Overdamped vs underdamped Langevin

1. Jan 16, 2014

LagrangeEuler

If overdamped equation looks like
$\dot{x}_i=x_{i+1}+x_{i-1}-2x_i-V'(x_i)+F(t)$
How to write down the underdamped Langevin equation
$\ddot{x}_i+\gamma\dot{x}_i=\gamma x_{i+1}+\gamma x_{i-1}-2 \gamma x_i-\gamma V'(x_i)+\gamma F(t)$
Am I right?

2. Jan 16, 2014

Staff: Mentor

It does look like the $\gamma$ has been absorbed into the terms. But I guess it could have been used to redefine the length scale, so I'm sure that multiplying everything by $\gamma$ is the right thing to do. You have to go back to the derivation of the equation.

And what does the index $i$ stand for?

3. Jan 16, 2014

LagrangeEuler

It labels particles. For example particle $i$ has neirest neighbours $i-1$ and $i+1$.

4. Jan 16, 2014

Staff: Mentor

Then I really need more information on the physical system you are considering before I can be of any help.

5. Jan 16, 2014

LagrangeEuler

http://allariz.uc3m.es/~anxosanchez/ep/prb_50_9652_94.pdf [Broken]

Last edited by a moderator: May 6, 2017
6. Jan 16, 2014

Staff: Mentor

Looking at equation (1) in that paper, they have the damping parameter $\alpha$. I skimmed through the article, and couldn't find any indication that they are considering an overdamped regime, or indeed the first equation you gave in the OP.

7. Jan 16, 2014

LagrangeEuler

I known. But I'm interesting in that relation. Do you know some reference where I can find it? How could you always get from overdamped, underdamped and vice versa?