# Fokker-Planck equation for overdamped motion: how to define the average velocity

Consider the Langevin equation in the overdamped regime,

$$0 = -\gamma \dot{\mathbf{x}} -\nabla U(\mathbf{x}) +\boldsymbol{\eta}(t) \,$$

where $$\boldsymbol{\eta}$$ is the usual white-noise term, $$U$$ a potential for the force and $$\gamma$$ the damping coefficient (or a "damping matrix"). Note: thanks to the good reference provided in the accepted answer, I found how to derive the associated Fokker-Planck equation for this system.

Assuming that we have our Fokker-Planck equation for the particle distribution $$P(\mathbf{x},t)$$, I imagine (but I am not sure) that the average velocity of particles is given by

$$\langle \dot{\mathbf{x}}(t) \rangle = -\int d^Nx \, P(\mathbf{x},t) \gamma^{-1} \nabla U(\mathbf{x}).$$

Note that I am talking about the average velocity: the velocity of a single particle is not well defined for the overdamped case (the Brownian motion is non-differentiable).

Now here is my doubt: at $$t=0$$ we could choose a certain $$P(\mathbf{x},0)$$, find $$P(\mathbf{x},t)$$ with the Fokker Planck and calculate $$\langle \dot{\mathbf{x}}(t) \rangle$$ as above. Alternatively, we could sample $$M$$ different initial conditions $$\mathbf{x}_i(0)$$ from $$P(\mathbf{x},0)$$, evolve each $$\mathbf{x}_i(t)$$ for $$i=1...M$$ with the Langevin equation and obtain

$$\langle \dot{\mathbf{x}}(t) \rangle \approx M^{-1} \sum_i \dot{\mathbf{x}}_i(t) .$$

If this is correct, which method in general is more convenient from the numerical point of view? I see a big difference: simulating a single PDE (the Fokker-Planck) and performing an integral VS simulating an high number $$M$$ of ODEs (but performing a simple sum).

• Thank you @Vadim ! Do you also have a reference to a method that is well suited for the escape problem? Moreover, just because I am not expert and I am not sure about what I wrote: do you think that the N-dimensional integral I wrote is correct? I didn't derive it, it is just an intuition (even though it should be like that if I understood correctly the meaning of $P$). – Quillo Jun 3 '20 at 15:49