A generalized theory of multiplicative stochastic processes using cumulant techniques

Abstract

The rules for the construction of the nth order cumulant for time−dependent, stochastic, matrices or operators which do not commute with themselves at unequal times are derived. The results are identical with van Kampen’s rules. In the Gaussian case, Kubo’s concept of a generalized Gaussian process is criticized. Under certain conditions Kubo’s idea becomes asymptotically valid, while the same conditions justify use of the author’s earlier delta function theory. A generalized density matrix equation is presented and its behavior during the approach to equilibrium is discussed. A finite correlation time, τc, does not necessarily invalidate a monotonic approach to equilibrium.