Editing Fokker–Planck equation (section)

===Asymptotic MFPT in Gradient Systems===

In systems with small noise and a drift given by the gradient of a potential, <math>b(x) = -\nabla\phi(x),</math> the stochastic process

:<math> dX_t = -\nabla \phi(X_t)\,dt + \sqrt{2\varepsilon}\,dW_t </math>

models the overdamped Langevin dynamics of a particle in a potential landscape <math>\phi(x)</math>. The associated mean first passage time <math>u(x),</math> which satisfies the backward Kolmogorov equation:

:<math> \varepsilon \Delta u(x) - \nabla \phi(x) \cdot \nabla u(x) = -1, </math>

subject to <math>u=0</math> on the exit boundary <math>\partial\Omega_a \subset \partial\Omega,</math>  has the following asymptotic solution in the limit <math>\varepsilon\to 0,</math> when <math>x</math> is near a local minimum <math>x_0</math> of <math>\phi</math> and escape occurs over a saddle point <math>x_s</math> of the potential:

:<math> \mathbb{E}[\tau] \sim \frac{2\pi}{\sqrt{|\det H(x_s)|}} \cdot \frac{e^{[\phi(x_s) - \phi(x_0)]/\varepsilon}}{\sqrt{\det H(x_0)}}, </math>

where: 

* <math>H(x_0)</math> is the Hessian matrix of <math>\phi</math> at the stable point <math>x_0</math>,
* <math>H(x_s)</math> is the Hessian at the saddle point <math>x_s,</math> with one negative eigenvalue,
* <math>\phi(x_s) - \phi(x_0)</math> is the energy barrier or quasi-potential difference the system must cross.

This formula generalizes Kramers' escape time to n-dimensional gradient systems and shows the exponential sensitivity of MFPT to potential barriers, with prefactors determined by second-order variations (local curvatures) of the potential at critical points. This result connects with large deviation theory and WKB asymptotics, where the action functional (or quasi-potential) governs the probability of rare events. It underpins modern approaches to metastability in physics, chemistry, and biology—such as chemical reaction rates, ion channel gating, or noise-induced switching in gene networks.