Damping of particles interacting with a vibrating medium. (English) Zbl 1386.82062

By slightly modifying the model given in [L. Bruneau and S. De Bièvre, Commun. Math. Phys. 229, No. 3, 511–542 (2002; Zbl 1073.37079)] the authors investigate the nonstationary solution of the Vlasov-Fokker-Planck equation for the inelastic Lorentz gas \[ \partial_tF+v\nabla_xF-\nabla_x(V+\Phi)\cdot\nabla_vF=\gamma\nabla_v(vF+\nabla_vF),\quad t\geq 0,\quad x\in\mathbb R^d,\quad v\in\mathbb R^d. \] The behavior of the nonstationary solution for this equation was investigated using a confining external potential \(V\) and the self-consistent potential \(\Phi\). The self-consistent potential \(\Phi\) is describing the interaction of the particles with vibrating environment. The authors consider that the environment is a medium vibrating in a direction transverse to particles’ motion. The system was considered for the special typed of the initial data and assuming that “the external potential and the self-consistent potential. have the same order of magnitude It was found that the considered model has the equilibrium states and have proofed that system demonstrate the asymptotic trend to equilibrium.”


82C70 Transport processes in time-dependent statistical mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
70F45 The dynamics of infinite particle systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
74A25 Molecular, statistical, and kinetic theories in solid mechanics


Zbl 1073.37079
Full Text: DOI


[1] Aguer, B.; De Bièvre, S.; Lafitte, P.; Parris, P. E., Classical motion in force fields with short range correlations, J. Stat. Phys., 138, 4-5, 780-814, (2010) · Zbl 1187.82106
[2] Arnold, A.; Markowich, P.; Toscani, G.; Unterreiter, A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Commun. Partial Differ. Equ., 26, 1-2, 43-100, (2001) · Zbl 0982.35113
[3] Batt, J.; Morrison, P. J.; Rein, G., Linear stability of stationary solutions of the Vlasov-Poisson system in three dimension, Arch. Ration. Mech. Anal., 130, 163-182, (1995) · Zbl 0828.76093
[4] Bouchut, F., Existence and uniqueness of a global smooth solution for the VPFP system in three dimensions, J. Funct. Anal., 111, 239-258, (1993) · Zbl 0777.35059
[5] Bouchut, F., Smoothing effect for the non-linear VPFP system, J. Differ. Equ., 122, 225-238, (1995)
[6] Bruneau, L.; De Bièvre, S., A Hamiltonian model for linear friction in a homogeneous medium, Commun. Math. Phys., 229, 3, 511-542, (2002) · Zbl 1073.37079
[7] Csiszar, I., Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hung., 2, 299-318, (1967) · Zbl 0157.25802
[8] De Bièvre, S.; Goudon, T.; Vavasseur, A., Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov-Poisson system, SIAM J. Math. Anal., 48, 6, 3984-4020, (2016) · Zbl 1357.82063
[9] De Bièvre, S.; Lafitte, P.; Parris, P. E., Normal transport at positive temperatures in classical Hamiltonian open systems, (Adventures in Mathematical Physics, Contemp. Math., vol. 447, (2007), Amer. Math. Soc. Providence, RI), 57-71 · Zbl 1251.82046
[10] De Bièvre, S.; Parris, P. E., Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases, J. Stat. Phys., 142, 2, 356-385, (2011) · Zbl 1216.82032
[11] De Bièvre, S.; Parris, P. E.; Silvius, A., Chaotic dynamics of a free particle interacting linearly with a harmonic oscillator, Phys. D, 208, 1-2, 96-114, (2005) · Zbl 1079.70015
[12] Di Perna, R.; Lions, P.-L.; Meyer, Y., \(L^p\) regularity of velocity average, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 8, 271-287, (1991) · Zbl 0763.35014
[13] Dolbeault, J.; Mouhot, C.; Schmeiser, C., Hypocoercivity for linear kinetic equations conserving mass, Trans. Am. Math. Soc., 367, 3807-3828, (2015) · Zbl 1342.82115
[14] El Ghani, N.; Masmoudi, N., Diffusion limit of the Vlasov-Poisson-Fokker-Planck system, Commun. Math. Sci., 8, 463-479, (2010) · Zbl 1193.35228
[15] Evans, L. C., Partial differential equations, Grad. Stud. Math., vol. 19, (1998), Am. Math. Soc.
[16] Goudon, T., Intégration: intégrale de Lebesgue et introduction à l’analyse fonctionnelle, (2011), Références Sciences. Ellipses · Zbl 1327.28002
[17] Goudon, T.; Vavasseur, A., Mean field limit for particles interacting with a vibrating medium, Ann. Univ. Ferrara, 62, 2, 231-273, (2016) · Zbl 1355.82043
[18] Helffer, B.; Nier, F., Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lect. Notes Math., vol. 1862, (2005), Springer · Zbl 1072.35006
[19] Hérau, F., Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46, 3, 4, 349-359, (2006) · Zbl 1096.35019
[20] Hérau, F.; Thomann, L., On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, J. Funct. Anal., 271, 5, 1301-1340, (2016) · Zbl 1347.35221
[21] Kullback, S., A lower bound for discrimination information in terms of variation, IEEE Trans. Inf. Theory, 4, 126-127, (1967)
[22] Lafitte, P.; Parris, P. E.; De Bièvre, S., Normal transport properties in a metastable stationary state for a classical particle coupled to a non-ohmic Bath, J. Stat. Phys., 132, 5, 863-879, (2008) · Zbl 1152.82021
[23] Lieb, E. H.; Loss, M., Analysis, Grad. Stud. Math., vol. 14, (2001), AMS Providence, RI · Zbl 0966.26002
[24] Lions, P.-L., Mathematical topics in fluid mechanics. vol. 2: compressible models, Oxf. Lect. Ser. Math. Appl., vol. 10, (1998), The Clarendon Press Oxford University Press New York, Oxford Science Publications · Zbl 0908.76004
[25] Masmoudi, N.; Tayeb, M. L., Diffusion limit of a semiconductor Boltzmann-Poisson system, SIAM J. Math. Anal., 38, 1788-1807, (2007) · Zbl 1206.82133
[26] Persson, A., Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand., 8, 143-153, (1960) · Zbl 0145.14901
[27] Perthame, B.; Souganidis, P., A limiting case for velocity averaging, Ann. Sci. Éc. Norm. Supér., 31, 591-598, (1998) · Zbl 0956.45010
[28] Sogge, C. D., Lectures on nonlinear wave equations, Monogr. Anal., (2008), Int. Press. of Boston · Zbl 1165.35001
[29] Sznitman, A.-S., Topics in propagation of chaos, (Ecole d’Eté de Probabilités de Saint-Flour XIX, 1989, Lect. Notes Math., vol. 1464, (1991), Springer), 165-251
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.