Fokker-Planck description of classical systems with application to critical dynamics

Classical systems with mixed canonical and dissipative dynamics are studied in terms of generalized Langevin equations for fluctuating variables. This class of systems embraces all models studied in recent papers on critical dynamics. The probability distributions satisfying the associated Fokker-Planck equation are used to build up a Green's function formalism which together with the diagram technique established earlier is applied to derive recursion relations of critical dynamics. A generalized fluctuation-dissipation theorem is proven and used to demonstrate the consistency between static and dynamic recursion relations. Special attention is given to a propagating soft mode for which a dynamical exponent close to one is obtained.