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Microdynamical Equations

  Frisch et al. [8] use a probabilistic approach which is traditional in statistical mechanics. Thus they consider the mean population

  equation962

the mean density

  equation969

the mean mass current

  equation976

and the mean velocity

  equation986

Note that the mean density and mass current are defined to be the mean quantities per site, and not per unit area as they normally are in the real world. Frisch et al. [8] have found a steady state equilibrium solution for the mean population,

  equation997

where f is the Fermi-Dirac function

equation1007

tex2html_wrap_inline13503 is the equilibrium mean population and h and tex2html_wrap_inline13129 are Lagrange multipliers which depend on the mean density tex2html_wrap_inline12075 and the mean velocity tex2html_wrap_inline12875 through equations (3.9) - (3.12). Note that although the solution is independent of the transition probabilities tex2html_wrap_inline13513 Frisch's proof requires that A satisfies semi-detailed balance: equation (3.8). The Fermi-Dirac distribution for the equilibrium state of the mean population is obtained because there is an exclusion principle applied to the lattice gas model: only one particle is allowed on any link at any time-step.

Explicit solutions for h and tex2html_wrap_inline13129 are only known for special cases, such as when tex2html_wrap_inline13521 where tex2html_wrap_inline13523 is clearly a solution. Expressions for h and tex2html_wrap_inline13129 can be found in the limit of small tex2html_wrap_inline12875 by expanding equation (3.13) about the zero velocity case to give

  equation1027

where

equation1043

and

equation1049

This restricts us to using speeds u which are small compared to tex2html_wrap_inline13535 , the speed of sound in the model. Given a solution for tex2html_wrap_inline13521 it is not possible to use a Galilean transformation to find the solution for non-zero tex2html_wrap_inline12875 because the solution found here only applies in an inertial frame in which the underlying grid is at rest. This lack of Galilean invariance is an important feature of the model and can lead to problems when the model is implemented.

It can also be shown [22] that the lattice gas model obeys an H-theorem. This states that the mean population must tend towards the equilibrium mean population tex2html_wrap_inline13503 .


next up previous contents
Next: Macrodynamical Equations Up: Equations for the Lattice Previous: Definitions

James Buick
Tue Mar 17 17:29:36 GMT 1998