Overdamped vs underdamped Langevin

[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?

 

[DrClaude]

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?

 

[LagrangeEuler]

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

 

[DrClaude]

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

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

 

[LagrangeEuler]

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

 

[DrClaude]

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.

 

[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?