Driven Diffusion — Fokker-Planck & Probability Currents

Nonequilibrium steady states (NESS) on a tilted periodic potential — visualizing probability density and currents

Probability density P(x)
Probability current J(x)
Potential V(x)

Controls

Fokker-Planck equation:
∂P/∂t = -∂J/∂x
J = -D·∂P/∂x - (V'(x) - F)·P/γ

At NESS: ∂P/∂t = 0, J = const ≠ 0.
Entropy production rate: σ = ∮J·(−V'+F)/DT dx ≥ 0