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The main focus of this book is on different topics in probability theory, partial differential equations and kinetic theory, presenting some of the latest developments in these fields. It addresses mathematical problems concerning applications in physics, engineering, chemistry and biology that were presented at the Third International Conference on Particle Systems and Partial Differential Equations, held at the University of Minho, Braga, Portugal in December 2014.

The purpose of the conference was to bring together prominent researchers working in the fields of particle systems and partial differential equations, providing a venue for them to present their latest findings and discuss their areas of expertise. Further, it was intended to introduce a vast and varied public, including young researchers, to the subject of interacting particle systems, its underlying motivation, and its relation to partial differential equations.

This book will appeal to probabilists, analysts and those mathematicians whose work involves topics in mathematical physics, stochastic processes and differential equations in general, as well as those physicists whose work centers on statistical mechanics and kinetic theory.



On Linear Hypocoercive BGK Models

We study hypocoercivity for a class of linear and linearized BGK models for discrete and continuous phase spaces. We develop methods for constructing entropy functionals that prove exponential rates of relaxation to equilibrium. Our strategies are based on the entropy and spectral methods, adapting Lyapunov’s direct method (even for “infinite matrices” appearing for continuous phase spaces) to construct appropriate entropy functionals. Finally, we also prove local asymptotic stability of a nonlinear BGK model.

Franz Achleitner, Anton Arnold, Eric A. Carlen

Hydrodynamic Limit of Quantum Random Walks

We discuss here the hydrodynamic limit of independent quantum random walks evolving on $$\mathbb {Z}$$Z. As main result, we obtain that the time evolution of the local equilibrium is governed by the convolution of the chosen initial profile with a rescaled version of the limiting probability density obtained in the law of large numbers for a single quantum random walk.

Alexandre Baraviera, Tertuliano Franco, Adriana Neumann

Sub-shock Formation in Reacting Gas Mixtures

The shock-wave structure problem is investigated for a gas mixture of four species, undergoing a reversible bimolecular reaction, modelled by a 10 moment Grad closure of reactive Boltzmann equations. The presence of jump discontinuities within the shock structure solution is discussed, the supersonic regime is characterized, and the critical values of Mach number allowing the formation of sub-shocks in the field variables of one or more components of the mixture are pointed out.

Marzia Bisi, Fiammetta Conforto, Giorgio Martalò

Compactness of Linearized Kinetic Operators

This article reviews various results on the compactness of the linearized Boltzmann operator and of its generalization to mixtures of non-reactive monatomic gases.

Laurent Boudin, Francesco Salvarani

Asymptotics for FBSDES with Jumps and Connections with Partial Integral Differential Equations

It is our intention to survey the asymptotic study of a certain class of coupled forward-backward stochastic differential equations (FBSDEs for short) when the noise terms in the forward diffusion have small intensities that converge to zero. The system of FBSDEs discussed can be used to give a probabilistic representation for the solution in the viscosity sense of an associated system of partial-integral differential equations (PIDEs) with a terminal condition. The asymptotic study of this PIDE is done probabilistically using the FBSDE system. Secondly we present a large deviations principle for the laws of the forward and backward processes of the stochastic system.

André de Oliveira Gomes

Entropy Dissipation Estimates for the Landau Equation: General Cross Sections

We present here an extension to the case of general cross sections of the lower bound obtained in Desvillettes (J Funct Anal 269:1359–1403, 2015, [14]) on the entropy dissipation of Landau’s collision kernel (in the case of soft potentials, including the Coulomb potential). We also simplify somewhat the proof of the lower bound proposed in Desvillettes (J Funct Anal 269:1359–1403, 2015, [14]).

Laurent Desvillettes

The Boltzmann Equation over : Dispersion Versus Dissipation

The Boltzmann equation of the kinetic theory of gases involves two competing processes. Dissipation—or entropy production—due to the collisions between gas molecules drives the gas towards local thermodynamic (Maxwellian) equilibrium. If the spatial domain is the Euclidean space $${{\mathbb R}^{{\mathrm {D}}}}$$RD, the ballistic transport of gas molecules between collisions results in a dispersion effect which enhances the rarefaction of the gas, and offsets the effect of dissipation. The competition between these two effects leads to a scattering regime for the Boltzmann equation over $${{\mathbb R}^{{\mathrm {D}}}}$$RD with molecular interaction satisfying Grad’s angular cutoff assumption. The present paper reports on results in this direction obtained in collaboration with Bardos, Gamba and Levermore [arxiv:1409.1430] and discusses a few open questions related to this work.

François Golse

The Gradient Flow Approach to Hydrodynamic Limits for the Simple Exclusion Process

We present a new approach to prove the macroscopic hydrodynamic behaviour for interacting particle systems, and as an example we treat the well-known case of the symmetric simple exclusion process (SSEP). More precisely, we characterize any possible limit of its empirical density measures as solutions to the heat equation by passing to the limit in the gradient flow structure of the particle system.

Max Fathi, Marielle Simon

Symmetries and Martingales in a Stochastic Model for the Navier-Stokes Equation

A stochastic description of solutions of the Navier-Stokes equation is investigated. These solutions are represented by laws of finite dimensional semi-martingales and characterized by a weak Euler-Lagrange condition. A least action principle, related to the relative entropy, is provided. Within this stochastic framework, by assuming further symmetries, the corresponding invariances are expressed by martingales, stemming from a weak Noether’s theorem.

Rémi Lassalle, Ana Bela Cruzeiro

Convergence of Diffusion-Drift Many Particle Systems in Probability Under a Sobolev Norm

In this paper we develop a new martingale method to show the convergence of the regularized empirical measure of many particle systems in probability under a Sobolev norm to the corresponding mean field PDE. Our method works well for the simple case of Fokker Planck equation and we can estimate a lower bound of the rate of convergence. This method can be generalized to more complicated systems with interactions.

Jian-Guo Liu, Yuan Zhang

From Market Data to Agent-Based Models and Stochastic Differential Equations

A survey of results from 2008 to 2014 on the construction of a stochastic market model, from the empirical data to its modelling interpretation and proof of mathematical consistency (no-arbitrage and completeness).

R. Vilela Mendes

Global Asymptotic Stability of a General Nonautonomous Cohen-Grossberg Model with Unbounded Amplification Functions

For a class of nonautonomous differential equations with infinite delay, we give sufficient conditions for the global asymptotic stability of an equilibrium point. This class is general enough to include, as particular cases, the most of famous neural network models such as Cohen-Grossberg, Hopfield, and bidirectional associative memory. It is relevant to notice that here we obtain global stability criteria without assuming bounded amplification functions. As illustrations, results are applied to several concrete models studied in some earlier publications and new global stability criteria are given.

José J. Oliveira

Phase Transitions and Coarse-Graining for a System of Particles in the Continuum

We revisit the proof of the liquid-vapor phase transition for systems with finite-range interaction by Lebowitz et al. (J. Stat. Phys. 94(5–6), 955–1025, 1999 [1]) and extend it to the case where we additionally include a hard-core interaction to the Hamiltonian. We establish the phase transition for the mean field limit and then we also prove it when the interaction range is long but finite, by perturbing around the mean-field theory. A key step in this procedure is the construction of a density (coarse-grained) model via cluster expansion. In this note we present the overall result but we mainly focus on this last issue.

Elena Pulvirenti, Dimitrios Tsagkarogiannis

Modelling of Systems with a Dispersed Phase: “Measuring” Small Sets in the Presence of Elliptic Operators

When modelling systems with a dispersed phase involving elliptic operators, as is the case of the Stokes or Navier-Stokes problem or the heat equation in a bounded domain, the geometrical structure of the space occupied by the dispersed phase enters in the homogenization process through its capacity, a quantity which can be used to define the equivalence classes in $$H^1$$H1. We shall review the relationship between capacity and homogenization terms in the limit when the number of inclusions becomes large, focusing in particular on the situation where the distribution of inclusions is not necessarily too regular (i.e. it is not periodic).

Valeria Ricci

Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond

We review some results concerning the derivation of the Boltzmann equation starting from the many-body classical Hamiltonian dynamics. In particular, the celebrated paper by Lanford III [21] and the more recent papers [13, 23] are discussed.

Chiara Saffirio

Duality Relations for the Periodic ASEP Conditioned on a Low Current

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of currents, parametrized by the driving strength $$s_K$$sK, $$K \ge 1$$K≥1, we prove duality relations which arise from the quantum algebra $$U_q[\mathfrak {gl}(2)]$$Uq[gl(2)] symmetry of the generator of the process with reflecting boundary conditions. Using these duality relations we prove on microscopic level a travelling-wave property of the conditioned process for a family of shock-antishock measures for $$N>K$$N>K particles: If the initial measure is a member of this family with K microscopic shocks at positions $$(x_1,\dots ,x_K)$$(x1,⋯,xK), then the measure at any time $$t>0$$t>0 of the process with driving strength $$s_K$$sK is a convex combination of such measures with shocks at positions $$(y_1,\dots ,y_K)$$(y1,⋯,yK), which can be expressed in terms of K-particle transition probabilities of the conditioned ASEP with driving strength $$s_N$$sN.

G. M. Schütz
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