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Fri, 06 Nov 2009 00:56:50 PST

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Fri, 06 Nov 2009 00:56:50 PST

Stochastic Acceleration in an Inhomogeneous Time Random Force Field

Goudon, T., Rousset, M. — Fri, 06 Nov 2009 00:56:50 PST

This paper studies the asymptotic behavior of a particle with large initial velocity and subject to a force field which is randomly time dependent and inhomogeneous in space. We analyze the diffusive limit of the position–velocity pair under the appropriate space–time rescaling: . Two alternative approaches are proposed. The first one is based on hydrodynamic limits and homogenization techniques for the underlying kinetic equation; the second one is based on homogenization of the random distribution of trajectories. Time randomness is embodied into an underlying Markov process. Space inhomogeneity is modeled by a periodic structure in the first approach, and by a random field in the second one. In the first case, the analysis relies on the dissipation properties of the Markov process, whereas in the second case, the mixing properties of the random field are used. We point out more analogies and differences of the two obtained results.

Minimum Energy Configurations of Classical Charges: Large N Asymptotics

Capet, S., Friesecke, G. — Fri, 06 Nov 2009 00:56:50 PST

We study the minimum energy configurations of N particles in of charge –1 ("electrons") in the potential of M particles of charges Z > 0 ("atomic nuclei"). In a suitable large-N limit, we determine the asymptotic electron distribution explicitly, showing in particular that the number of electrons surrounding each nucleus is asymptotic to the nuclear charge ("screening"). The proof proceeds by establishing, via Gamma-convergence, a coarse-grained variational principle for the limit distribution, which can be solved explicitly.

Semiclassical Resolvent Estimates in Chaotic Scattering

Nonnenmacher, S., Zworski, M. — Fri, 06 Nov 2009 00:56:50 PST

We prove resolvent estimates for semiclassical operators such as – h² + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h^{– M} in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrödinger propagation and to energy decay of solutions to wave equations.

A Coupled System of PDEs and ODEs Arising in Electrocardiograms Modeling

Boulakia, M., Fernandez, M. A., Gerbeau, J.-F., Zemzemi, N. — Tue, 01 Apr 2008 12:57:04 PDT

We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction–diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell–Schaeffer, FitzHugh–Nagumo, Aliev–Panfilov, and McCulloch. Uniqueness is proved in the case of the FitzHugh–Nagumo ionic model. The proof is based on a regularization argument with a Faedo–Galerkin/compactness procedure.

Stochastic Diffeomorphisms and Homogenization of Multiple Integrals

Gloria, A. — Thu, 28 Feb 2008 07:34:33 PST

In [4], Blanc, Le Bris, and Lions have introduced the notion of stochastic diffeomorphism together with a variant of stochastic homogenization theory for linear and monotone elliptic operators. Their proofs rely on the ergodic theorem and on the analysis of the associated corrector equation. In the present article, we provide another proof of their results using the formalism of integral functionals. We also extend the analysis to cover the case of quasiconvex integrands.