# What does convergence to equilibrium for the Fokker-Planck equation mean?

I am a math major who recently started to study thermodynamics seriously. I have some confusing points while studying it, so I'd appreciate it if you'd correct me and give me some answers.

(1) As far as I know, a small particle immersed in a fluid is in Brownian motion, which takes place in thermal equilibrium (if the temperature is maintained constant).

(2) The special Fokker-Plank equation describes this Brownian motion. (For example, one dimensional Fokker-Plank equation is $$\frac{\partial W}{\partial t} = \gamma \frac{\partial (vW)}{\partial v} + \gamma \frac{kT}{m}\frac{\partial^2 W}{\partial v^2}$$, where $$W(v,t)$$ is the distribution for the particle.)

However, the solution of the (general) Fokker-Planck equation $$W(v,t)$$ is not in equilibrium? If so, what sense of equilibrium is this?

(3) I've seen the phrase "convergence to equilibrium" in the literature about the Fokker-Planck equation. Here, even though $$W(v,t)$$ is not in equilibrium, it converges to equilibrium?

The stationary solution of your Fokker-Planck equation is an equilibrium distribution. Here, assuming that $$v$$ is the variable for the velocity, that mean a Maxwell-Boltzmann distribution.
So if your initial condition for the distribution $$W(v,t)$$ is not an equilibrium distribution, your system will not be at equilibrium. However the evolution of your distribution via the Fokker-Planck equation will brings it towards an equilibrium distribution, hence the convergence towards the equilibrium.