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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.”

MSC:

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

Citations:

Zbl 1073.37079
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References:

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