p 79

Time-correlation function

Let  q ( t ) be a (statistically steady) physical quantity at time  t.  For simplicity let us assume that its ensemble average is zero.  C ( t ) ¥equiv 〈 q ( t )  q (0)〉, where 〈 〉 is taken over the initial conditions with respect to the probability measure ¥mu, is called the time correlation function of  q . If the system is stationary, |C(t)|<= C(0) as can be seen from 〈(q(t)-q(0))^2〉 >= 0 (note that 〈q(t)^2〉 = 〈q(0)^2〉 due to stationarity). Therefore, it tends to decay as time passes. Time-correlation functions appear as vital quantities when we wish to understand transport phenomena statistical-mechanically. (The author does not know any nonequilibrium statistical mechanics textbook that he can recommend.) 

Estimation of Poincare’s recursion time