Overdamped vs underdamped Langevin

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Discussion Overview

The discussion revolves around the formulation of overdamped and underdamped Langevin equations, focusing on the mathematical representation and the physical implications of these equations in a particle system context. Participants explore the relationships between the two regimes and seek references for further understanding.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the overdamped Langevin equation and asks how to write the underdamped version, suggesting a specific form for it.
  • Another participant notes that the damping parameter ##\gamma## appears to have been absorbed into the terms and discusses the implications of redefining length scales.
  • Clarification is provided regarding the index ##i##, which labels particles and indicates their nearest neighbors.
  • A request for more information about the physical system is made, indicating a need for context to provide further assistance.
  • A link to a paper is shared, which contains an equation with a damping parameter ##\alpha##, but it is noted that the paper does not explicitly consider an overdamped regime.
  • One participant expresses interest in the relationship between overdamped and underdamped regimes and seeks references for understanding how to transition between them.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the formulation of the equations or the context of the physical system being discussed. Multiple viewpoints and uncertainties remain regarding the definitions and implications of the damping regimes.

Contextual Notes

There is a lack of clarity regarding the specific physical system under consideration, and the discussion includes unresolved questions about the derivation and application of the equations presented.

LagrangeEuler
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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?
 
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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?
 
It labels particles. For example particle ##i## has neirest neighbours ##i-1## and ##i+1##.
 
LagrangeEuler said:
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.
 
http://allariz.uc3m.es/~anxosanchez/ep/prb_50_9652_94.pdf
 
Last edited by a moderator:
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.
 
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?
 

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