91 research outputs found
On the ground state energy for a magnetic Shcr\"odinger operator and the effect of the De Gennes Boundary condition
Motivated by the Ginzburg-Landau theory of superconductivity, we estimate in
the semi-classical limit the ground state energy of a magnetic Schr\"odinger
operator with De Gennes boundary condition and we study the localization of the
ground state. We exhibit cases when the De Gennes boundary condition has strong
effects on this localization.Comment: content revise
Some ideas about quantitative convergence of collision models to their mean field limit
We consider a stochastic -particle model for the spatially homogeneous
Boltzmann evolution and prove its convergence to the associated Boltzmann
equation when . For any time we bound the distance between
the empirical measure of the particle system and the measure given by the
Boltzmann evolution in some homogeneous negative Sobolev space. The control we
get is Gaussian, i.e. we prove that the distance is bigger than
with a probability of type . The two main ingredients are first a
control of fluctuations due to the discrete nature of collisions, secondly a
Lipschitz continuity for the Boltzmann collision kernel. The latter condition,
in our present setting, is only satisfied for Maxwellian models. Numerical
computations tend to show that our results are useful in practice.Comment: 27 pages, references added and style improve
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics
International audienceWe provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function
Collisionless kinetic theory of rolling molecules
We derive a collisionless kinetic theory for an ensemble of molecules
undergoing nonholonomic rolling dynamics. We demonstrate that the existence of
nonholonomic constraints leads to problems in generalizing the standard methods
of statistical physics. In particular, we show that even though the energy of
the system is conserved, and the system is closed in the thermodynamic sense,
some fundamental features of statistical physics such as invariant measure do
not hold for such nonholonomic systems. Nevertheless, we are able to construct
a consistent kinetic theory using Hamilton's variational principle in
Lagrangian variables, by regarding the kinetic solution as being concentrated
on the constraint distribution. A cold fluid closure for the kinetic system is
also presented, along with a particular class of exact solutions of the kinetic
equations.Comment: Revised version; 31 pages, 1 figur
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