A consistent method for the solution to reduced FPK equation in statistical mechanics

The solution of reduced Fokker–Planck–Kolmogorov (FPK) equation resulted from the problems in statistical mechanics is formulated as an exponential function of polynomials in state variables. Special measure is taken to satisfy the FPK equation in the weak sense of integration with the assumed function. Examples are given to show the application of the method to the non-linear systems with additive random excitations and those with both additive and multiplicative random excitations. The PDF solutions obtained with the proposed method and conventional Gaussian closure method are compared with the exact solutions in special cases. Numerical results showed that the solutions obtained with the proposed method are much close to the exact solutions even for highly non-linear system and the system with both additive and multiplicative excitations. The convergence of the approximate PDF solutions obtained with the method is also shown with numerical results.