Abstract
We derive, in the “hydrodynamic” limit (large space and time scales), an evolution equation for the particle density in physical space from the (special) relativistic Ornstein–Uhlenbeck process introduced by Debbasch, Mallick, and Rivet. This equation turns out to be identical with the classical diffusion equation, without any relativistic correction. We prove that, in the “hydrodynamic” limit, this result is indeed compatible with special relativity.
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REFERENCES
F. Debbasch, K. Mallick, and J. P. Rivet, J. Stat. Phys. 88(3/4):945-966 (1997).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products(Academic Press, 1965), p. 952, formula 8.407.1, and p. 958, formula 8.432.1.
N. G. van Kampen, Stochastic Processes in Physics and Chemistry(North-Holland, 1992).
F. Reif, Fundamentals of Statistical and Thermal Physics(McGraw-Hill, 1965).
L. Landau and E. Lifschitz, Fluid Mechanics, 2nd ed. (Pergamon Press, 1987).
W. Israel, in Relativistic Fluid Dynamics, A. Anile and Y. Choquet-Bruhat, eds. (Springer-Verlag, 1989).
I. Mueller and T. Ruggeri, Extended Thermodynamics(Springer Verlag, 1993).
G. Wilemski, J. Stat. Phys. 14(2):153-169 (1976).
U. M. Titulaer, Physica A 91:321-344 (1978).
U. M. Titulaer, Physica A 10A:234-250 (1978).
E. Nelson, Dynamical Theories of Brownian Motion(Princeton University Press, 1967).
S. Chapman and T. G. Cowling, The mathematical Theory of Non-Uniform Gases, 2nd ed. (Cambridge University Press, 1952).
C. Cattaneo, Atti Sem. Mat. Fis. Univ. Modena 3:3 (1948).
A. Bressan, Relativistic Theories of Materials, in Springer Tracts in Natural Philosophy, Vol. 29 (Springer Verlag, 1978).
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Debbasch, F., Rivet, J.P. A Diffusion Equation from the Relativistic Ornstein–Uhlenbeck Process. Journal of Statistical Physics 90, 1179–1199 (1998). https://doi.org/10.1023/A:1023275210656
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DOI: https://doi.org/10.1023/A:1023275210656