What is the name of $\mu \dot x = -\nabla W(x)$? In many biological models, the mass is negligible compared to friction and force. In this friction dominated regime, the equations of motion are therefore
$$
\mu \dot x = -\nabla W(x)
$$
where $\mu$ is the friction coefficient and $W$ is the potential energy.
Is there a canonical name for this equation?
(Gradient flow seems to be the mathematical term. But I don't know if it's the correct name in a physical context.)
 A: For what its worth, such systems behave as Aristotelian mechanics (AM), cf. e.g. my Phys.SE answer here.
A: To me $\mu$ here is not the friction coefficient, but a damping coefficient relating forces to speeds.
$$ \mu \, \underbrace{ \dot x }_\text{speed} = - \underbrace{ \nabla W(x)}_\text{force} $$
The higher the relative speed $\dot{x}$ the higher the forces. This is a typical dashpot type of equation.
A: It is the overdamped limit of the usual Newton's second law. You may call it "overdamped equation of motion", meaning that the drag is so strong that the inertia is negligible.
If you add a noise term to it, you may call it "overdamped Langevin equation".
The associated Fokker-Planck equation that governs the diffusion of Brownian particles described by the overdamped Langevin equation is called "Smoluchowski equation".
(Yes, "gradient flow" is a mathematically accurate terminology and I think that there is nothing bad in using it, especially if you want to stress the fact that the velocity of your particle follows the steepest descent trajectory).
