Page:Elementary Principles in Statistical Mechanics (1902).djvu/60

and the system ${\displaystyle B}$ as one of an ensemble of systems of ${\displaystyle n}$ degrees of freedom distributed in phase with a probability-coefficient

${\displaystyle e^{\frac {\psi _{B}-\epsilon _{B}}{\Theta }}}$

which has the same modulus. Let ${\displaystyle q_{1},\ldots q_{m}}$, ${\displaystyle p_{1},\ldots p_{m}}$ be the coördinates and momenta of ${\displaystyle A}$, and ${\displaystyle q_{m+1},\ldots q_{m+n}}$, ${\displaystyle p_{m+1},\ldots p_{m+n}}$ those of ${\displaystyle B}$. Now we may regard the systems ${\displaystyle A}$ and ${\displaystyle B}$ as together forming a system ${\displaystyle C}$, having ${\displaystyle m+n}$ degrees of freedom, and the coördinates and momenta ${\displaystyle q_{1},\ldots q_{m+n}}$, ${\displaystyle p_{1},\ldots p_{m+n}}$. The probability that the phase of the system ${\displaystyle C}$, as thus defined, will fall within the limits

${\displaystyle dp_{1},\ldots dp_{m+n}\,dq_{1},\ldots dq_{m+n}}$

is evidently the product of the probabilities that the systems ${\displaystyle A}$ and ${\displaystyle B}$ will each fall within the specified limits, viz.,

${\displaystyle e^{\frac {\psi _{A}+\psi _{B}-\epsilon _{A}-\epsilon _{B}}{\Theta }}dp_{1},\ldots dp_{m+n}\,dq_{1},\ldots dq_{m+n}.}$

(94)

We may therefore regard ${\displaystyle C}$ as an undetermined system of an ensemble distributed with the probability-coefficient

${\displaystyle e^{\frac {\psi _{A}+\psi _{B}-\epsilon _{A}-\epsilon _{B}}{\Theta }},}$

(95)

an ensemble which might be defined as formed by combining each system of the first ensemble with each of the second. But since ${\displaystyle \epsilon _{A}+\epsilon _{B}}$ is the energy of the whole system, and ${\displaystyle \psi _{A}}$ and ${\displaystyle \psi _{B}}$ are constants, the probability-coefficient is of the general form which we are considering, and the ensemble to which it relates is in statistical equilibrium and is canonically distributed. This result, however, so far as statistical equilibrium is concerned, is rather nugatory, since conceiving of separate systems as forming a single system does not create any interaction between them, and if the systems combined belong to ensembles in statistical equilibrium, to say that the ensemble formed by such combinations as we have supposed is in statistical equilibrium, is only to repeat the data in different