258 research outputs found
1D nonlinear Fokker–Planck equations for fermions and bosons
AbstractWe obtain equilibration rates for nonlinear Fokker–Planck equations modelling the relaxation of fermion and boson gases. We show how the entropy method applies for quantifying explicitly the exponential decay towards Fermi–Dirac and Bose–Einstein distributions in the one-dimensional case
Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with
quadratic drift as a PDE model for the dynamics of bosons in the spatially
homogeneous setting. It is an open question whether this equation has solutions
exhibiting condensates in finite time. The main analytical challenge lies in
the continuation of exploding solutions beyond their first blow-up time while
having a linear diffusion term. We present a thoroughly validated time-implicit
numerical scheme capable of simulating solutions for arbitrarily large time,
and thus enabling a numerical study of the condensation process in the
Kaniadakis--Quarati model. We show strong numerical evidence that above the
critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model
in 3D form a condensate in finite time and converge in entropy to the unique
minimiser of the natural entropy functional at an exponential rate. Our
simulations further indicate that the spatial blow-up profile near the origin
follows a universal power law and that transient condensates can occur for
sufficiently concentrated initial data.Comment: To appear in Kinet. Relat. Model
Condensation phenomena in nonlinear drift equations
We study nonnegative, measure-valued solutions to nonlinear drift type
equations modelling concentration phenomena related to Bose-Einstein particles.
In one spatial dimension, we prove existence and uniqueness for measure
solutions. Moreover, we prove that all solutions blow up in finite time leading
to a concentration of mass only at the origin, and the concentrated mass
absorbs increasingly the mass converging to the total mass as time goes to
infinity. Our analysis makes a substantial use of independent variable scalings
and pseudo-inverse functions techniques
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
A finite volume scheme for nonlinear degenerate parabolic equations
We propose a second order finite volume scheme for nonlinear degenerate
parabolic equations. For some of these models (porous media equation,
drift-diffusion system for semiconductors, ...) it has been proved that the
transient solution converges to a steady-state when time goes to infinity. The
present scheme preserves steady-states and provides a satisfying long-time
behavior. Moreover, it remains valid and second-order accurate in space even in
the degenerate case. After describing the numerical scheme, we present several
numerical results which confirm the high-order accuracy in various regime
degenerate and non degenerate cases and underline the efficiency to preserve
the large-time asymptotic
Scaling and Crossover Functions for the Conductance in the Directed Network Model of Edge States
We consider the directed network (DN) of edge states on the surface of a
cylinder of length L and circumference C. By mapping it to a ferromagnetic
superspin chain, and using a scaling analysis, we show its equivalence to a
one-dimensional supersymmetric nonlinear sigma model in the scaling limit, for
any value of the ratio L/C, except for short systems where L is less than of
order C^{1/2}. For the sigma model, the universal crossover functions for the
conductance and its variance have been determined previously. We also show that
the DN model can be mapped directly onto the random matrix (Fokker-Planck)
approach to disordered quasi-one-dimensional wires, which implies that the
entire distribution of the conductance is the same as in the latter system, for
any value of L/C in the same scaling limit. The results of Chalker and Dohmen
are explained quantitatively.Comment: 10 pages, REVTeX, 2 eps figure
Singularities in -supercritical Fokker--Planck equations: A qualitative analysis
A class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L1-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis -- the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model
Localization in disordered superconducting wires with broken spin-rotation symmetry
Localization and delocalization of non-interacting quasiparticle states in a
superconducting wire are reconsidered, for the cases in which spin-rotation
symmetry is absent, and time-reversal symmetry is either broken or unbroken;
these are referred to as symmetry classes BD and DIII, respectively. We show
that, if a continuum limit is taken to obtain a Fokker-Planck (FP) equation for
the transfer matrix, as in some previous work, then when there are more than
two scattering channels, all terms that break a certain symmetry are lost. It
was already known that the resulting FP equation exhibits critical behavior.
The additional symmetry is not required by the definition of the symmetry
classes; terms that break it arise from non-Gaussian probability distributions,
and may be kept in a generalized FP equation. We show that they lead to
localization in a long wire. When the wire has more than two scattering
channels, these terms are irrelevant at the short distance (diffusive or
ballistic) fixed point, but as they are relevant at the long-distance critical
fixed point, they are termed dangerously irrelevant. We confirm the results in
a supersymmetry approach for class BD, where the additional terms correspond to
jumps between the two components of the sigma model target space. We consider
the effect of random fluxes, which prevent the system localizing. We show
that in one dimension the transitions in these two symmetry classes, and also
those in the three chiral symmetry classes, all lie in the same universality
class
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