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This demo notebook shows how to modify the LB collision operator to account for microscopic fluctuations. This technique is also called thermalized or randomized LBM. In these methods a random fluctuation is added to the equilibrium moments. In this simple example we implement a thermalized model by writing our own simple linear congruent random number generator, the draws indepedent random numbers on each cell. A seed is stored for each cell since all cells are processed in parallel.
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$\displaystyle \left[ {{seed}_{(0,0)}^{0}} \leftarrow 1664525 {{seed}_{(0,0)}^{0}} + 1013904223, \ {{seed}_{(0,0)}^{1}} \leftarrow 1664525 {{seed}_{(0,0)}^{1}} + 1013904223, \ {{seed}_{(0,0)}^{2}} \leftarrow 1664525 {{seed}_{(0,0)}^{2}} + 1013904223, \ rand_{0} \leftarrow \frac{{{seed}_{(0,0)}^{0}}}{4294967295}, \ rand_{1} \leftarrow \frac{{{seed}_{(0,0)}^{1}}}{4294967295}, \ rand_{2} \leftarrow \frac{{{seed}_{(0,0)}^{2}}}{4294967295}, \ {{dbg}_{(0,0)}} \leftarrow \frac{{{seed}_{(0,0)}^{0}}}{4294967295}\right]$
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