Generalized Fokker–Planck equation for quantum systems

A dynamical equation (of Fokker–Planck type) is obtained for the quantum distribution function of an arbitrary set of coarse-grain variables used to describe the evolution of a strongly fluctuating nonequilibrium system. In the general case, this equation is an integrodifferential equation, and its “nonlocality” is due not only to the contribution of small-scale fluctuations but also to the noncommutativity of the basis operators corresponding to the coarse-grain variables. The conditions under which a transition to a local approximation is possible are considered. If the basis operators form a complete set, the obtained generalized Fokker–Planck equation goes over into a “continuity equation” for the Weyl distribution function, and in this case it is equivalent to an exact Liouville equation.