THE FULL ORDER WEAK SCHEMES OF THE LANGEVIN SIMULATIONS

New partial differential equations (PDEs) for the full order weak sche mes of the Langevin simulations are formulated. They are solved recurs ively in full order series solutions with respect to root t (the full order weak Taylor schemes). Arbitrariness involved in the solutions is analyzed and clarified in detail. Specific solutions within some orde rs are presented as examples of the weak Taylor schemes. These PDEs an d their solutions will serve for further developments of efficient hig her order Runge-Kutta-like schemes. The similar formulation is possibl e for the imaginary time Hamiltonian evolution kernels as well.