Page:Scientific Papers of Josiah Willard Gibbs.djvu/92

two different kinds of action upon external systems, the two functions which express the twofold capabilities of the system afford an almost equally simple criterion of equilibrium.

Criteria of Equilibrium and Stability.

The criterion of equilibrium for a material system which is isolated from all external influences may be expressed in either of the following entirely equivalent forms:—

I. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations of the state of the system which do not alter its energy, the variation of its entropy shall either vanish or be negative. If $\epsilon$ denote the energy, and $\eta$ the entropy of the system, and we use a subscript letter after a variation to indicate a quantity of which the value is not to be varied, the condition of equilibrium may be written

$(\delta \eta )_{\epsilon }\leqq 0.$

(1)

II. For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system which do not alter its entropy, the variation of its energy shall either vanish or be positive. This condition may be written

$(\delta \epsilon )_{\eta }\geqq 0.$

(2)

That these two theorems are equivalent will appear from the consideration that it is always possible to increase both the energy and the entropy of the system, or to decrease both together, viz., by imparting heat to any part of the system or by taking it away. For, if condition (1) is not satisfied, there must be some variation in the state of the system for which

$\delta \eta >0$ and $\delta \epsilon =0$;

therefore, by diminishing both the energy and the entropy of the system in its varied state, we shall obtain a state for which (considered as a variation from the original state)

$\delta \eta =0$ and $\delta \epsilon <0$;

therefore condition (2) is not satisfied. Conversely, if condition (2) is not satisfied, there must be a variation in the state of the system for which

$\delta \epsilon <0$ and $\delta \eta =0$;

hence there must also be one for which

$\delta \epsilon =0$ and $\delta \eta >0$;

therefore condition (1) is not satisfied.

The equations which express the condition of equilibrium, as also its statement in words, are to be interpreted in accordance with the