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

Question

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 a high number $M$ of ODEs (but performing a simple sum).

Related posts: Average velocity of overdamped particles in external field (MathSE), Understanding mean rate of change in Brownian motion, What does this observation of instantaneous velocity in Brownian particles mean?, Correlation of position and velocity in Brownian motion.

Answer 1

The Fokker-Planck equation by Risken is a standard reference that you might be looking for.

Calculating the average velocity via the probability density is the principled and mathematically exact approach, whereas the alternative approach obviously has all the disadvantages related to having a finite number of samples. However numerically solving a PDE in time and space is rather tricky (even with a solver), whereas the sampling approach is straightforward and easy to realize (although it has its own pitfalls). In the end it depends on the problem. E.g., in my experience neither approach is good for calculating the escape times over a potential barrier, due to the presence of a slow time scale.