Relativistic diffusion
Abstract
We discuss relativistic diffusion in proper time in the approach of Schay (Ph.D. thesis, Princeton University, Princeton, NJ, 1961) and Dudley [Ark. Mat. 6, 241 (1965)]. We derive (Langevin) stochastic differential equations in various coordinates. We show that in some coordinates the stochastic differential equations become linear. We obtain momentum probability distribution in an explicit form. We discuss a relativistic particle diffusing in an external electromagnetic field. We solve the Langevin equations in the case of parallel electric and magnetic fields. We derive a kinetic equation for the evolution of the probability distribution. We discuss drag terms leading to an equilibrium distribution. The relativistic analog of the OrnsteinUhlenbeck process is not unique. We show that if the drag comes from a diffusion approximation to the master equation then its form is strongly restricted. The drag leading to the Tsallis equilibrium distribution satisfies this restriction whereas the one of the Jüttner distribution does not. We show that any function of the relativistic energy can be the equilibrium distribution for a particle in a static electric field. A preliminary study of the time evolution with friction is presented. It is shown that the problem is equivalent to quantum mechanics of a particle moving on a hyperboloid with a potential determined by the drag. A relation to diffusions appearing in heavy ion collisions is briefly discussed.
 Publication:

Physical Review E
 Pub Date:
 February 2009
 DOI:
 10.1103/PhysRevE.79.021128
 arXiv:
 arXiv:0809.1340
 Bibcode:
 2009PhRvE..79b1128H
 Keywords:

 02.50.Ey;
 05.10.Gg;
 25.75.q;
 Stochastic processes;
 Stochastic analysis methods;
 Relativistic heavyion collisions;
 High Energy Physics  Theory;
 High Energy Physics  Phenomenology;
 Mathematical Physics;
 Physics  Plasma Physics
 EPrint:
 9 pages,some numerical factors corrected