On a relativistic Fokker-Planck equation in kinetic theory

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June 2011, 4(2): 401-426. doi: 10.3934/krm.2011.4.401

On a relativistic Fokker-Planck equation in kinetic theory

José Antonio Alcántara ^1, and Simone Calogero ^2,

1.	Departamento de Matemática Aplicada, Facultad de ciencias, Universidad de Granada, 18071 Granada, Spain
2.	Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada

Received November 2010 Revised January 2011 Published April 2011

Abstract
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A relativistic kinetic Fokker-Planck equation that has been recently proposed in the physical literature is studied. It is shown that, in contrast to other existing relativistic models, the one considered in this paper is invariant under Lorentz transformations in the absence of friction. A similar property (invariance by Galilean transformations in the absence of friction) is verified in the non-relativistic case. In the first part of the paper some fundamental mathematical properties of the relativistic Fokker-Planck equation are established. In particular, it is proved that the model is compatible with the finite propagation speed of particles in relativity. In the second part of the paper, two non-linear relativistic mean-field models are introduced. One is obtained by coupling the relativistic Fokker-Planck equation to the Maxwell equations of electrodynamics, and is therefore of interest in plasma physics. The other mean-field model couples the Fokker-Planck dynamics to a relativistic scalar theory of gravity (the Nordström theory) and is therefore of interest in gravitational physics. In both cases the existence of steady states for all possible prescribed values of the mass is established. In the gravitational case this result is better than for the corresponding non-relativistic model, the Vlasov-Poisson-Fokker-Planck system, for which existence of steady states is known only for small mass.

Keywords: Vlasov-Nordström-Fokker-Planck, Vlasov-Maxwell-Fokker-Planck, existence of steady states..

Mathematics Subject Classification: Primary: 35Q75, 74A26; Secondary: 74G3.

Citation: José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

References:

[1]	F. Andreu, V. Caselles, J. M. Mazón and S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal., 182 (2006), 269-297. doi: 10.1007/s00205-006-0428-3.
[2]	M. Bostan and T. Goudon, Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case, Kinet. Relat. Models, 1 (2008), 139-170.
[3]	F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with coulombic and Newtonian potentials, Diff. Integ. Eqs., 8 (1995), 487-514.
[4]	S. Calogero, Spherical symmetric steady states of galactic dynamics in scalar gravity, Class. Quant. Grav., 20 (2003), 1729-1741. doi: 10.1088/0264-9381/20/9/310.
[5]	S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system, Comm. Math. Phys., 266 (2006), 343-353. doi: 10.1007/s00220-006-0029-x.
[6]	S. Calogero, O. Sánchez and J. Soler, Asymptotic behavior and orbital stability of galactic dynamics in relativistic scalar gravity, Arch. Rat. Mech. Anal., 194 (2009), 743-773. doi: 10.1007/s00205-008-0173-x.
[7]	S. Calogero, J. Calvo, O. Sánchez and J. Soler, Virial inequalities for steady states in relativistic galactic dynamics, Nonlinearity, 23 (2010), 1851-1871. doi: 10.1088/0951-7715/23/8/004.
[8]	S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations on Riemannian manifolds, preprint, arXiv:1009.5086.
[9]	J. A. Carrillo, P. Laurençot and J. Rosado, Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics, J. Diff. Eqns., 247 (2009), 2209-2234. doi: 10.1016/j.jde.2009.07.018.
[10]	G. Chac\'on-Acosta and G. M. Kramer, Fokker-Planck-type equations for a simple gas and for a semirelativistic Brownian motion from a relativistic kinetic theory, Phys. Rev. E., 76 (2007), 021201. doi: 10.1103/PhysRevE.76.021201.
[11]	S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev Mod. Phys., 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1.
[12]	F. Debbasch and C. Chevalier, Relativistic stochastic processes: A review, AIP Conf. Proc., 913 (2007), 42-48. doi: 10.1063/1.2746722.
[13]	J. Dolbeault, Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: External potential and confinement (large time behavior and steady states), J. Math. Pures Appl., 78 (1999), 121-157. doi: 10.1016/S0021-7824(01)80006-4.
[14]	K. Dressler, Steady states in plasma physics-the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci., 12 (1990), 471-487. doi: 10.1002/mma.1670120603.
[15]	K. Dressler, Stationary solutions of the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci., 9 (1987), 169-176. doi: 10.1002/mma.1670090113.
[16]	J. Dunkel and P. Hänggi, Theory of the relativistic Brownian motion: The (1+3)-dimensional case, Phys. Rev. E, 72 (2005), 036106. doi: 10.1103/PhysRevE.72.036106.
[17]	J. Dunkel and P. Hänggi, Relativistic Brownian motion, Phys. Rep., 471 (2009), 1-73. doi: 10.1016/j.physrep.2008.12.001.
[18]	D. T. Frank, "Nonlinear Fokker-Planck Equations: Fundamentals and Applications," Springer Series in Synergetics 25, Springer-Verlag, New York, 2005.
[19]	R. T. Glassey, J. Schaeffer and Y. Zheng, Steady states of the Vlasov-Poisson-Fokker-Planck system, J. Math. An. Appl., 202 (1996), 1058-1075. doi: 10.1006/jmaa.1996.0360.
[20]	Z. Haba, Relativistic diffusion, Phys. Rev. E, 79 (2009), 021128. doi: 10.1103/PhysRevE.79.021128.
[21]	Z. Haba, Relativistic diffusion of elementary particles with spin, Journ. Phys. A, 42 (2009), 445401. doi: 10.1088/1751-8113/42/44/445401.
[22]	Z. Haba, Energy and entropy of relativistic diffusing particles, Mod. Phys. Lett. A, 25 (2010), 2683-2695. doi: 10.1142/S0217732310033992.
[23]	Z. Haba, Relativistic diffusive transport, preprint arXiv:0911.3126.
[24]	B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators And Witten Laplacians," Lecture Notes in Mathematics 1862, Springer-Verlag, New York, 2000.
[25]	L. Hörmander, Pseudodifferential operators and non-elliptic boundary problems, Ann. of Math., 83 (1966), 129-209. doi: 10.2307/1970473.
[26]	F. John, Blow-up for quasi linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51. doi: 10.1002/cpa.3160340103.
[27]	I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Graduate Texts in Mathematics 113 2nd edition, Springer-Verlag, New York, 1991.
[28]	C. R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Math. Meth. Appl. Sci., 21 (1998), 1287-1296. doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G.
[29]	E. H. Lieb and M. Loss, "Analysis," American Math. Soc. 14, Providence, 1997.
[30]	C.-P. Ma and E. Bertschinger, A cosmological kinetic theory for the evolution of cold dark matter halos with substructure: Quasi-linear theory, The Astroph. J., 612 (2004), 28-49.
[31]	M. Risken, "The Fokker-Planck Equation: Methods of Solutions and Applications," Springer Series in Synergetics 18, Springer-Verlag, Berlin, 1996.
[32]	M. Schunck, M. Hegmann and E. Sedlmayr, The influence of stochastic density fluctuations on the infrared emissions of interstellar dark clouds, Mon. Noti. Royal Astron. Soc., 374 (2007), 949-959. doi: 10.1111/j.1365-2966.2006.11215.x.
[33]	S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity, Phys. Rev. D, 47 (1993), 1529-1540. doi: 10.1103/PhysRevD.47.1529.
[34]	C. Sogge, "Lectures on Nonlinear Wave Equations," International Press, Cambridge, 1995.
[35]	J. L. Vázquez, "The Porous Medium Equation: Mathematical Theory," Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, Oxford 2007.
[36]	T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488. doi: 10.1137/090755796.
[37]	C. Villani, Hypocoercivity, Memoirs of the AMS, 202 (2009), n. 950.

show all references

References:

[1]	F. Andreu, V. Caselles, J. M. Mazón and S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal., 182 (2006), 269-297. doi: 10.1007/s00205-006-0428-3.
[2]	M. Bostan and T. Goudon, Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case, Kinet. Relat. Models, 1 (2008), 139-170.
[3]	F. Bouchut and J. Dolbeault, On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with coulombic and Newtonian potentials, Diff. Integ. Eqs., 8 (1995), 487-514.
[4]	S. Calogero, Spherical symmetric steady states of galactic dynamics in scalar gravity, Class. Quant. Grav., 20 (2003), 1729-1741. doi: 10.1088/0264-9381/20/9/310.
[5]	S. Calogero, Global classical solutions to the 3D Nordström-Vlasov system, Comm. Math. Phys., 266 (2006), 343-353. doi: 10.1007/s00220-006-0029-x.
[6]	S. Calogero, O. Sánchez and J. Soler, Asymptotic behavior and orbital stability of galactic dynamics in relativistic scalar gravity, Arch. Rat. Mech. Anal., 194 (2009), 743-773. doi: 10.1007/s00205-008-0173-x.
[7]	S. Calogero, J. Calvo, O. Sánchez and J. Soler, Virial inequalities for steady states in relativistic galactic dynamics, Nonlinearity, 23 (2010), 1851-1871. doi: 10.1088/0951-7715/23/8/004.
[8]	S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations on Riemannian manifolds, preprint, arXiv:1009.5086.
[9]	J. A. Carrillo, P. Laurençot and J. Rosado, Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics, J. Diff. Eqns., 247 (2009), 2209-2234. doi: 10.1016/j.jde.2009.07.018.
[10]	G. Chac\'on-Acosta and G. M. Kramer, Fokker-Planck-type equations for a simple gas and for a semirelativistic Brownian motion from a relativistic kinetic theory, Phys. Rev. E., 76 (2007), 021201. doi: 10.1103/PhysRevE.76.021201.
[11]	S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev Mod. Phys., 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1.
[12]	F. Debbasch and C. Chevalier, Relativistic stochastic processes: A review, AIP Conf. Proc., 913 (2007), 42-48. doi: 10.1063/1.2746722.
[13]	J. Dolbeault, Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: External potential and confinement (large time behavior and steady states), J. Math. Pures Appl., 78 (1999), 121-157. doi: 10.1016/S0021-7824(01)80006-4.
[14]	K. Dressler, Steady states in plasma physics-the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci., 12 (1990), 471-487. doi: 10.1002/mma.1670120603.
[15]	K. Dressler, Stationary solutions of the Vlasov-Fokker-Planck equation, Math. Meth. Appl. Sci., 9 (1987), 169-176. doi: 10.1002/mma.1670090113.
[16]	J. Dunkel and P. Hänggi, Theory of the relativistic Brownian motion: The (1+3)-dimensional case, Phys. Rev. E, 72 (2005), 036106. doi: 10.1103/PhysRevE.72.036106.
[17]	J. Dunkel and P. Hänggi, Relativistic Brownian motion, Phys. Rep., 471 (2009), 1-73. doi: 10.1016/j.physrep.2008.12.001.
[18]	D. T. Frank, "Nonlinear Fokker-Planck Equations: Fundamentals and Applications," Springer Series in Synergetics 25, Springer-Verlag, New York, 2005.
[19]	R. T. Glassey, J. Schaeffer and Y. Zheng, Steady states of the Vlasov-Poisson-Fokker-Planck system, J. Math. An. Appl., 202 (1996), 1058-1075. doi: 10.1006/jmaa.1996.0360.
[20]	Z. Haba, Relativistic diffusion, Phys. Rev. E, 79 (2009), 021128. doi: 10.1103/PhysRevE.79.021128.
[21]	Z. Haba, Relativistic diffusion of elementary particles with spin, Journ. Phys. A, 42 (2009), 445401. doi: 10.1088/1751-8113/42/44/445401.
[22]	Z. Haba, Energy and entropy of relativistic diffusing particles, Mod. Phys. Lett. A, 25 (2010), 2683-2695. doi: 10.1142/S0217732310033992.
[23]	Z. Haba, Relativistic diffusive transport, preprint arXiv:0911.3126.
[24]	B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators And Witten Laplacians," Lecture Notes in Mathematics 1862, Springer-Verlag, New York, 2000.
[25]	L. Hörmander, Pseudodifferential operators and non-elliptic boundary problems, Ann. of Math., 83 (1966), 129-209. doi: 10.2307/1970473.
[26]	F. John, Blow-up for quasi linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51. doi: 10.1002/cpa.3160340103.
[27]	I. Karatzas and S. E. Shreve, "Brownian Motion and Stochastic Calculus," Graduate Texts in Mathematics 113 2nd edition, Springer-Verlag, New York, 1991.
[28]	C. R. Lai, On the one-and-one-half-dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity, Math. Meth. Appl. Sci., 21 (1998), 1287-1296. doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G.
[29]	E. H. Lieb and M. Loss, "Analysis," American Math. Soc. 14, Providence, 1997.
[30]	C.-P. Ma and E. Bertschinger, A cosmological kinetic theory for the evolution of cold dark matter halos with substructure: Quasi-linear theory, The Astroph. J., 612 (2004), 28-49.
[31]	M. Risken, "The Fokker-Planck Equation: Methods of Solutions and Applications," Springer Series in Synergetics 18, Springer-Verlag, Berlin, 1996.
[32]	M. Schunck, M. Hegmann and E. Sedlmayr, The influence of stochastic density fluctuations on the infrared emissions of interstellar dark clouds, Mon. Noti. Royal Astron. Soc., 374 (2007), 949-959. doi: 10.1111/j.1365-2966.2006.11215.x.
[33]	S. L. Shapiro and S. A. Teukolsky, Scalar gravitation: A laboratory for numerical relativity, Phys. Rev. D, 47 (1993), 1529-1540. doi: 10.1103/PhysRevD.47.1529.
[34]	C. Sogge, "Lectures on Nonlinear Wave Equations," International Press, Cambridge, 1995.
[35]	J. L. Vázquez, "The Porous Medium Equation: Mathematical Theory," Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, Oxford 2007.
[36]	T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488. doi: 10.1137/090755796.
[37]	C. Villani, Hypocoercivity, Memoirs of the AMS, 202 (2009), n. 950.

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