20 research outputs found
Langevin Equation for the Rayleigh model with finite-ranged interactions
Both linear and nonlinear Langevin equations are derived directly from the
Liouville equation for an exactly solvable model consisting of a Brownian
particle of mass interacting with ideal gas molecules of mass via a
quadratic repulsive potential. Explicit microscopic expressions for all kinetic
coefficients appearing in these equations are presented. It is shown that the
range of applicability of the Langevin equation, as well as statistical
properties of random force, may depend not only on the mass ratio but
also by the parameter , involving the average number of molecules in
the interaction zone around the particle. For the case of a short-ranged
potential, when , analysis of the Langevin equations yields previously
obtained results for a hard-wall potential in which only binary collisions are
considered. For the finite-ranged potential, when multiple collisions are
important (), the model describes nontrivial dynamics on time scales
that are on the order of the collision time, a regime that is usually beyond
the scope of more phenomenological models.Comment: 21 pages, 1 figure. To appear in Phys. Rev.
A limit model for thermoelectric equations
We analyze the asymptotic behavior corresponding to the arbitrary high
conductivity of the heat in the thermoelectric devices. This work deals with a
steady-state multidimensional thermistor problem, considering the Joule effect
and both spatial and temperature dependent transport coefficients under some
real boundary conditions in accordance with the Seebeck-Peltier-Thomson
cross-effects. Our first purpose is that the existence of a weak solution holds
true under minimal assumptions on the data, as in particular nonsmooth domains.
Two existence results are studied under different assumptions on the electrical
conductivity. Their proofs are based on a fixed point argument, compactness
methods, and existence and regularity theory for elliptic scalar equations. The
second purpose is to show the existence of a limit model illustrating the
asymptotic situation.Comment: 20 page