Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one, show that the nucleation time divided by its average converges to an exponential random variable, express the proportionality constant for the average nucleation time in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion.     

Global convergence of a kinetic model of chemotaxis to a perturbed Keller–Segel model

We consider a class of kinetic models of chemotaxis with two positive non-dimensional parameters coupled to a parabolic equation of the chemo-attractant. If both parameters are set equal zero, we have the classical Keller–Segel model for chemotaxis. We prove global existence of solutions of this two-parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero. In one case, we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed Keller–Segel model with global existence of solutions.     

Non-Markovianity of the Boltzmann-Grad limit of a system of random obstacles in a given force field

In this paper we consider a particle moving in a random distribution of obstacles. Each obstacle is absorbing and a fixed force field is imposed. We show rigorously that certain (very smooth) fields prevent the process obtained by the Boltzmann-Grad limit from being Markovian. Then, we propose a slightly different setting which allows this difficulty to be removed.     

Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.     

Asymptotic description of Dirac mass formation in kinetic equations for quantum particles

In this paper, the detailed asymptotic behaviour of the solutions of a kinetic equation for quantum particles is studied. It is shown that this behaviour is sensitive not only to the total mass of the initial data but also to its precise behaviour near the origin. In some cases, solutions develop a Dirac mass at the origin for long times in a self-similar manner that is analysed in detail.     

Application of extended kinetic theory to particle transport with inelastic scattering

In the frame of extended kinetic theory, the linear Boltzmann equation for test particles in an absorbing and inelastically scattering background leads to a partial-integral-difference equation which is studied in the proper mathematical setting. As an application, penetration of a beam of particles in a plane slab is considered in steady state conditions, and the relevant problem is solved by a rigorous algorithm. Accurate results for particle and energy distributions, and for transmission, reflection, and absorption coefficients are provided and briefly discussed.

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Riassunto Viene studiata, nell'ambito della teoria cinetica estesa, 1'equazione di Boltzmann lineare per il trasporto di particelle in un mezzo assorbente e scatterante anelasticamente. L'equazione integro-differenziale alle differenze è applicata al problema stazionario della penetrazione di un fascio di particelle in una lastra piana. Vengono presentati risultati numerici rigorosi per le distribuzioni di particelle ed energia, e per i coefficient: di trasmissione, riflessione ed assorbimento.

     

The swapping algorithm for the Hopfield model with two patterns

The so-called swapping algorithm was designed to simulate from spin glass distributions, among others. In this note we show that it mixes rapidly, in a very simple disordered system, the Hopfield model with two patterns.     

Lagrangian dynamics on an infinite-dimensional torus; a Weak KAM theorem

The space L²(0,1) has a natural Riemannian structure on the basis of which we introduce an L²(0,1)-infinite-dimensional torus T. For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential.     

Time dependent critical fluctuations of a one dimensional local mean field model

Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.     

A LOWER BOUND FOR FIRST EIGENVALUE WITH MIXED BOUNDARY CONDITIOIN

Let M be an n-dimensional compact Riemannian manifold with or without boundary,and its Ricci curvature RicM≥n- 1. The paper obtains an inequality for the first eigenvalue η1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz,Reilly, Escobar and Xia. It is also proved that η1≥ n for certain n-dimensional compact Riemannian manifolds with boundary,which is an extension of the work of Cheng,Li and Yau.     

The random-cluster model on a homogeneous tree

Summary The random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1q2, the percolation probability in the maximal random-cluster measure is continuous inp, while forq>2 it has a discontinuity at the critical valuep=p _c (q). It is also shown that forq>2, there is nonuniqueness of random-cluster measures for an entire interval of values ofp. The latter result is in sharp contrast to what happens on the integer lattice Z ^d .Research partially supported by a grant from the Royal Swedish Academy of Sciences     

Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials

We give an explicit bound for the Wasserstein distance with quadratic cost between the solutions of the Boltzmann and Landau equations in the case of soft and Coulomb potentials. This gives an explicit rate of convergence for the grazing collisions limit. Our result is local in time for very soft and Coulomb potentials and global in time for moderately soft potentials.     

Mathematical properties of a kinetic transport model for carriers and phonons in semiconductors

We present studies on the mathematical properties of a multigroup formulation of the Bloch–Boltzmann–Peierls equations. The considered model equations are based on a general carrier dispersion law and contain the full quantum statistics of both the carriers and the phonons. Moreover, the transport model allows the investigation of particle distributions with arbitrary anisotropy with respect to the main direction. We prove the boundedness of the solution according to the Pauli principle and study the conservational properties of the multigroup equations. In addition, the existence of a Lyapounov functional to the proposed model equations is proved and expressions for the equilibrium solution are given. Numerical results are presented for the stationary state distributions of a coupled system of electrons and longitudinal optical phonons in GaAs.     

Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes

We prove the hydrostatics of boundary driven gradient exclusion processes, Fick’s law and we present a simple proof of the dynamical large deviations principle which holds in any dimension.     

Kinetic limits for a class of interacting particle systems

Summary We study one dimensional particle systems in which particles travel as independent random walks and collide stochastically. The collision rates are chosen so that each particle experiences finitely many collisions per unit time. We establish the kinetic limit and derive the discrete Boltzmann equation for the macroscopic particle density.     

The linear problem of a vibrator in a subsonic boundary layer

The subsonic flow over a flat plate with a fitted to it triangular vibrator which effects harmonic oscillations is studied. The plate and vibrator are assumed heatinsulated, and the vibrator dimensions and oscillation frequency is such that the flow can be defined by equations of the boundary layer with self-induced pressure. The oscillation amplitude is assumed small, making it possible to linearize these equations. The solution is obtained by double application of the Fourier transform with respect to time and longitudinal coordinate. Inverse transformation is achieved by numerical methods. Analysis is carried out for the vibrator frequency ω lower than the critical ω_* predicted by the classical theory of stability. It is shown that vibrator-induced perturbations become rapidly damped upstream. Damping downstream is rapid for ω considerably lower than ω_* and slows down as ω approaches ω_*.     

Large deviation principles for the Hopfield model and the Kac-Hopfield model

Summary We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distribution of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.Work partially supported by the Commission of the European Communities under contract No. SC1-CT91-0695     

Two scales hydrodynamic limit for a model of malignant tumor cells

We consider a model introduced in [S. Luckhaus, L. Triolo, The continuum reaction-diffusion limit of a stochastic cellular growth model, Rend. Acc. Lincei (S.9) 15 (2004) 215-223] with two species (η and ξ) of particles, representing respectively malignant and normal cells. The basic motions of the η particles are independent random walks, scaled diffusively. The ξ particles move on a slower time scale and obey an exclusion rule among themselves and with the η particles. The competition between the two species is ruled by a coupled birth and death process. We prove convergence in the hydrodynamic limit to a system of two reaction-diffusion equations with measure valued initial data.     

Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles

Dissipative solutions [12] of the Euler equations of incompressible fluids are obtained as the hydrodynamic limit of a properly scaled BGK equation. This stability result comes from refined entropy and entropy dissipation bounds. It uses in a crucial way the local conservation laws which are known to hold for weak solutions of this simplified model of the Boltzmann equation.     

A Monte Carlo method for the Broadwell model with relaxation

A Monte Carlo method is proposed for numerical solution of the Broadwell model. Developing a probabilistic interpretation of the equations, the transport and collision parts are treated separately in the method. Particles are advected according to their velocities and collisions are performed between randomly chosen particles. We numerically test the algorithm for a variety of examples. In particular we are interested in situations which generate structures that have nonsmooth fronts. Our simulations show that this Monte Carlo method is capable of capturing the nonlinear regime in presence of shocks and interactions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)