dorsal/arxiv
View SchemaKinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria
| Authors | E. Carlen, R. Esposito, J. L. Lebowitz, R. Marra, A. Rokhlenko |
|---|---|
| Categories | |
| ArXiv ID | physics/9706027 |
| URL | https://arxiv.org/abs/physics/9706027 |
| DOI | 10.1007/s002050050090 |
Abstract
We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field $E$. The density $f$ satisfies a Boltzmann type kinetic equation containing a full nonlinear electron-electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the $L^1$ distance between $f$ and a certain time dependent Maxwellian stays small uniformly in $t$. Moreover, the mean and variance of this time dependent Maxwellian satisfy a coupled set of nonlinear ODE's that constitute the ``hydrodynamical'' equations for this kinetic system. This remain true even when these ODE's have non-unique equilibria, thus proving the existence of multiple stabe stationary solutions for the full kinetic model. Our approach relies on scale independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globably in time.
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"abstract": "We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially\nhomogeneous system that models a system of electrons in a weakly ionized\nplasma, subjected to a constant external electric field $E$. The density $f$\nsatisfies a Boltzmann type kinetic equation containing a full nonlinear\nelectron-electron collision term as well as linear terms representing\ncollisions with reservoir particles having a specified Maxwellian distribution.\nWe show that when the constant in front of the nonlinear collision kernel,\nthought of as a scaling parameter, is sufficiently strong, then the $L^1$\ndistance between $f$ and a certain time dependent Maxwellian stays small\nuniformly in $t$. Moreover, the mean and variance of this time dependent\nMaxwellian satisfy a coupled set of nonlinear ODE\u0027s that constitute the\n``hydrodynamical\u0027\u0027 equations for this kinetic system. This remain true even\nwhen these ODE\u0027s have non-unique equilibria, thus proving the existence of\nmultiple stabe stationary solutions for the full kinetic model. Our approach\nrelies on scale independent estimates for the kinetic equation, and entropy\nproduction estimates. The novel aspects of this approach may be useful in other\nproblems concerning the relation between the kinetic and hydrodynamic scales\nglobably in time.",
"arxiv_id": "physics/9706027",
"authors": [
"E. Carlen",
"R. Esposito",
"J. L. Lebowitz",
"R. Marra",
"A. Rokhlenko"
],
"categories": [
"physics.plasm-ph"
],
"doi": "10.1007/s002050050090",
"title": "Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria",
"url": "https://arxiv.org/abs/physics/9706027"
},
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