Relaxation of a Single Conserved Variable

To illustrate the general applicability of the relaxation equations of Section 11.4 let us study the simple case of a single conserved variable A(q, t) which has the form given by Eq. (11.5.32). The property aj of the jth molecule is presumed to have definite time-reversal symmetry and parity. 

First we note that according to Theorem 2 of Section 11.5 

Thus Z(q) is an even function of q. This follows from considerations of inversion symmetry [cf. (Eq. (11.5.37b)]. The chief consequence of this is that 

These considerations also show [cf. Eq. (11.5.37d)] that the random-force autocorrelation function and thereby Aq) are also even functions of q that is, 

Because 7 (q) is even in q a Taylor expansion of Aq) around 7 = 0 should then have the form 

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