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

From: Ulf Schiller
Subject: Re: [ESPResSo-users] Brownian motion with lattice Boltzmann
Date: Sat, 3 Mar 2012 12:34:30 +0100
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On 03/02/2012 11:25 PM, Stefan Kesselheim wrote:
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 

In general, three out of the four parameters fix your mass, length and time units. The fourth parameter, typically the friction, is used to fix the diffusion coefficient.

The parameters above correspond to a Schmidt number of roughly Sc ~ 0.5 which I would expect to be low enough to see diffusion in a 2d colloidal suspension.

Note that technically the particles are point particles, whose apparent size is determined by the grid resolution. This may lead to problems in determining a hydrodynamic radius (typically defined in terms of hard spheres), as Stephan has pointed out. For colloids of finite size one could also use the raspberry model by Lobaskin and Duenweg.


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


Dr. Salvador Herrera Velarde
Division de Ciencias e Ingenierias
Campus Leon
Universidad de Guanajuato


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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