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## Re: equilibration time

 From: Peter Košovan Subject: Re: equilibration time Date: Tue, 28 Jun 2022 17:36:22 +0200

This is a well-known issue in polymer solutions. Most likely, you have one of the two problems or a combination of them:
1. Your chains are rather long, so that their relaxation times are longer than your simulation time
2. Your simulation time is longer than the relaxation time but you do not have sufficient statistics to accurately compute MSD(t) at long times, so that it is not perfectly linear.

Ad 1. Depending on the density of your system and chain length N, your Relaxation times may be given by the Zimm time O(N**1.8), Rouse time O(N**2) or Reptation time O(N**3). In any case, these relaxation times very steeply increase with chain length. On intermediate time scales, MSD(t) of monomers is subdiffusive. For more details, check Rubinstein's Polymer Physics textbook, chapters 8 and 9.

To ensure that your system is equilibrated, your simulation time must be many times longer than the relaxation time. For example, it is relatively easy to equilibrate N=200, it is hard to equilibrate N=250 and very hard to equilibrate N=300. The relaxation can be even slower if you have some attractive interactions. There is no easy way to circumvent it, just to simulate for a longer time or to use shorter chains or weaker attraction.

Ad 2. To check your statistics, you can compute MSD in (x, y z) directions separately. If your system is isotropic, then these directions are equivalent and yield equivalent results within the statistical uncertainty. Thus, the difference between MSD(x,y,z) gives you a clue on this uncertainty. Because MSD is a time-correlation function, it has the nasty property that it may look smooth although the statistical accuracy is low. As a rule of thumb, you get reasonable statistics on MSD only for time scales which are about 100 times shorter than the total simulation time.

Finally, it is highly recommended to plot MSD(t) on a double-logarithmic scale because it typically spans many orders of magnitude. Then, you can put lines with various slopes as a guide to the eye to check if your MSD is linear or follows some other scaling. To see what I mean, check for example Fig.5 in  http://link.aps.org/supplemental/10.1103/PhysRevLett.111.088301
If you plot it on a linear scale, then you see only the long times which have the worst statistics.

I hope this helps.

With regards,

Peter

On Tue, Jun 28, 2022 at 4:14 PM Ahmad Reza Motezakker <armot@kth.se> wrote:

Dear EspressoMD users,

I hope you are fine.

I have a crowded system of polymers coupled with LBM fluid. I have a concern related to the proper time for system to reach equilibrium and then extracting correct data.

for lower concentrations, MSD graph was almost linear but with increasing concentration it seems that the system does not converge and msd behaviour is not normal. Could you please help me with this issue and share with me your experience?

really appreciate that,

--
Dr. Peter Košovan

Department of Physical and Macromolecular Chemistry
Faculty of Science, Charles University in Prague, Czech Republic

Katedra fyzikální a makromolekulární chemie
Přírodovědecká fakulta Univerzity Karlovy v Praze

www.natur.cuni.cz/chemistry/fyzchem/
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