The actual mass of a particle in coarse-grained molecular dynamics

If I use Lennard-Jones unit with the fix langevin command, and set both the mass and damping parameter as 1, does it mean the actual mass of a particle in this system is $$m=\sigma^2\lambda^2/\epsilon$$? If the $$\sigma$$, $$\epsilon$$ and $$\lambda$$ (damping coefficient for the solvent) of the system are given. (Since $$t=\sqrt{m\sigma^2/\epsilon}$$ and damp $$=m/ \lambda$$, it follows $$m=\sigma^2\lambda^2/\epsilon$$, where $$\sigma$$, $$\epsilon$$ and $$\lambda$$ are distance, energy and damping coefficient in the Langevin equation respectively.)

I really want to make sure of it because many papers for coarse-grained model just set the mass and damping parameter as $$1$$, or without a mention of these and rarely have explanations.

• Welcome to Physics SE! Please format any mathematical content using MathJax. I've attempted to do this for you as a guide here, please check in case I've made any errors. – user197851 Feb 26 at 10:28

You are thinking along the right lines, but I believe that it is more usual to start from $$m$$, $$\epsilon$$, and $$\sigma$$ as defining the units of mass, energy, and length respectively. You said that Lennard-Jones units have been adopted: this implies $$\epsilon=1$$, and $$\sigma=1$$. And you also said that $$m=1$$. So this indeed seems to be your starting point. Then the unit of time follows, as you have noted: $$\tau=\sqrt{m\sigma^2/\epsilon}$$.

In your simulation, perhaps you have particles whose masses are defined to be $$1$$ in these units, and which interact via a Lennard-Jones potential characterized by unit values of $$\epsilon$$ and $$\sigma$$. It is also possible that you've defined particles whose masses, and interaction parameters, are not unity: but they are still measured in those same units.

Then you should be in good shape to discuss the other quantities defined as simulation parameters, as well as any results that come out of the simulation. You seem to be using the LAMMPS package. The documentation for that package gives you the information you need. In this case, on the fix langevin command page it says

The damp parameter is specified in time units and determines how rapidly the temperature is relaxed. For example, a value of 100.0 means to relax the temperature in a timespan of (roughly) 100 time units (tau or fmsec or psec - see the units command).

This is consistent with the equation they give for the frictional force $$\vec{F}_f = -\frac{m}{\text{damp}} \vec{v} = -\lambda \vec{v}$$ where I have inserted the definition of $$\lambda$$ that you mentioned in your question.

So, choosing $$\text{damp}=1$$ means that in SI units, where the atoms used to define the units have mass $$m$$ kilograms, and interact with a LJ potential characterized by $$\epsilon$$ Joules and $$\sigma$$ metres, the actual damping factor appearing in the real-world Langevin dynamics would be $$\sqrt{m\sigma^2/\epsilon}$$ seconds. It doesn't mean that the masses of the particles have been redefined. You have specified those masses somewhere else, in terms of your chosen unit of mass.

It is only necessary to specify three of the quantities $$m$$, $$\sigma$$, $$\epsilon$$, $$\tau$$ in real-world units, in order to define the fourth one. The papers by Cooke et al Phys Rev E, 72, 011506 (2005), which defines the model, and by Reynwar et al Nature, 447, 461 (2007), which uses it to model aggregation, are good examples. They define $$\sigma \approx 1$$ nm so as to reproduce the observed membrane thickness. They define $$\tau\approx 15$$ ns so that the lipid self diffusion coefficient roughly matches the experimental value. The value of $$\epsilon$$ is not spelt out, but one chooses a value of temperature not far from $$k_BT/\epsilon\approx1$$, and this presumably corresponds to ambient temperature for which, as they say, $$k_BT\approx 4.1\times 10^{-21}$$ J. Hence, there is an implied value for the coarse-grained particle mass $$m=\tau^2\epsilon/\sigma^2$$. Yes, in both papers (one using the Langevin thermostat, the other using the DPD thermostat) they choose the thermostat damping parameter to be $$1$$ in reduced units. But one hopes the precise choice of this parameter will not affect the physical results (such as the diffusion coefficient) too much. If it did, I would worry about the physical significance of the results. So in these examples, the particle mass $$m$$ is a consequence of the choice of $$\tau$$, but it is not the damping parameter that determines $$\tau$$: some more physical aspect of the simulation does that.

Hopefully this will be enough to answer your question. Detailed enquiries about the LAMMPS package, of course, are better directed to the package authors or user community, rather than here.

• Thanks, professor. Very detailed and helpful. In principle it should start from m, but sometimes it is not easy to determine "the mass in SI units" of one particle in a coarse-grained polymer model. And I found papers for CGMD simulations using Langevin approach are commonly just set the mass and damping parameter as 1 in LJ units. So if I really want to determine the actual mass of the particle in their systems, can I just calculated as m=σ^2λ^2/ϵ? – Jade Gobinet Feb 27 at 6:33
• Can you give references to some of these papers, so I can take a look at them? – user197851 Feb 27 at 6:47
• pubs.rsc.org/en/content/articlepdf/2014/SM/C4SM00885E (Page. 7662 lower right), nature.com/articles/nature05840.pdf (SI, using DPD model but I think the question is similar) – Jade Gobinet Feb 27 at 7:11
• I have edited my answer, concentrating on the Nature paper from the Kremer-Deserno group (and a preceding paper of theirs, which defines the model and uses the Langevin thermostat). Similar comments apply to the paper by Zhang and Zhang. So, I would agree that you can set $m=\tau^2\epsilon/\sigma^2$ (you made a small error in your version of this formula in your question) but I would say that the physical value of $\tau$ to insert in this formula comes from more physical aspects of the system (such as the measured dynamical propoerties), not from the thermostat damping parameter. – user197851 Feb 27 at 9:57
• I see. The timescale of MD study can always be a question from reviewers and it is hard to match the experimental value. To simulate the physical phenomena is more important. Anyway, thank you very much for patient answers and have a nice day! – Jade Gobinet Feb 27 at 11:49