84 research outputs found
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity
We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian
Penalty Methods for the Hyperbolic System Modelling the Wall-Plasma Interaction in a Tokamak
The penalization method is used to take account of obstacles in a tokamak,
such as the limiter. We study a non linear hyperbolic system modelling the
plasma transport in the area close to the wall. A penalization which cuts the
transport term of the momentum is studied. We show numerically that this
penalization creates a Dirac measure at the plasma-limiter interface which
prevents us from defining the transport term in the usual sense. Hence, a new
penalty method is proposed for this hyperbolic system and numerical tests
reveal an optimal convergence rate without any spurious boundary layer.Comment: 8 pages; International Symposium FVCA6, Prague : Czech Republic
(2011
Continuum model for linked fibers with alignment interactions
International audienceWe introduce an individual-based model for fiber elements having the ability to cross-link or unlink each other and to align with each other at the cross links. We first formally derive a kinetic model for the fiber and cross-links distribution functions. We then consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only and investigate its diffusion limit. The resulting macroscopic model consists of a system of nonlinear diffusion equations for the fiber density and mean orientation. In the case of a homogeneous fiber density, we show that the model is elliptic
Integral representation of the linear Boltzmann operator for granular gas dynamics with applications
We investigate the properties of the collision operator associated to the
linear Boltzmann equation for dissipative hard-spheres arising in granular gas
dynamics. We establish that, as in the case of non-dissipative interactions,
the gain collision operator is an integral operator whose kernel is made
explicit. One deduces from this result a complete picture of the spectrum of
the collision operator in an Hilbert space setting, generalizing results from
T. Carleman to granular gases. In the same way, we obtain from this integral
representation of the gain operator that the semigroup in L^1(\R \times \R,\d
\x \otimes \d\v) associated to the linear Boltzmann equation for dissipative
hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic
Classical motion in force fields with short range correlations
We study the long time motion of fast particles moving through time-dependent
random force fields with correlations that decay rapidly in space, but not
necessarily in time. The time dependence of the averaged kinetic energy and
mean-squared displacement is shown to exhibit a large degree of universality;
it depends only on whether the force is, or is not, a gradient vector field.
When it is, p^{2}(t) ~ t^{2/5} independently of the details of the potential
and of the space dimension. Motion is then superballistic in one dimension,
with q^{2}(t) ~ t^{12/5}, and ballistic in higher dimensions, with q^{2}(t) ~
t^{2}. These predictions are supported by numerical results in one and two
dimensions. For force fields not obtained from a potential field, the power
laws are different: p^{2}(t) ~ t^{2/3} and q^{2}(t) ~ t^{8/3} in all dimensions
d\geq 1
Analysis of a diffusive effective mass model for nanowires
We propose in this paper to derive and analyze a self-consistent model
describing the diffusive transport in a nanowire. From a physical point of
view, it describes the electron transport in an ultra-scaled confined
structure, taking in account the interactions of charged particles with
phonons. The transport direction is assumed to be large compared to the wire
section and is described by a drift-diffusion equation including effective
quantities computed from a Bloch problem in the crystal lattice. The
electrostatic potential solves a Poisson equation where the particle density
couples on each energy band a two dimensional confinement density with the
monodimensional transport density given by the Boltzmann statistics. On the one
hand, we study the derivation of this Nanowire Drift-Diffusion Poisson model
from a kinetic level description. On the other hand, we present an existence
result for this model in a bounded domain
Stability of flows associated to gradient vector fields and convergence of iterated transport maps
In this paper we address the problem of stability of flows
associated to a sequence of vector fields under minimal regularity requirements
on the limit vector field, that is supposed to be a gradient. We apply this
stability result to show the convergence of iterated compositions of optimal
transport maps arising in the implicit time discretization (with respect to the
Wasserstein distance) of nonlinear evolution equations of a diffusion type.
Finally, we use these convergence results to study the gradient flow of a
particular class of polyconvex functionals recently considered by Gangbo, Evans
ans Savin. We solve some open problems raised in their paper and obtain
existence and uniqueness of solutions under weaker regularity requirements and
with no upper bound on the jacobian determinant of the initial datum
Large scale dynamics of the Persistent Turning Walker model of fish behavior
International audienceThis paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results
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