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Calculation of collisions
Equation
In the case of collisions, two particles with velocity
$\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_1',\vec{v}_2')d\vec{v}_1'd\vec{v}_2')$
\\n\\nAs the probability that the particles entering the collision are
$f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)$
and the displacement occurs as a function of the relative velocity
| $f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v}_22)d\vec{v}_12d\vec{v}_22$ |
ID:(9078, 0)
Collisions leaving the cell
Equation
In the case that they leave the cell it is considered
| $f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v}_22)d\vec{v}_12d\vec{v}_22$ |
Integrating on one of the speeds that initiate the collision and both resulting since the other is the contribution to the local distribution function
| $\displaystyle\frac{1}{\tau}f_{out}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_12d\vec{v}_22f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v},t)|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}_12,\vec{v}_22)$ |
ID:(9080, 0)
Collisions that contribute
Equation
In the case of contributions to the cell, consider
| $f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v}_22)d\vec{v}_12d\vec{v}_22$ |
Integrating on the speeds that initiate the collision and one of the resulting ones since the other is the contribution to the local distribution function
| $\displaystyle\frac{1}{\tau}f_{in}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_2d\vec{v}_12f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v})$ |
ID:(9079, 0)
Distribution in Balance (Gas of Particles)
Equation
The equilibrium distribution can be approximated by a distribution of Maxwell Boltzmann
| $f_i^{eq}=\displaystyle\frac{m}{2\pi kT}e^{-m|c\vec{e}_i-\vec{u}|^2/2kT}$ |
Where
ID:(8490, 0)
LBM equation in the relaxation approximation
Equation
In the relaxation approximation, it is assumed that the distribution
$\displaystyle\frac{df_i}{dt}=-\displaystyle\frac{f_i-f_i^{eq}}{\tau}$
which has in the discrete approximation the equation
| $f_i(\vec{x}+c\vec{e_i}\delta t,t+\delta t)=f_i(\vec{x},t)+\displaystyle\frac{1}{\tau}(f_i^{eq}(\vec{x},t)-f_i(\vec{x},t))\delta t$ |
where the term of the differences in the distribution functions represents the collisions.
ID:(8489, 0)
Total collisions
Equation
With the term collisions that contribute
| $f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v}_22)d\vec{v}_12d\vec{v}_22$ |
and those that reduce particles
| $\displaystyle\frac{1}{\tau}f_{in}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_2d\vec{v}_12f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v})$ |
you get the total exchange factor
| $\displaystyle\frac{1}{\tau}(f_{in}-f_{out})=\displaystyle\int d\vec{v}_1d\vec{v}2d\vec{v}_12(f(\vec{x},\vec{v}2,t)f(\vec{x},\vec{v}_12,t)-f(\vec{x},\vec{v},t)f(\vec{x},\vec{v}_1,t))|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}2,\vec{v}_12)$ |
ID:(9081, 0)
Collisions
Equation
In case the particles collide, the distribution function
$\displaystyle\frac{df}{dt}\neq 0$
Collisions cause particles of neighboring cells to undergo a collision that takes them to the cell under consideration and particles within the cell being expelled. The first leads to an increase of
| $\displaystyle\frac{df}{dt}=\displaystyle\frac{1}{\tau}(f_{in}-f_{out})$ |
ID:(9077, 0)