q}'),}$

we may write

${\displaystyle e^{\phi }=e^{\phi _{p}'+\phi _{q}'}\int \limits _{-\infty }^{+\infty }e^{{\Big [}{\bigg (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\bigg )}'+{\bigg (}{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\bigg )}'{\Big ]}{\frac {(\epsilon _{q}-\epsilon _{q}')^{2}}{2}}}\,d\epsilon _{q},}$

(312)

where the limits have been made ${\displaystyle \pm \infty }$ for analytical simplicity. This is allowable when the quantity in the square brackets has a very large negative value, since the part of the integral