TY - JOUR

T1 - Propagation of Chaos for a Thermostated Kinetic Model

AU - Bonetto, F.

AU - Carlen, E. A.

AU - Esposito, R.

AU - Lebowitz, J. L.

AU - Marra, R.

N1 - Funding Information:
Work of E.A. Carlen is partially supported by U.S. National Science Foundation grant DMS 1201354. Work of J.L. Lebowitz is partially supported by U.S. National Science Foundation grant PHY 0965859. Work of R. Marra is partially supported by MIUR and GNFM-INdAM.

PY - 2014/1

Y1 - 2014/1

N2 - We consider a system of N point particles moving on a d-dimensional torus Td. Each particle is subject to a uniform field E and random speed conserving collisions vi → vi′ with {pipe}vi{pipe}{pipe}vi′{pipe}. This model is a variant of the Drude-Lorentz model of electrical conduction (Ashcroft and Mermin in Solid state physics. Brooks Cole, Pacific Grove, 1983). In order to avoid heating by the external field, the particles also interact with a Gaussian thermostat which keeps the total kinetic energy of the system constant. The thermostat induces a mean-field type of interaction between the particles. Here we prove that, starting from a product measure, in the limit N→∞, the one particle velocity distribution f(q,v,t) satisfies a self consistent Vlasov-Boltzmann equation, for all finite time t. This is a consequence of "propagation of chaos", which we also prove for this model.

AB - We consider a system of N point particles moving on a d-dimensional torus Td. Each particle is subject to a uniform field E and random speed conserving collisions vi → vi′ with {pipe}vi{pipe}{pipe}vi′{pipe}. This model is a variant of the Drude-Lorentz model of electrical conduction (Ashcroft and Mermin in Solid state physics. Brooks Cole, Pacific Grove, 1983). In order to avoid heating by the external field, the particles also interact with a Gaussian thermostat which keeps the total kinetic energy of the system constant. The thermostat induces a mean-field type of interaction between the particles. Here we prove that, starting from a product measure, in the limit N→∞, the one particle velocity distribution f(q,v,t) satisfies a self consistent Vlasov-Boltzmann equation, for all finite time t. This is a consequence of "propagation of chaos", which we also prove for this model.

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U2 - 10.1007/s10955-013-0861-2

DO - 10.1007/s10955-013-0861-2

M3 - Article

AN - SCOPUS:84893743636

SN - 0022-4715

VL - 154

SP - 265

EP - 285

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 1-2

ER -