## FANDOM

138 Pages

In the non relativistic picture of diffusion, a particle moves a distance comparable to the mean free path $l$ it is scattered to a random direction. If the velocity of the particle is $v$, then the diffusion coefficient is

$D \approx v l$

so increasing the velocity increases the diffusion coefficient. One could naively think that in the relativistic limit the largest possible diffusion coefficient is $D < c l$, and increasing the energy of the particle further will not increase the diffusion coefficient because the velocity cannot exceed the speed of light $c$. Moreover, the mean velocity would never be relativistic because even if the particle does move with velocity $c$, after $N$ collisions the average velocity would be $c/\sqrt{N}$, which is barely relativistic after just two collisions.

These difficulties can be reconciled by relativistic beaming. This effect limits the scattering angle to $\gamma^{-2}$, where $\gamma$ is the Lorentz factor. Thus, it takes $\gamma^{4}$ collision to scatter a particle by an angle of order unity. The relativistic diffusion coefficient is therefore

$D \approx c l \gamma^4$