Probability density evolution under drift and diffusion: noise-driven escape from metastable states
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Left well pop: —
Right well pop: —
Mean ⟨x⟩: —
Variance: —
The Fokker-Planck equation ∂P/∂t = −∂(FP)/∂x + D∂²P/∂x² describes how a probability density evolves under drift force F = −dV/dx and diffusion D ∝ temperature. For the double-well V(x) = ax⁴ − 2ax² + εx, the stationary solution is P_st ∝ e−V/D (Boltzmann). Kramers' escape rate: τ⁻¹ ∝ e−ΔV/D. The asymmetry ε tilts the wells, shifting the stationary distribution.