Microscopic expression for the time correlation function

Given the Smoluchowski equation, time correlation fimctions can be calculated. For the sake of simplicity we use x to denote the whole set of coordinatesXi,x, , Xn appearing in the Smoluchowski equation. Let G x, x t) be the probability that the system which was in the state x at time r s 0 is in the state x at time t. Qearly such probability is obtained 

Though eqn (3.47) gives a general method for calculating the time correlation function, it is not easy to carry out this procedure since G(x,x t) is difficult to obtain. Usually, more convenient methods are available, which will be demonstrated in subsequent sections. Howeva, the initial slope of the time correlation function can be calculated directly from eqn (3.47). The time derivative of eqn (3.47) is calculated as 

The average in the final expression is for the equilibrium distribution function Peq(x). The initial decay rate defined by 

Consider a time-dependent external field h(t) (magnetic field, electric field, or velocity gradient field) applied to a system in equilibrium. In general, the field perturbs the system, and changes the average values of physical quantities from those in the equflibrium state. If the field is weak, the change in any physical quantity is a linear functional of the field, and is written as 

In many cases, the effect of the field on the system is expressed by a potential such as 

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