Commit a439d0f7 authored by s.zitz's avatar s.zitz
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Some theory on equilibria

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Computes the equilibrium distribution `dists.feq` of a given macroscopic state `mom`.
# Theory
One of the reasons why the lattice Boltzmann method became so widely spread is due to its simple assumptions.
Among them is that any system simulated with lattice Boltzmann is a system close to equilibrium.
For an ideal gas one knows that the momentum distribution is given by a maxwellian distribution with temperature T,
`` f^{eq} = \\frac{\\rho}{2\\pi T}^{D/2}\\exp{-\\frac{(\\mathbf{c}-\\mathbf{u})^2}{2T}}. ``
A similar approach can be found for the shallow water system where `` f^{eq} `` is given by
`` f^{eq} = \\frac{\\rho}{\\pi g \\rho}^{D/2}\\exp{-\\frac{(\\mathbf{c}-\\mathbf{u})^2}{g\\rnho}}. ``
Above it is assumed that `` T = \\frac{1}{2}g\\rho``, with equation of state `` P(\\rho) = \\frac{1}{2}g\\rho^2 ``.
# Arguments
- `dists::distributions`: container for the three distribution functions.
- `mom::moments`: struct that contains a macroscopic state, e.g. height and velocity.
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# References
Paul Dellar and Rick Salmon did a lot of work for the shallow water lattice Boltzmann method, bellow are two of the best references
- [The lattice Boltzmann method as a basis for ocean circulation modeling](https://www.ingentaconnect.com/contentone/jmr/jmr/1999/00000057/00000003/art00005#)
- [Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations](https://journals.aps.org/pre/pdf/10.1103/PhysRevE.65.036309)
See also: [`moments`](@ref), [`velocity`](@ref), [`params`](@ref)
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