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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