Page:Elementary Principles in Statistical Mechanics (1902).djvu/61

words. Let us therefore suppose that in forming the system $C$ we add certain forces acting between $A$ and $B$, and having the force-function $-\epsilon _{AB}$. The energy of the system $C$ is now $\epsilon _{A}+\epsilon _{B}+\epsilon _{AB}$ and an ensemble of such systems distributed with a density proportional to

$e^{\frac {-(\epsilon _{A}+\epsilon _{B}+\epsilon _{AB}}{\Theta }}$

(96)

would be in statistical equilibrium. Comparing this with the probability-coefficient of $C$ given above (95), we see that if we suppose $\epsilon _{AB}$ (or rather the variable part of this term when we consider all possible configurations of the systems $A$ and $B$) to be infinitely small, the actual distribution in phase of $C$ will differ infinitely little from one of statistical equilibrium, which is equivalent to saying that its distribution in phase will vary infinitely little even in a time indefinitely prolonged.^[1] The case would be entirely different if $A$ and $B$ belonged to ensembles having different moduli, say $\Theta _{A}$ and $\Theta _{B}$. The probability-coefficient of $C$ would then be

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

(97)

which is not approximately proportional to any expression of the form (96). Before proceeding farther in the investigation of the distribution in phase which we have called canonical, it will be interesting to see whether the properties with respect to

1. It will be observed that the above condition relating to the forces which act between the different systems is entirely analogous to that which must hold in the corresponding case in thermodynamics. The most simple test of the equality of temperature of two bodies is that they remain in equilibrium when brought into thermal contact. Direct thermal contact implies molecular forces acting between the bodies. Now the test will fail unless the energy of these forces can be neglected in comparison with the other energies of the bodies. Thus, in the case of energetic chemical action between the bodies, or when the number of particles affected by the forces acting between the bodies is not negligible in comparison with the whole number of particles (as when the bodies have the form of exceedingly thin sheets), the contact of bodies of the same temperature may produce considerable thermal disturbance, and thus fail to afford a reliable criterion of the equality of temperature.