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A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field, using a Fock space approach

Sébastien Breteaux (2014)

Annales de l’institut Fourier

In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.

A numerical scheme for the quantum Boltzmann equation with stiff collision terms

Francis Filbet, Jingwei Hu, Shi Jin (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010) 7625–7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann equation. To define the quantum Maxwellian...

A numerical scheme for the quantum Boltzmann equation with stiff collision terms⋆

Francis Filbet, Jingwei Hu, Shi Jin (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic-preserving schemes was introduced in [F. Filbet and S. Jin,J. Comput. Phys. 229 (2010) 7625–7648] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own difficulty when applied to the quantum Boltzmann...

A Stochastic Solver of the Generalized Born Model

Robert C. Harris, Travis Mackoy, Marcia O. Fenley (2013)

Molecular Based Mathematical Biology

A stochastic generalized Born (GB) solver is presented which can give predictions of energies arbitrarily close to those that would be given by exact effective GB radii, and, unlike analytical GB solvers, these errors are Gaussian with estimates that can be easily obtained from the algorithm. This method was tested by computing the electrostatic solvation energies (ΔGsolv) and the electrostatic binding energies (ΔGbind) of a set of DNA-drug complexes, a set of protein-drug complexes, a set of protein-protein...

An introduction to probabilistic methods with applications

Pierre Del Moral, Nicolas G. Hadjiconstantinou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This special volume of the ESAIM Journal, Mathematical Modelling and Numerical Analysis, contains a collection of articles on probabilistic interpretations of some classes of nonlinear integro-differential equations. The selected contributions deal with a wide range of topics in applied probability theory and stochastic analysis, with applications in a variety of scientific disciplines, including physics, biology, fluid mechanics, molecular chemistry, financial mathematics and bayesian statistics....

Asymptotic stability of a linear Boltzmann-type equation

Roksana Brodnicka, Henryk Gacki (2014)

Applicationes Mathematicae

We present a new necessary and sufficient condition for the asymptotic stability of Markov operators acting on the space of signed measures. The proof is based on some special properties of the total variation norm. Our method allows us to consider the Tjon-Wu equation in a linear form. More precisely a new proof of the asymptotic stability of a stationary solution of the Tjon-Wu equation is given.

Compensated compactness and time-periodic solutions to non-autonomous quasilinear telegraph equations

Eduard Feireisl (1990)

Aplikace matematiky

In the present paper, the existence of a weak time-periodic solution to the nonlinear telegraph equation U t t + d U t - σ ( x , t , U x ) x + a U = f ( x , t , U x , U t , U ) with the Dirichlet boundary conditions is proved. No “smallness” assumptions are made concerning the function f . The main idea of the proof relies on the compensated compactness theory.

Dynamic von Kármán equations involving nonlinear damping: Time-periodic solutions

Eduard Feireisl (1989)

Aplikace matematiky

In the paper, time-periodic solutions to dynamic von Kármán equations are investigated. Assuming that there is a damping term in the equations we are able to show the existence of at least one solution to the problem. The Faedo-Galerkin method is used together with some basic ideas concerning monotone operators on Orlicz spaces.

Equations of magnetohydrodynamics of compressible fluid: Periodic solutions

Milan Štědrý, Otto Vejvoda (1985)

Aplikace matematiky

The authors prove the global existence and exponential stability of solutions of the given system of equations under the condition that the initial velocities and the external forces are small and the initial density is not far from a constant one. If the external forces are periodic, then solutions periodic with the same period are obtained. The investigated system of equations is a bit non-standard - for example the displacement current in the Maxwell equations is not neglected.

Kac’s chaos and Kac’s program

Stéphane Mischler (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

In this note I present the main results about the quantitative and qualitative propagation of chaos for the Boltzmann-Kac system obtained in collaboration with C. Mouhot in [33] which gives a possible answer to some questions formulated by Kac in [25]. We also present some related recent results about Kac’s chaos and Kac’s program obtained in [34, 23, 13] by K. Carrapatoso, M. Hauray, C. Mouhot, B. Wennberg and myself.

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