Fokker-Planck Equation
∂P/∂t = -∂[A(x)·P]/∂x + ½·D·∂²P/∂x² — probability flows like a fluid
Probability Density P(x,t) + Trajectories
Potential Energy V(x)
Probability Current J(x,t)
Entropy H(t) = -∫P·ln(P)dx
Fokker-Planck equation: ∂P/∂t = -∂[A(x)P]/∂x + (D/2)∂²P/∂x² describes the evolution of the probability density P(x,t) for a particle undergoing Brownian motion in a force field A(x) = -dV/dx.
Ornstein-Uhlenbeck (harmonic): V = ½kx² gives A = -kx. The stationary solution is Gaussian: P_∞ = N(0, D/k). Mean reverts to zero with rate k.
Double well: V = x⁴/4 - x²/2. Two stable fixed points x=±1. Kramers escape rate: k = (ω₀·ω_b/2π)·exp(-ΔV/D) — thermal noise drives rare transitions between wells.
Tilted washboard: V = cos(x) - F·x (potential plus linear tilt). Relevant to Josephson junctions and molecular motors. At large tilt F, the particle runs freely; near F=1 (depinning), dynamics are critical.