Page:Elementary Principles in Statistical Mechanics (1902).djvu/199

$\epsilon _{1}$, and $\phi _{2}=-\infty$ for the least possible value of $\epsilon _{2}$. Between these limits $\phi _{1}$ and $\phi _{2}$ will be finite and continuous. Hence $\phi _{1}+\phi _{2}$ will have a maximum satisfying the equation (488).

But if $n_{1}\leq 2$, or $n_{2}\leq 2$, $d\phi _{1}/d\epsilon _{1}$ or $d\phi _{2}/d\epsilon _{2}$ may be negative, or zero, for all values of $\epsilon _{1}$ or $\epsilon _{2}$, and can hardly be regarded as having properties analogous to temperature.

It is also worthy of notice that if a system which is microcanonically distributed in phase has three parts with separate energies, and each with more than two degrees of freedom, the most probable division of energy between these parts satisfies the equation

${\frac {d\phi _{1}}{d\epsilon _{1}}}={\frac {d\phi _{2}}{d\epsilon _{2}}}={\frac {d\phi _{3}}{d\epsilon _{3}}}.$

That is, this equation gives the most probable set of values of $\epsilon _{1}$, $\epsilon _{2}$, and $\epsilon _{3}$. But it does not give the most probable value of $\epsilon _{1}$, or of $\epsilon _{2}$, or of $\epsilon _{3}$. Thus, if the energies are quadratic functions of the $p$'s and $q$'s, the most probable division of energy is given by the equation

${\frac {n_{1}-1}{\epsilon _{1}}}={\frac {n_{2}-1}{\epsilon _{1}}}={\frac {n_{3}-1}{\epsilon _{3}}}.$

But the most probable value of $\epsilon _{1}$ is given by

${\frac {n_{1}-1}{\epsilon _{1}}}={\frac {n_{2}+n_{3}-1}{\epsilon _{2}+\epsilon _{3}}},$

while the preceding equations give

${\frac {n_{1}-1}{\epsilon _{1}}}={\frac {n_{2}+n_{3}-2}{\epsilon _{2}+\epsilon _{3}}}.$

These distinctions vanish for very great values of $n_{1}$, $n_{2}$, $n_{3}$. For small values of these numbers, they are important. Such facts seem to indicate that the consideration of the most probable division of energy among the parts of a system does not afford a convenient foundation for the study of thermodynamic analogies in the case of systems of a small number of degrees of freedom. The fact that a certain division of energy is the most probable has really no especial physical importance, except when the ensemble of possible divisions are grouped so