Structure-Preserving Detailed-Balance Master-Equation Discretizations for Fokker--Planck Equations

We develop a variational–Markov construction of detailed-balance master-equation discretizations for Fokker–Planck equations directly from their energy–dissipation laws. Rather than discretizing the differential operator, we represent mass transfer between neighboring grid points by two directional jump rates. On each edge, the discrete energy–dissipation law determines the net flux, while detailed balance fixes the rate ratio; a local reconstruction of edge quantities from neighboring grid-point values then uniquely determines both rates. A logarithmic-mean reconstruction recovers the classical Scharfetter–Gummel/Wang–Peskin–Elston rates, while alternative reconstructions yield other reversible schemes. The resulting semi-discrete systems conserve mass, preserve nonnegativity, satisfy detailed balance, and dissipate a discrete free energy. The same edgewise construction extends to state-dependent and degenerate mobilities, nonlocal interaction energies and higher-dimensional problems. The corresponding implicit and semi-implicit fully discrete schemes are linear, mass-conservative, positivity-preserving, and energy-stable, with a time-step restriction required only for nonlocal interaction energies whose kernel is not negative semidefinite. Numerical experiments confirm the expected spatial convergence rates of all detailed-balance schemes and verify their equilibrium accuracy, positivity preservation, and free-energy decay with discontinuous potentials, saturation, nonlocal interactions, and two-dimensional problems.