# Mapping from tdamp in the actual viscous fluid to tdamp in fix Langevin

Dear all,

I want to simulate a mesoscopic polymer model of DNA in the standard condition (298K, 0.01 NaCl Molar, 7 phH) based on the parameters given in this article. I follow the steps in the article and map from the SI unit to LJ unit in the following way:

• \sigma=2.5nm\sim7.0 \text{base pair}
• \epsilon\sim298*k_B
• m\sim4777\text{dalton}
• \tau\sim0.1ns

The authors then say if the Stokes-Einstein formula
D=k_BT/\zeta=k_BT/m\gamma=k_BT/3\pi\eta\sigma
is used, then one simulation time unit is \tau_D=\sigma^2/D=35ns=350\tau (diffusion time) in water with viscosity \eta=0.001 \text{Pa.s} at T=298. They then set dt=0.01\tau_D=0.35ns=3.26\tau.

However, the authorsâ€™ explanation is vague when they talk about the friction coefficient \zeta in the Langevin thermostat: They first say damping factor \zeta=m\gamma\sim0.5 with m=1 and then say \zeta=k_Bt/D. If we use the above numbers, then zeta\sim2.36\times10^{-11}kg/s, \gamma=2.97\times10^{13}s^{-1} and damping time \tau_{\gamma}=1/\gamma=3.36*10^{-4}ns\sim0.003\tau and if dt=0.01\tau_D, then it is \tau_{\gamma}\sim10^{-2}dt â€“ \tau_{\gamma} is the tdamp in Langevin thermostat.

However, I know that \tau_{\gamma} should be in range from 0.1\tau^{-1} to \tau^{-1} if we want to study a single polymer in a good solvent; moreover, we should use dt\ll\tau_{\gamma} ; for instance, dt=0.005\tau. I also know that we need to remove the overall diffusion caused by the Langevin thermostat if we want to study diffusion of polymers and for instance, test the Rouse dynamics (See Grest and Kremer 1990).

I am now confused; what is the value of tdamp (or \tau_{\gamma})?

Does one measure \tau_{\gamma}\sim0.003\tau in the sampling phase when the contribution due to \tau_{\gamma}=0.5\tau of the Langevin thermostat is removed?

I know the detailed calculation above is probably confusing itself, so it is probably better to ask my question this way:

If I want to be able to compare the diffusion coefficient I measure in an MD simulation with an experimental one; how I have to set the tdamp in the Langevin thermostat?

Kind regards,

Amir

This question is much more about simulation theory than about how to use LAMMPS specifically, so you may not get the answers you are looking for on this forum.

Having said that, you should first make sure you have a firm theoretical basis. From â€śbottom-upâ€ť the generalised Langevin equation comes from using the Mori-Zwanzig formalism to average out solvent dynamics into a memory friction term, and if you use the simplest possible friction kernel (a constant, to multiply with the current velocity) you end up with Langevin dynamics.

From â€śtop-downâ€ť, if you assume the equations of motion of Langevin dynamics (from Mori-Zwanzig theory, or equally well from idealized fluid dynamics as Einstein might have), you can use stochastic calculus to derive the Smoluchowski equation for evolving the position probability distribution (aka the Fokker-Planck equation or forward Kolmogorov equation). You can then see that this equation obeys Fickâ€™s law of diffusion, giving a link from theory to an observable diffusivity.

I donâ€™t think this answers your question directly â€“ but once you grasp the theory you should be able to answer your own questions.

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