1,070 research outputs found
Entropy of Open Lattice Systems
We investigate the behavior of the Gibbs-Shannon entropy of the stationary
nonequilibrium measure describing a one-dimensional lattice gas, of L sites,
with symmetric exclusion dynamics and in contact with particle reservoirs at
different densities. In the hydrodynamic scaling limit, L to infinity, the
leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be
that of a product measure corresponding to strict local equilibrium; we compute
the first correction, which is O(1). The computation uses a formal expansion of
the entropy in terms of truncated correlation functions; for this system the
k-th such correlation is shown to be O(L^{-k+1}). This entropy correction
depends only on the scaled truncated pair correlation, which describes the
covariance of the density field. It coincides, in the large L limit, with the
corresponding correction obtained from a Gaussian measure with the same
covariance.Comment: Latex, 28 pages, 4 figures as eps file
Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria
We study, globaly in time, the velocity distribution of a spatially
homogeneous system that models a system of electrons in a weakly ionized
plasma, subjected to a constant external electric field . The density
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
distance between and a certain time dependent Maxwellian stays small
uniformly in . 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.Comment: 30 pages, in TeX, to appear in Archive for Rational Mechanics and
Analysis: author's email addresses: [email protected],
[email protected], [email protected],
[email protected], [email protected]
Propagation of Chaos for a Thermostated Kinetic Model
We consider a system of N point particles moving on a d-dimensional torus.
Each particle is subject to a uniform field E and random speed conserving
collisions. This model is a variant of the Drude-Lorentz model of electrical
conduction. 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 large N limit, the one particle velocity distribution satisfies a self
consistent Vlasov-Boltzmann equation.. This is a consequence of "propagation of
chaos", which we also prove for this model.Comment: This version adds affiliation and grant information; otherwise it is
unchange
Shift Equivalence of Measures and the Intrinsic Structure of Shocks in the Asymmetric Simple Exclusion Process
We investigate properties of non-translation-invariant measures, describing
particle systems on \bbz, which are asymptotic to different translation
invariant measures on the left and on the right. Often the structure of the
transition region can only be observed from a point of view which is
random---in particular, configuration dependent. Two such measures will be
called shift equivalent if they differ only by the choice of such a viewpoint.
We introduce certain quantities, called translation sums, which, under some
auxiliary conditions, characterize the equivalence classes. Our prime example
is the asymmetric simple exclusion process, for which the measures in question
describe the microscopic structure of shocks. In this case we compute
explicitly the translation sums and find that shocks generated in different
ways---in particular, via initial conditions in an infinite system or by
boundary conditions in a finite system---are described by shift equivalent
measures. We show also that when the shock in the infinite system is observed
from the location of a second class particle, treating this particle either as
a first class particle or as an empty site leads to shift equivalent shock
measures.Comment: Plain TeX, 2 figures; [email protected], [email protected],
[email protected], [email protected]
Product Measure Steady States of Generalized Zero Range Processes
We establish necessary and sufficient conditions for the existence of
factorizable steady states of the Generalized Zero Range Process. This process
allows transitions from a site to a site involving multiple particles
with rates depending on the content of the site , the direction of
movement, and the number of particles moving. We also show the sufficiency of a
similar condition for the continuous time Mass Transport Process, where the
mass at each site and the amount transferred in each transition are continuous
variables; we conjecture that this is also a necessary condition.Comment: 9 pages, LaTeX with IOP style files. v2 has minor corrections; v3 has
been rewritten for greater clarit
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