Revaluation of the first-order upwind difference scheme to solve coarse-grained master equations

This study addresses the initial-boundary value problem of coarse-grained probability measure on the state space in which a differentiable vector field v is given and, as a consequence, the differenced continuity equation using the first-order upwind difference scheme (UDS) based on the finite volume method appears as the physical substance on the coarse-grained dynamics. At first, the UDS is theoretically shown to be equivalent to a class of coarse-grained master equations (CGME), brought by a principle that we cannot distinguish state points in the same partition with each other. The principle is based on the formulation of non-equilibrium statistical mechanics to resolve the macroscopic irreversibility. Moreover the entropy production evaluated by the UDS is also shown to be in accord with the average volume contraction rate in the steady state. This is essential for the non-equilibrium statistical dynamics and was numerically confirmed. Under the principle of coarse graining the UDS is very superior to the conventional Monte-Carlo method in computer time and storage and is very useful to solve the CGME.