Newtonian limit and trend to equilibrium for the relativistic Fokker-Planck equation. (English) Zbl 1288.82048

The Fokker-Plank equation is considered in the paper. The behavior of solutions to the relativistic Fokker-Planck equation for the case when the speed of light \(c \to \infty\) is studied. Under some additional assumptions on the initial data it is shown that its solutions converge in \(L^1\)-norm to solutions of the non-relativistic Fokker-Planck equation.
Another remarkable fact concerning the behavior of solutions to the relativistic Fokker-Planck equation is its exponential convergence as \(t \to \infty\) to the global thermodynamical equilibrium state in \(L^2\)-norm.
As a remark, the reviewer would like to suggest that the restriction \(\gamma >7\), \(w>9\) on the initial data is purely technical and apparently could be avoided.


82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35Q84 Fokker-Planck equations
35Q75 PDEs in connection with relativity and gravitational theory
83A05 Special relativity
Full Text: DOI arXiv


[1] DOI: 10.3934/krm.2011.4.401 · Zbl 1219.35312
[2] DOI: 10.1063/1.3659685 · Zbl 1272.82033
[3] Bakry D., C. R. Acad. Sci., Ser. I: Math. 299 pp 775– (1984)
[4] DOI: 10.1016/j.jfa.2007.11.002 · Zbl 1146.60058
[5] Csiszár I., Stud. Sci. Math. Hung. 2 pp 299– (1967)
[6] DOI: 10.1103/PhysRevE.72.036106
[7] DOI: 10.1016/j.physrep.2008.12.001
[8] DOI: 10.1512/iumj.1990.39.39009 · Zbl 0674.60097
[9] DOI: 10.1103/PhysRevE.79.021128
[10] DOI: 10.1103/PhysRevE.80.051110
[11] Risken H., Springer Series in Synergetics 18, in: The Fokker-Planck Equation: Methods of Solution and Applications (1996) · Zbl 0866.60071
[12] DOI: 10.1007/BF02412223 · Zbl 1075.35540
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.