Page:Elementary Principles in Statistical Mechanics (1902).djvu/205

Rh ensemble will be increased, but not without limit. The anomalies of the energy of the bath, considered in comparison with its whole energy, diminish indefinitely as the quantity of the bath is increased, and become in a sense negligible, when the quantity of the bath is sufficiently increased. The ensemble of phases of the body, and of the thermometer, approach a standard form as the quantity of the bath is indefinitely increased. This limiting form is easily shown to be what we have described as the canonical distribution.

Let us write $$\epsilon$$ for the energy of the whole system consisting of the body first mentioned, the bath, and the thermometer (if any), and let us first suppose this system to be distributed canonically with the modulus $$\Theta$$. We have by (205) and since   If we write $$\Delta \epsilon$$ for the anomaly of mean square, we have  If we set  $$\Delta \Theta$$ will represent approximately the increase of $$\Theta$$ which would produce an increase in the average value of the energy equal to its anomaly of mean square. Now these equations give which shows that we may diminish $$\Delta \Theta$$ indefinitely by increasing the quantity of the bath.

Now our canonical ensemble consists of an infinity of microcanonical ensembles, which differ only in consequence of the different values of the energy which is constant in each. If we consider separately the phases of the first body which