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Non-linear relativistic diffusions
We obtain a non-linear generalization of the relativistic diffusion of
particles with spin. We discuss diffusion equations whose non-linearity is a
consequence of quantum statistics. We show that the assumptions of the
relativistic invariance and an interpretation of the solution as a probability
distribution substantially restrict the class of admissible non-linear
diffusion equations. We consider relativistic invariant as well as covariant
frame dependent diffusion equations with a drift. In the latter case we show
that there can exist stationary solutions of the diffusion equation besides the
equilibrium solution corresponding to the quantum or Tsallis distributions. We
define the relative entropy as a function of the diffusion probability and
prove that it is monotonically decreasing in time when the diffusion tends to
the equilibrium. We discuss its relation to the thermodynamic behavior of
diffusing particles.Comment: 20 pages, the version to appear in Physica
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