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I am researching about granular gases and their collisions, and have come across this Boltzmann transport equation: $$\frac { \partial f ( v ) } { \partial t } = \iint d u _ { 1 } d u _ { 2 } f \left( u _ { 1 } \right) f \left( u _ { 2 } \right) \left| u _ { 1 } - u _ { 2 } \right| ^ { \lambda } \left[ \delta \left( v - p u _ { 1 } - q u _ { 2 } \right) - \delta \left( v - u _ { 2 } \right) \right]$$ I have figured out that $\left| u_1 - u_2 \right|^\lambda$ refers to the collision rate, and $\lambda$ is the "homogeneity index", but have no idea what that is.

Can anyone explain what the entire equation and each section of it means, and how it was derived?

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    $\begingroup$ Without knowing anything about the problem, one might assume the index implies a deviation from isotropic probability of scattering directions (e.g., think of hard spheres colliding). I might also assume this index relates to elasticity vs. inelasticity of the collisions (I know inelasticity and non-isotropic scattering matters for granular media like sand, but not sure if it extends to gases). It would probably be useful to check out these possibilities. $\endgroup$
    – honeste_vivere
    Commented Jan 24, 2019 at 21:06
  • $\begingroup$ It would help if you can provide the reference for this equation ... $\endgroup$
    – FraSchelle
    Commented Jan 30, 2019 at 8:13

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