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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.)

Qmechanic
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Steffen Plunder
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    Physicists will want to complain that, with the usual definition of $\mu$ as a dimensionless ratio of forces, this zero-acceleration instance of Newton's Second Law has inconsistent units. I suspect that in actual use $\mu$ is here a dimensionful drag parameter, but a link might be helpful. – rob Oct 21 '20 at 16:59
  • @rob Thanks for the correction. Yes, it should be a dimensionful drag parameter. – Steffen Plunder Oct 21 '20 at 17:06
  • One example is the viscous damping coefficient here: https://en.wikipedia.org/wiki/Harmonic_oscillator#Damped_harmonic_oscillator In the limit, $m \to 0$ the equation in the question appears. – Steffen Plunder Oct 21 '20 at 17:13
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    In this context $\mu$ appears like a damping coefficient relating speed to forces. – JAlex Oct 21 '20 at 18:59

3 Answers3

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For what its worth, such systems behave as Aristotelian mechanics (AM), cf. e.g. my Phys.SE answer here.

Qmechanic
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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.

JAlex
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  • Yes, calling it friction was a mistake, you are right, it should be damping! Thanks also for the reference to dashpot type equations! – Steffen Plunder Oct 22 '20 at 09:18
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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).

Quillo
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