It follows that if a stable equilibrium state has been attained [i.e., one in which no further spontaneous processes (other than random fluctuations) can take place] then this state is such that is maximized. In other words, it is the most probable state of the system, subject to the given constraints. Thus, we can make the following statement:

For a thermally isolated system, the stable equilibrium state is such that

Now, in a thermally isolated system, the first law of thermodynamics implies that

(9.2) |

or

If the external parameters of the system (e.g., its volume) are kept fixed, so that no work is done (i.e., ), then

(9.4) |

as evolves toward its maximum value.

We can phrase the previous argument in more explicit statistical terms. Suppose that an isolated system is described by a parameter (or by several such parameters), but that its total energy is constant. Let denote the number of microstates accessible to the system when this parameter lies between and ( being some fixed small interval). The corresponding entropy of the system is . (See Section 5.6.) If the parameter is free to vary then the principle of equal a priori probabilities asserts that, in an equilibrium state, the probability, , of finding the system with the parameter between and is given by

(See Section 3.3.) The previous expression shows explicitly that the most probable state is one in which attains a maximum value, and also allows us to determine the relative probability of fluctuations about this state.