Page:Elementary Principles in Statistical Mechanics (1902).djvu/201

To make the problem definite, let us consider a system consisting of the original system together with another having the coördinates $a_{1}$, $a_{2}$, etc., and forces $A_{1}'$, $A_{2}'$ etc., tending to increase those coördinates. These are in addition to the forces $A_{1}$, $A_{2}$, etc., exerted by the original system, and are derived from a force-function ($-\epsilon _{q}'$) by the equations

$A_{1}'=-{\frac {d\epsilon _{q}'}{da_{1}}},\quad A_{2}'=-{\frac {d\epsilon _{q}'}{da_{2}}},\quad \mathrm {etc.}$

For the energy of the whole system we may write

$E=\epsilon +\epsilon _{q}'+{\tfrac {1}{2}}m_{1}{\dot {a}}_{1}^{2}+{\tfrac {1}{2}}m_{2}{\dot {a}}_{2}^{2}+\mathrm {etc.} ,$

and for the extension-in-phase of the whole system within any limits

$\int \ldots \int dp_{1}\ldots dq_{n}\,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots$

or

$\int e^{\phi }\,d\epsilon \,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots ,$

or again

$\int e^{\phi }\,dE\,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots ,$

since $d\epsilon =dE$, when $a_{1}$, ${\dot {a}}_{1}$, $a_{2}$, ${\dot {a}}_{2}$, etc., are constant. If the limits are expressed by $E$ and $E+dE$, $a_{1}$ and $a_{1}+da_{1}$, ${\dot {a}}_{1}$ and $a_{1}+d{\dot {a}}_{1}$, etc., the integral reduces to

$e^{\phi }\,dE\,da_{1}\,m_{1}\,d{\dot {a}}_{1}\,da_{2}\,m_{2}\,d{\dot {a}}_{2}\ldots$

The values of $a_{1}$, ${\dot {a}}_{1}$, $a_{2}$, ${\dot {a}}_{2}$, etc., which make this expression a maximum for constant values of the energy of the whole system and of the differentials $dE$, $da_{1}$, $d{\dot {a}}_{1}$, etc., are what may be called the most probable values of $a_{1}$, ${\dot {a}}_{1}$, etc., in an ensemble in which the whole system is distributed microcanonically. To determine these values we have

$de^{\phi }=0,$

when

$d(\epsilon +\epsilon _{q}'+{\tfrac {1}{2}}m_{1}{\dot {a}}_{1}^{2}+{\tfrac {1}{2}}m_{2}{\dot {a}}_{2}^{2}+\mathrm {etc.} )=0.$

That is,

$d\phi =0,$