Equilibrium of Isolated System

Consider a thermally isolated system, $A$ . According to Section 5.6, any spontaneously occurring process is such that the system's entropy tends to increase in time. In statistical terms, this means that the system evolves toward a situation of greater intrinsic probability. Thus, in any spontaneous process, the change in entropy satisfies

$${\mit\Delta} S \geq 0.$$ (9.1)

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 $S$ 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

$$S = {\rm maximum}.$$

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

$$0=Q = W+{\mit\Delta}\overline{E},$$ (9.2)

or

$$W =\left(-{\mit\Delta}\overline{E}\right).$$ (9.3)

If the external parameters of the system (e.g., its volume) are kept fixed, so that no work is done (i.e., $W=0$ ), then

$$\overline{E} = {\rm constant}$$ (9.4)

as $S$ 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 $y$ (or by several such parameters), but that its total energy is constant. Let ${\mit\Omega}(y)$ denote the number of microstates accessible to the system when this parameter lies between $y$ and $y+\delta y$ ($\delta y$ being some fixed small interval). The corresponding entropy of the system is $S=k \ln {\mit\Omega}$ . (See Section 5.6.) If the parameter $y$ is free to vary then the principle of equal a priori probabilities asserts that, in an equilibrium state, the probability, $P(y)$ , of finding the system with the parameter between $y$ and $y+\delta y$ is given by

$$P(y)\propto {\mit\Omega} (y)= \exp\left[\frac{S(y)}{k}\right].$$ (9.5)

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