# Other physics applications

$\newcommand\inv{^{-1}}\newcommand\invt{^{-t}} \newcommand\bbP{\mathbb{P}} \newcommand\bbR{\mathbb{R}} \newcommand\defined{ \mathrel{\lower 5pt \hbox{{\equiv\atop\mathrm{\scriptstyle D}}}}}$ 15.1 : Lattice Boltzmann methods
15.2 : Hartree-Fock / Density Functional Theory
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# 15.1 Lattice Boltzmann methods

Top > Lattice Boltzmann methods LBM \footnote{This chapter owes much to the presentation in  [ChenDoolen:LBM] .} offer a different way of computing a discretized solution to a PDE , based on dividing a domain in cells. Its basic idea is then to wonder `if there is a particle in this cell, and given the force on this cell, where does he particle move to?'. Rather than reasoning about actual particles, the LBM considers the probability that there is a particle in a cell, and how that probability is updated.

Distribution function $f$ (or particle velocity distribution, just under equation 2?). For each cell $i$ $$f_i(\mathbf{x}+\mathbf{e}_i\Delta x,t+\Delta t) = f_i(\mathbf{x},t)+\Omega_i(f(\mathbf{x},t)).$$ where $\Omega_i$ is the collision operator giving the rate of change of the distribution function $f_i$.

Density: $\rho=\sum_i f_i$, momentum density: $\rho\mathbf{u}=\sum_if_i\mathbf{e}_i$.

Conservation of mass $\sum_i\Omega_i=0$ conservation of momentum $\sum_i\Omega_i\mathbf{e}_i=0$

Basic equation, second order in $\epsilon$: $$\frac{\partial f_i}{\partial t} + \mathbf{e}_i\cdot \nabla f_i + \epsilon\left( \frac12 \mathbf{e}_e\mathbf{e}_i\colon \nabla\nabla f_i + \mathbf{e}_i\cdot\frac{\partial f_i}{\partial t} + \frac12 \frac{\partial^2f_i}{\partial t^2} \right) = \frac{\Omega_i}{\epsilon}$$

# 15.2 Hartree-Fock / Density Functional Theory

Top > Hartree-Fock / Density Functional Theory

Replace particle-particle interactions with computing a field: repeat until field is self-consistent.

Field is matrix-valued

Repeated diagonalization of the Kohn-sham matrix.

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