Page:Elementary Principles in Statistical Mechanics (1902).djvu/173

Rh $$t''$$. In the part of $$DV''$$ occupied by systems which at the time $$t'$$ were in $$DV'$$ the value of $$\eta$$ will be the same as its value in $$DV'$$ at the time $$t'$$, which we shall call $$\eta'$$. In the parts of $$DV''$$ occupied by systems which at $$t'$$ were in elements very near to $$DV'$$ we may suppose the value of $$\eta$$ to vary little from $$\eta'$$. We cannot assume this in regard to parts of $$DV''$$ occupied by systems which at $$t'$$ were in elements remote from $$DV'$$. We want, therefore, some idea of the nature of the extension-in-phase occupied at $$t'$$ by the systems which at $$t$$ will occupy $$DV$$. Analytically, the problem is identical with finding the extension occupied at $$t''$$ by the systems which at $$t'$$ occupied $$DV'$$. Now the systems in $$DV$$ which lie on the same path as the system first considered, evidently arrived at $$DV$$ at nearly the same time, and must have left $$DV'$$ at nearly the same time, and therefore at $$t'$$ were in or near $$DV'$$. We may therefore take $$\eta'$$ as the value for these systems. The same essentially is true of systems in $$DV''$$ which lie on paths very close to the path already considered. But with respect to paths passing through $$DV'$$ and $$DV$$, but not so close to the first path, we cannot assume that the time required to pass from $$DV'$$ to $$DV$$ is nearly the same as for the first path. The difference of the times required may be small in comparison with $$t''-t'$$, but as this interval can be as large as we choose, the difference of the times required in the different paths has no limit to its possible value. Now if the case were one of statistical equilibrium, the value of $$\eta$$ would be constant in any path, and if all the paths which pass through $$DV$$ also pass through or near $$DV'$$, the value of $$\eta$$ throughout $$DV$$ will vary little from $$\eta'$$. But when the case is not one of statistical equilibrium, we cannot draw any such conclusion. The only conclusion which we can draw with respect to the phase at $$t'$$ of the systems which at $$t$$ are in $$DV$$ is that they are nearly on the same path.

Now if we should make a new estimate of indices of probability of phase at the time $$t''$$, using for this purpose the elements $$DV$$,—that is, if we should divide the number of