Registered: Aug 2004
Posts: 5

Please tell me where I can find information about Lattice Boltzmann Model. It's desirable to see, how implement simple models on computer code (c, fortran, pascal is acceptible). If someone has link to FREE books, articles or websites give it to me.
Here some questions.
What does it mean 'local equilibrium distribution' and 'non-equilibrium distribution'.
How it connected with macroscopic fluids properties.

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Registered: Aug 2003
Posts: 712

LBMs are an evolution of lattice gas or CA methods for fluid flow, as discussed in NKS around page 378. The basic idea is to allow a continuous value at each location instead of the "element present or not" of the original CA idea. Effectively one has already done one step of the "coarse graining" or local averaging step one does later with pure lattice gas methods. The motivation is computational - you don't need as many cells to get realistic approximations as you would simulating down to the particle level, so simulations are easier to run for large time steps etc.

Then the idea is to have a local velocity and typically also a local density or pressure as a continuous value at each cell in a lattice. The velocity parameter is typically a probability distribution rather than a single value - the Boltzmann distribution describes the expected mix of velocities of particles in a fluid at a given temperature.

One then writes entirely local rules for changes in these, so that the values of the cell on the next step depend on the "incoming" and "outgoing" values for its neighbors. Care must be taken to ensure the local rules preserve material and momentum, that local gradients diffuse believably etc. Then one can run the localised rules in parallel the same way one updates a conventional cellular automaton. At each time step you get a new field of velocities and concentrations.

You can get more complicated than this e.g. if you want to simulate unusual fluid properties (mixtures, suspensions, etc), but that is the basic idea. You can find out more about them at the first few of the following links, and more detail at the later ones.

As for "local equilibrium distribution", collisions within a fluid create the characteristic curve of a bulk of particles around or bit below the mean velocity, and a tail of faster moving ones. If one assumes that locally that is true within each cell, then you can say the mix of particle speeds is a local equilibrium. But allowing for differences in velocity at different cells in the lattice, effectively means they need not be in any overall equilibrium (yet), throughout the whole fluid rather than in a single cell.

As momentum spreads around in the simulation, the total distribution of velocities over all the cells taken together, may approximate a single large Boltzmann style distribution. The fluid can be said to have one temperature, after a while, if/when that happens. Before it is that well mixed, the dynamics can matter.

One can in principle allow arbitrary velocity distributions within each cell, provided you have some way of wrapping them into a parameter, and updating that parameter based on the values in the cells around you at the previous time step. Usually people assume a regular distribution locally, within each cell, rather than allowing arbitrary velocity mixtures within each. But that may depend on your actual problem, whether it is a reasonable assumption (or approximation), or not.

I hope this helps. Here are a bunch of web available resources to learn more.

http://library.wolfram.com/infocenter/Articles/2107/

http://math.nist.gov/mcsd/savg/parallel/lb/

http://www.sandia.gov/eesector/gs/gc/movies.htm

http://segovia.mit.edu/

http://www.science.uva.nl/research/...bm_web/lbm.html

http://arjournals.annualreviews.org....fluid.30.1.329

http://www.une.edu.au/Physics/Staff...sis/node69.html

http://www.ph.ed.ac.uk/~jmb/thesis/tot.html

http://ladd.che.ufl.edu/publications/publications.htm

http://www.latticeboltzmann.com/

http://www.physics.ndsu.nodak.edu/wagner/LB.html

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Registered: Aug 2004
Posts: 5

I have found some terms in the literature. I do not know what does it mean. Here they are:
Galilean invariance. Why it so important to obey it in LB fluids. What it menas "lack of Galilean invariance". How mathimatically it can be expressed.
What does "moments" mean. Where I can find some information "for dummies" about it. What does it mean: zeros order moments, first order moments and so forth.
What does this equation mean, how it read, what derivatives are taken. I do not understand derivative and supperscript (in brackets) notation here.

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Registered: Aug 2004
Posts: 5

Here some equqtions:

Attachment: eq1.bmp
This has been downloaded 1451 time(s).

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Registered: Aug 2003
Posts: 712

Galilean invariance means the laws of physics must be the same in every constantly moving frame. That is, it doesn't matter whether you consider a system at rest or one in which every component is moving in some direction at the same velocity. Every such frame also implies the laws are the same if merely translated or rotated in space.

In mathematical terms, if every position x is replaced by a new position x' = x + dx, whatever relations constraints or equations were obeyed before have to still be obeyed - and similarly if first derivatives of space (velocities) are replaced by a new variable adding a constant dv to all velocities in the system.

Moments refers to statistical aspects of a distribution, and is being used in a statistical mechanics sense, to describe the distribution of model elements about an expected value (positions and velocities e.g.). The first moment of any distribution is its mean, the second moment is its variance, third is called skew and fourth kurtosis. They are numbered that way because the formula for all of them is the same except for the exponent - what power the difference between each value and its overall expectation is raised to.

Mean is the average value (each value to first power aka itself divided by number of elements), variance is calculated by subtracting the mean from each element and then squaring (2nd power, 2nd moment) before averaging, etc.

As a general modeling point, one would like to get correct not only the mean values for some quantity one is modeling, but also to get the higher moments right. If particles in a fluid show some distribution of velocities, one wants to get not only the mean of that distribution but its whole shape etc. Although normally we assume the velocity distribution will follow a Boltzman distribution, as the equilibrium distribution all sorts of different initial ensembles of (positions, velocities) will tend toward, under internal collisions etc. That assumption is what generally allows us to characterise all the velocities with one number, the temperature.

As for the equation, it looks like Euler's formula, which comes from what is called the action integral. A quantity called the action is defined in a phase space of a system - how much of its energy is kinetic and how much potential at each moment in time, thought of as two axes for the system, as a simple example. The mechanical principle of least action says the equations governing the motion will be such that the action is minimized. That involves finding the minimum (or extrema, more generally) of a functional. And Euler's theorem (and equation) tells us what that extrema will be - or more strictly, tells us a necessary condition for an extrema of the action functional.

That typically gives a second order differential equation which one can solve for the equations of motion, with the various constants of integration determined from known boundary conditions etc.

But much of that standard statistical mechanics machinery is unnecessary for a typical CA fluid model, because it can be shown (at least for typical assumptions about incompressibility etc) that it is sufficient that the neighbor interaction rules conserve particle number, momentum, energy. The basic paper showing this is Cellular Automaton Fluids: Basic Theory, 1986. Another useful treatment can be found in Ilachinski's Cellular Automaton, chapter 9 "CA models of fluid dynamics", pp 463-506.

Which among other things gives a derivation of the fluid system Euler equation and the well known Navier-Stokes equation just from a micro-assumption of the Boltzman equation plus the conservation laws - to first order at least. If one is thinking of non-equilibrium velocity distributions, then admittedly things can get more complex and one may want to go back to the equations to get more exact requirements for the local interaction rules to obey.

I hope this helps. Needless to say, it helps to know classical and statistical mechanics thoroughly if one is going to rejigger the standard lattice-Boltzman models to tailor them to a particular problem.

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