2.7 Time averages

For time differences that are small, that is, for time intervals that are comparable to the fluctuation times of the subsystems, p _n( t) is a rapidly varying function of time. But since the interactions, and therefore the fluctuations, are random, the fluctuations over a period of time that is large compared with the interaction times are distributed about some mean. Therefore, let us consider a time interval ( t ₂ ? t ₁) that is long compared to the interaction time but short relative to macroscopic times. That is, if the macroscopic properties of the subsystem are changing with time, then ( t ₂ ? t ₁) is short with respect to the time it takes for the macroscopic properties to change appreciably. This means that the macroscopic properties must be changing slowly relative to the times of fluctuations at the micro level.

Now take a time average of equation (2.6.3) to get

The integral is taken over the time interval ( t ₂ ? t ₁), and the time t on the left-hand side of (2.7.1) is the midpoint of the time interval ( t ₂ ? t ₁); At the macroscopic level, the average defined by equation (2.7.1) is very nearly continuous for subsystems whose macroscopic properties are stationary or changing slowly with time.

During the time interval ( t ₂ ? t ₁), the subsystem spends a time t _n in the nth...