Time Reversal Asymmetry of Fluctuation in Detailed Balance

Abstract

A fluctuation in equilibrium is naturally symmetric with respect to time reverse. A nonequilibrium steady state is heretofore ex­ plained in terms of the cyclic balance character­ ized by the invalidity of detailed balance. 1) However, the nonequilibrium steady state does not necessarily mean the invalidity of detailed balance. In the present paper, we show simple examples of nonequilibrium steady state such that the time reversal invariance does hold with respect to two different times (detailed balance) but not with respect to more than two different times. To this end, we describe an operater method which enables us to obtain expectation values for stationary random processes. We consider random variable x described by a general random process which need not be Markoffian. The realization value of x at time n is denoted by x (n). Each x (n) is assumed to take one of r values: