55,329 research outputs found
Chapman-Enskog derivation of the generalized Smoluchowski equation
We use the Chapman-Enskog method to derive the Smoluchowski equation from the
Kramers equation in a high friction limit. We consider two main extensions of
this problem: we take into account a uniform rotation of the background medium
and we consider a generalized class of Kramers equations associated with
generalized free energy functionals. We mention applications of these results
to systems with long-range interactions (self-gravitating systems, 2D vortices,
bacterial populations,...). In that case, the Smoluchowski equation is
non-local. In the limit of short-range interactions, it reduces to a
generalized form of the Cahn-Hilliard equation. These equations are associated
with an effective generalized thermodynamical formalism.Comment: In pres
Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
We study a general class of nonlinear mean field Fokker-Planck equations in
relation with an effective generalized thermodynamical formalism. We show that
these equations describe several physical systems such as: chemotaxis of
bacterial populations, Bose-Einstein condensation in the canonical ensemble,
porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model,
Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian
particles, Debye-Huckel theory of electrolytes, two-dimensional turbulence...
In particular, we show that nonlinear mean field Fokker-Planck equations can
provide generalized Keller-Segel models describing the chemotaxis of biological
populations. As an example, we introduce a new model of chemotaxis
incorporating both effects of anomalous diffusion and exclusion principle
(volume filling). Therefore, the notion of generalized thermodynamics can have
applications for concrete physical systems. We also consider nonlinear mean
field Fokker-Planck equations in phase space and show the passage from the
generalized Kramers equation to the generalized Smoluchowski equation in a
strong friction limit. Our formalism is simple and illustrated by several
explicit examples corresponding to Boltzmann, Tsallis and Fermi-Dirac entropies
among others
General properties of nonlinear mean field Fokker-Planck equations
Recently, several authors have tried to extend the usual concepts of
thermodynamics and kinetic theory in order to deal with distributions that can
be non-Boltzmannian. For dissipative systems described by the canonical
ensemble, this leads to the notion of nonlinear Fokker-Planck equation (T.D.
Frank, Non Linear Fokker-Planck Equations, Springer, Berlin, 2005). In this
paper, we review general properties of nonlinear mean field Fokker-Planck
equations, consider the passage from the generalized Kramers to the generalized
Smoluchowski equation in the strong friction limit, and provide explicit
examples for Boltzmann, Tsallis and Fermi-Dirac entropies.Comment: Paper presented at the international conference CTNEXT07, 1-5 july
2007, Catania, Ital
Dynamics of the Ericksen-Leslie Equations with General Leslie Stress I: The Incompressible Isotropic Case
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain
with general Leslie and isotropic Ericksen stress is studied in the case of a
non-isothermal and incompressible fluid. This system is shown to be locally
well-posed in the -setting, and a dynamic theory is developed. The
equilibria are identified and shown to be normally stable. In particular, a
local solution extends to a unique, global strong solution provided the initial
data are close to an equilibrium or the solution is eventually bounded in the
topology of the natural state manifold. In this case, the solution converges
exponentially to an equilibrium, in the topology of the state manifold. The
above results are proven {\em without} any structural assumptions on the Leslie
coefficients and in particular {\em without} assuming Parodi's relation
Path integrals on manifolds by finite dimensional approximation
Let M be a compact Riemannian manifold without boundary and let H be a
self-adjoint generalized Laplace operator acting on sections in a bundle over
M. We give a path integral formula for the solution to the corresponding heat
equation. This is based on approximating path space by finite dimensional
spaces of geodesic polygons. We also show a uniform convergence result for the
heat kernels. This yields a simple and natural proof for the
Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of
the heat operator.Comment: 23 page
H-Theorem and Generalized Entropies Within the Framework of Non Linear Kinetics
In the present effort we consider the most general non linear particle
kinetics within the framework of the Fokker-Planck picture. We show that the
kinetics imposes the form of the generalized entropy and subsequently we
demonstrate the H-theorem. The particle statistical distribution is obtained,
both as stationary solution of the non linear evolution equation and as the
state which maximizes the generalized entropy. The present approach allows to
treat the statistical distributions already known in the literature in a
unifying scheme. As a working example we consider the kinetics, constructed by
using the -exponential recently proposed which reduces to
the standard exponential as the deformation parameter approaches to
zero and presents the relevant power law asymptotic behaviour
. The -kinetics obeys the H-theorem and in the case of
Brownian particles, admits as stationary state the distribution
which can be obtained also
by maximizing the entropy with after properly constrained.Comment: To appear in Phys. Lett.
Generalized Fokker-Planck equations and effective thermodynamics
We introduce a new class of Fokker-Planck equations associated with an
effective generalized thermodynamical framework. These equations describe a gas
of Langevin particles in interaction. The free energy can take various forms
which can account for anomalous diffusion, quantum statistics, lattice
models... When the potential of interaction is long-ranged, these equations
display a rich structure associated with canonical phase transitions and
blow-up phenomena. In the limit of short-range interactions, they reduce to
Cahn-Hilliard equations
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