A quantum stochastic Fokker–Planck theory for adiabatic processes in condensed phases

A maximum entropy approach to time‐dependent phenomena is employed to construct a quantum stochastic theory for adiabatic processes in condensed phases. This theory is illustrated by considering the relative motion of a two‐particle system interacting with a heat bath. We present a fully quantum mechanical phase space master equation, which is new. In the limit where the length of thermal spatial fluctuations along the relative coordinates is much greater than half the wavelength for thermal momentum fluctuations associated with relative motion, our master equation reduces to a simpler form, and the principle of detailed balance emerges. This limit is shown to be equivalent to fulfilling a ’’thermal Heisenberg uncertainty’’ relation: $\text{Δp̄Δq̄>>ℏ/2.}$ The simpler master equation is utilized in the development of a quantum mechanical nonlinear Fokker–Planck equation. For cases in which correlations involving spatial and momentum fluxes are short ranged, the nonlinear Fokker–Planck equation reduces to the usual linear form, with ’’collision’’ and ’’streaming’’ terms that contain quantum corrections to all orders in ℏ. The relationship between the classical limit of the present theory and Brownian motion theory based on the Chapman–Kolmogorov equation is established.