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Collision Equation
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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
In the case that they leave the cell it is considered
Integrating on one of the speeds that initiate the collision and both resulting since the other is the contribution to the local distribution function
In the case of contributions to the cell, consider
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
The equilibrium distribution can be approximated by a distribution of Maxwell Boltzmann
Where
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
where the term of the differences in the distribution functions represents the collisions.
With the term collisions that contribute
and those that reduce particles
you get the total exchange factor
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
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