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Re: [ESPResSo-users] Brownian motion with lattice Boltzmann

From: Stefan Kesselheim
Subject: Re: [ESPResSo-users] Brownian motion with lattice Boltzmann
Date: Fri, 2 Mar 2012 23:25:48 +0100

Dear Salvador, dear Espresso-Users,

Am 02.03.2012 um 17:36 schrieb Salvador H-V:

> Dear All,
> I have been reading the Thesis of Dominic Røhm where he discuss the 
> implementation of Lattice Boltzmann with GPU's in ESPResSo.
> In chapter 8.2, he presents a test case about brownian motion and diffusion 
> of a single particle. 
> Some of the parameters used in the simulation were:
> density           1.0
> viscosity         0.1
> friction            5.0
> temperature    1.0
> I would like to ask for an educated guess of were this parameters come from? 
> Somebody could give me an idea of why these parameters are good enough to 
> reproduce brownian motion and if they could be used in a whole 2D colloidal 
> suspension?

As far as I know, this parameters were chosen not for a particular purpose 
beyond LB works stable with this set (missing is the more important LB 
timestep, most likely 0.01, that actually determines the speed of sound and the 
Boltzmann Number (cf. Duenweg/Ladd Review article).

Brownian Motion can, in principle, be reproduced by any parameter sets, under 
which the algorithm is stable, because the term "Brownian Motion" has no 
meaning beyond random motion with a well defined temperature. The question is 
just: What will be the Diffusion constant you get. 

Hidden in the Diffusion constant is the effective hydrodynamic particle radius: 
One would define it as k_B T /6/pi/eta/D : The radius of a Stokes sphere with 
the same Diffusion coefficient in the same solvent (eta is visc). If your 
parameter set creates high Diffusion coefficients at high viscosity, your 
particles become small (from a hydrodynamic point of view). This can reduce the 
hydrodynamic interactions (\propto r_h) until the disappear fully. Early works 
by Duenweg and Ahlrichs suggested that good parameter sets have hydrodynamic 
radii ~ 0.5 lattice constant. 

A careful analysis of this is still missing in the literature, and we are 
currently working on that (hopefully it will eventually come out). My 
suggestion for a good parameter set would be agrid 1 density 1 viscosity 0.8 
friction 10 tau 0.01 . This gave good results for a 3D system of uncharged 
spheres and probably works in well in many cases. The hydrodynamic radius is 
~0.45 lattice constants then. To me it is not very clear, why it actually is 
good and "superior" to other parameter sets, but it might however be a good 
starting point. 

Sorry for not being more clear in the answers, but it is still research in 
Cheers and good luck
PS: If you have any trouble with the LB implementation, please let us (mostly 
Dominic and me) know. 

> Thanks a lot in advance,
> Herrera-Velarde S 
> -- 
> =o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o
> Dr. Salvador Herrera Velarde
> Division de Ciencias e Ingenierias
> Campus Leon
> Universidad de Guanajuato
> =o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o
> Este correo ha sido editado para evitar el uso de acentos y guardar 
> compatibilidad entre 
> diferentes distribuciones

Stefan Kesselheim
Institute for Computational Physics
University of Stuttgart

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