Functional Fokker–Planck Equations

Abstract

Abstract This chapter deals with deriving functional Fokker–Planck equations (FFPEs) that govern the behaviour of phase space distribution functionals (normalised and unnormalised) for boson and fermion systems due to dynamical or thermal evolution. The approach used is to derive correspondence rules, which map changes in the density operator when multiplied from left or right by field annihilation or creation operators (as in evolution equations) onto corresponding changes in distribution functionals, these being functional differentiations and/or multiplications by field functions (left or right for fermions). Derivations are carried out by applying mode expansions to previously established results for distribution functions. Specific forms are obtained for FFPEs relating drift vector and diffusion matrix elements in the FFPE to corresponding quantities in equivalent Fokker–Planck equations and to sums over mode functions. Symmetry (bosons) and antisymmetry (fermions) properties of the diffusion matrix are established. Results are generalised to treat cases with several sets of field operators.