Artículo

Cañizo, J.A.; Carrillo, J.A.; Laurençot, P.; Rosado, J. "The Fokker-Planck equation for bosons in 2D: Well-posedness and asymptotic behavior" (2016) Nonlinear Analysis, Theory, Methods and Applications. 137:291-305
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Abstract:

We show that solutions of the 2D Fokker-Planck equation for bosons are defined globally in time and converge to equilibrium, and this convergence is shown to be exponential for radially symmetric solutions. The main observation is that a variant of the Hopf-Cole transformation relates the 2D equation in radial coordinates to the usual linear Fokker-Planck equation. Hence, radially symmetric solutions can be computed analytically, and our results for general (non radially symmetric) solutions follow from comparison and entropy arguments. In order to show convergence to equilibrium we also prove a version of the Csiszár-Kullback inequality for the Bose-Einstein-Fokker-Planck entropy functional. © 2015 Elsevier Ltd. All rights reserved.

Registro:

Documento: Artículo
Título:The Fokker-Planck equation for bosons in 2D: Well-posedness and asymptotic behavior
Autor:Cañizo, J.A.; Carrillo, J.A.; Laurençot, P.; Rosado, J.
Filiación:Departamento de Matemática Aplicada, Universidad de Granada, Granada, 18071, Spain
Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom
Institut de Mathématiques de Toulouse, UMRA 5219, Université de Toulouse, CNRS, Toulouse Cedex 9, F-31062, France
Departamento de Matemática, Universidad de Buenos Aires, Argentina
Palabras clave:Bose-Einstein; Entropy method; Long-time asymptotics; Bosons; Linear transformations; Mathematical transformations; Statistical mechanics; Timing jitter; Asymptotic behaviors; Bose-Einstein; Convergence to equilibrium; Entropy functional; Entropy methods; Hopf-cole transformations; Long-time asymptotics; Radially symmetric solution; Fokker Planck equation
Año:2016
Volumen:137
Página de inicio:291
Página de fin:305
DOI: http://dx.doi.org/10.1016/j.na.2015.07.030
Título revista:Nonlinear Analysis, Theory, Methods and Applications
Título revista abreviado:Nonlinear Anal Theory Methods Appl
ISSN:0362546X
CODEN:NOAND
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v137_n_p291_Canizo

Referencias:

  • Aftalion, A., Helffer, B., On mathematical models for Bose-Einstein condensates in optical lattices (2009) Rev. Math. Phys., 21 (2), pp. 229-278
  • Allemand, T., Toscani, G., The grazing collision limit of Kac caricature of Bose-Einstein particles (2011) Asymptot. Anal., 72 (3-4), pp. 201-229
  • Ben Abdallah, N., Gamba, I.M., Toscani, G., On the minimization problem of sub-linear convex functionals (2011) Kinet. Relat. Models, 4 (4), pp. 857-871
  • Benedetto, D., Caglioti, E., Carrillo, J.A., Pulvirenti, M., A non-Maxwellian steady distribution for one-dimensional granular media (1998) J. Stat. Phys., 91 (5-6), pp. 979-990
  • Cáceres, M.J., Carrillo, J.A., Dolbeault, J., Nonlinear stability in Lp for a confined system of charged particles (2002) SIAM J. Math. Anal., 34 (2), pp. 478-494
  • Carrillo, J.A., Cordier, S., Toscani, G., Over-populated tails for conservative-in-the-mean inelastic Maxwell models (2009) Discrete Contin. Dyn. Syst., 24 (1), pp. 59-81
  • Carrillo, J.A., Laurençot, Ph., Rosado, J., Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics (2009) J. Differential Equations, 247 (8), pp. 2209-2234
  • Carrillo, J.A., Rosado, J., Salvarani, F., 1D nonlinear Fokker-Planck equations for fermions and bosons (2008) Appl. Math. Lett., 21 (2), pp. 148-154
  • Carrillo, J.A., Toscani, G., Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations (1998) Math. Methods Appl. Sci., 21 (13), pp. 1269-1286
  • Cordier, S., Pareschi, L., Toscani, G., On a kinetic model for a simple market economy (2005) J. Stat. Phys., 120 (1-2), pp. 253-277
  • Desvillettes, L., Dolbeault, J., On long time asymptotics of the Vlasov-Poisson-Boltzmann equation (1991) Comm. Partial Differential Equations, 16 (2-3), pp. 451-489
  • Desvillettes, L., Ricci, V., A rigorous derivation of a linear kinetic equation of Fokker-Planck type in the limit of grazing collisions (2001) J. Stat. Phys., 104 (5-6), pp. 1173-1189
  • Erdös, L., Schlein, B., Yau, H.T., Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate (2010) Ann. of Math., 172, pp. 291-370
  • Escobedo, M., Mischler, S., On a quantum Boltzmann equation for a gas of photons (2001) J. Math. Pures Appl., 80 (5), pp. 471-515
  • Escobedo, M., Mischler, S., Velazquez, J., Asymptotic description of Dirac mass formation in kinetic equations for quantum particles (2004) J. Differential Equations, 202 (2), pp. 208-230
  • Fokker, A.D., Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld (1914) Ann. Phys., 43, pp. 810-820
  • Ford, G.W., Lewis, J.T., O'Connell, R.F., Quantum Langevin equation (1988) Phys. Rev. A, 37, pp. 4419-4428
  • Frank, T.D., Classical Langevin equations for the free electron gas and blackbody radiation (2004) J. Phys. A, 37, pp. 3561-3567
  • Frank, T., Nonlinear Fokker-Planck Equations: Fundamentals and Applications (2006) Springer Series in Synergetics, , Springer
  • Gamba, I.M., Haack, J.R., A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit (2014) J. Comput. Phys., 270, pp. 40-57
  • Gardiner, C.W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (2004) Springer Series in Synergetics, 13. , third ed. Springer-Verlag Berlin
  • Gardiner, C.W., Collett, M.J., Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation (1985) Phys. Rev. A, 31, pp. 3761-3774
  • Goudon, T., On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions (1997) J. Stat. Phys., 89 (3-4), pp. 751-776
  • Hu, B.L., Paz, J.P., Zhang, Y., Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise (1992) Phys. Rev. D, 45, pp. 2843-2861
  • Kaniadakis, G., Generalized Boltzmann equation describing the dynamics of bosons and fermions (1995) Phys. Lett. A, 203, pp. 229-234
  • Kaniadakis, G., Quarati, P., Kinetic equation for classical particles obeying an exclusion principle (1993) Phys. Rev. E, 48, pp. 4263-4270
  • Lu, X., On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles (2001) J. Stat. Phys., 105 (1-2), pp. 353-388
  • Planck, M., Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie (1917) Sitzungsber. Preuss. Akad. Wiss. (Berlin), pp. 324-341
  • Risken, H., The Fokker-Planck Equation (1989) Springer Series in Synergetics, 18. , second ed. Springer-Verlag Berlin
  • Rossani, A., Kaniadakis, G., A generalized quasi-classical Boltzmann equation (2000) Physica A, 277, pp. 349-358
  • Sopik, J., Sire, C., Chavanis, P.-H., Dynamics of the Bose-Einstein condensation: Analogy with the collapse dynamics of a classical self-gravitating Brownian gas (2006) Phys. Rev. E, 74
  • Staliunas, K., Gross-Pitaevskii model for nonzero temperature Bose-Einstein condensates (2006) Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (16), pp. 2713-2719
  • Toscani, G., Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles (2011) Comm. Partial Differential Equations, 37 (1), pp. 77-87
  • Yano, R., On quantum Fokker-Planck equation (2015) J. Stat. Phys., 158 (1), pp. 231-247

Citas:

---------- APA ----------
Cañizo, J.A., Carrillo, J.A., Laurençot, P. & Rosado, J. (2016) . The Fokker-Planck equation for bosons in 2D: Well-posedness and asymptotic behavior. Nonlinear Analysis, Theory, Methods and Applications, 137, 291-305.
http://dx.doi.org/10.1016/j.na.2015.07.030
---------- CHICAGO ----------
Cañizo, J.A., Carrillo, J.A., Laurençot, P., Rosado, J. "The Fokker-Planck equation for bosons in 2D: Well-posedness and asymptotic behavior" . Nonlinear Analysis, Theory, Methods and Applications 137 (2016) : 291-305.
http://dx.doi.org/10.1016/j.na.2015.07.030
---------- MLA ----------
Cañizo, J.A., Carrillo, J.A., Laurençot, P., Rosado, J. "The Fokker-Planck equation for bosons in 2D: Well-posedness and asymptotic behavior" . Nonlinear Analysis, Theory, Methods and Applications, vol. 137, 2016, pp. 291-305.
http://dx.doi.org/10.1016/j.na.2015.07.030
---------- VANCOUVER ----------
Cañizo, J.A., Carrillo, J.A., Laurençot, P., Rosado, J. The Fokker-Planck equation for bosons in 2D: Well-posedness and asymptotic behavior. Nonlinear Anal Theory Methods Appl. 2016;137:291-305.
http://dx.doi.org/10.1016/j.na.2015.07.030