Conservative Particle Systems Research Articles

Large deviations of Kac’s conservative particle system and energy nonconserving solutions to the Boltzmann equation: A counterexample to the predicted rate function

Large deviations of Kac’s conservative particle system and energy nonconserving solutions to the Boltzmann equation: A counterexample to the predicted rate function

Traction chain networks: Insights beyond force chain networks for non-spherical particle systems

Force chain networks are generally applied in granular materials to gain insight into inter-particle granular contact. For conservative spherical particle systems, i.e. frictionless and undamped, force chains are information complete due to symmetries resulting from isotropy and constant curvature of a sphere. In fact, for conservative spherical particle systems, given the geometry and material, the force chain network uniquely defines the contact state that includes elastic forces, penetration distance, overlap volume, contact areas and contact pressures in a particle system. This is, however, not the case for conservative non-spherical particle systems. The reason is that a force chain network is not sufficient to uniquely define the contact state in a conservative non-spherical particle system. Additional information is required to define the contact state of non-spherical granular systems. Traction chain networks are proposed to complement force chain networks for the improved quantification of the state of contact of a granular system.

Invariant measures of mass migration processes

We introduce the “mass migration process” (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\geq 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We derive necessary and sufficient conditions for the existence of translation invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation invariant, invariant probability measures. We also consider attractiveness properties (through couplings), condensation phenomena, and their links for MMP. We illustrate our results on many examples; we prove the coexistence of condensation and attractiveness in one of them.

From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules

We prove a quantitative result of convergence of a conservative stochastic particle system to the solution of the homogeneous Landau equation for hard potentials. There are two main difficulties: (i) the known stability results for this class of Landau equations concern regular solutions and seem difficult to extend to study the rate of convergence of some empirical measures; (ii) the conservativeness of the particle system is an obstacle for (approximate) independence. To overcome (i), we prove a new stability result for the Landau equation for hard potentials concerning very general measure solutions. Due to (ii), we have to couple, our particle system with some non independent nonlinear processes, of which the law solves, in some sense, the Landau equation. We then prove that these nonlinear processes are not so far from being independent. Using finally some ideas of Rousset [25], we show that in the case of Maxwell molecules, the convergence of the particle system is uniform in time.

A stochastic Burgers equation from a class of microscopic interactions

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^{-\gamma})$ for $1/2<\gamma\leq1$, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein-Uhlenbeck process. However, at the critical weak asymmetry when $\gamma=1/2$, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp "Boltzmann-Gibbs" estimate which improves on earlier bounds.

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.

Stationary product measures for conservative particle systems and ergodicity criteria

We study conservative particle systems on W^S, where S is countable and W = {0, ..., N} or the natural numbers. The rate of a particle moving from site x to site y is given by p(x,y) b(eta_x, eta_y), where eta_z is the number of particles at site z. Under assumptions on b and the assumption that p is finite range, which allow for the exclusion, zero range and misanthrope processes, we show exactly what the stationary product measures are. Furthermore we show that a stationary measure mu is ergodic if and only if the tail sigma algebra of the partial sums is trivial under mu. This is a consequence of a more general result on interacting particle systems that shows that a stationary measure is ergodic if and only if the sigma algebra of sets invariant under the transformations of the process is trivial. We apply this result combined with a coupling argument on the stationary product measures to determine which product measures are ergodic. For the case that W is finite this gives a complete characterisation. For the case that W is the set of natural numbers we show that for nearly all functions b a stationary product measure is ergodic if and only if it is supported by configurations with an infinite amount of particles. We show that this picture is not complete, we give an example of a system where b is such that there is a stationary product measure which is not ergodic, even though it concentrates on configurations with an infinite number of particles.

Immortal particle for a catalytic branching process

We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth coefficients for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive are given. In the Brownian motions case the domain of evolution can be non-smooth, including Lipschitz, with integrable Martin kernel. The results are valid for an arbitrary number of particles and non-uniform redistribution after branching. Additionally, with probability one, it is shown that only one ancestry line survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times.

Couplings, attractiveness and hydrodynamics for conservative particle systems

L’attractivité est un outil fondamental d’étude des systèmes à une infinité de particules en interaction; la construction du couplage de base est la méthode habituelle pour démontrer cette propriété (par exemple pour l’exclusion simple). Le processus couplé markovien (ξt, ζt)t≥0 obtenu vérifie: (A) si ξ0≤ζ0 (coordonnée par coordonnée), alors pour tout t≥0, ξt≤ζt p.s. Nous considérons dans cet article des systèmes de particules conservatifs sur ℤd qui généralisent le processus des misanthropes en ce que, à chaque transition, k particules peuvent sauter d’un site x vers un autre site y, avec k≥1. Ces modèles incluent l’exclusion simple, où k=1, mais, au-delà de cette valeur, le couplage de base n’est plus valide et il faut une autre construction. Nous obtenons des conditions nécessaires et suffisantes pour l’attractivité sur les taux de transition; nous construisons un processus couplé markovien qui à la fois satisfait (A), et fait décroitre les discrépances entre ses deux marginales. Nous déterminons les probabilités invariantes et invariantes par translation extrémales sous des conditions générales d’irréductibilité. Nous appliquons nos résultats à des exemples incluant un modèle d’exclusion asymétrique à deux espèces avec conservation de la charge (où k≤2) issu d’une dynamique d’interfaces ‘solid-on-solid’, et un modèle de batons (où k n’est pas borné) en correspondance avec un processus de Hammersley–Aldous–Diaconis discret généralisé. Nous obtenons la limite hydrodynamique de ces deux modèles unidimensionnels.

Hydrodynamic limit for a particle system with degenerate rates

We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of kinetically constrained lattice gases (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists blocked configurations which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.

Euler hydrodynamics of one-dimensional attractive particle systems

We consider attractive irreducible conservative particle systems on $\mathbb{Z}$, without necessarily nearest-neighbor jumps or explicit invariant measures. We prove that for such systems, the hydrodynamic limit under Euler time scaling exists and is given by the entropy solution to some scalar conservation law with Lipschitz-continuous flux. Our approach is a generalization of Bahadoran et al. [Stochastic Process. Appl. 99 (2002) 1--30], from which we relax the assumption that the process has explicit invariant measures.

Blockage hydrodynamics of one-dimensional driven conservative systems

We consider an arbitrary one-dimensional conservative particle system with finite-range interactions and finite site capacity, governed on the hydrodynamic scale by a scalar conservation law with Lipschitz-continuous flux h. A finite-size perturbation restricts the local current to some maximum value ϕ. We show that the perturbed hydrodynamic behavior is entirely determined by ϕ if inf(h;ϕ) is first nondecreasing and then nonincreasing, which we believe is a necessary condition.

Existence of quasi-stationary measures for asymmetric attractive particle systems on $\ZZ^d$

We show the existence of nontrivial quasi-stationary measures for conservative attractive particle systems on $\ZZ^d$ conditioned on avoiding an increasing local set $\A$. Moreover, we exhibit a sequence of measures $\{\nu_n\}$, whose $\omega$-limit set consists of quasi-stationary measures. For zero-range processes, with stationary measure $\nur$, we prove the existence of an $L^2(\nur)$ nonnegative eigenvector for the generator with Dirichlet boundary on $\A$, after establishing a priori bounds on the $\{\nu_n\}$.

Exclusion Processes with Degenerate Rates: Convergence to Equilibrium and Tagged Particle

Stochastic lattice gases with degenerate rates, namely conservative particle systems where the exchange rates vanish for some configurations, have been introduced as simplified models for glassy dynamics. We introduce two particular models and consider them in a finite volume of size $\ell$ in contact with particle reservoirs at the boundary. We prove that, as for non--degenerate rates, the inverse of the spectral gap and the logarithmic Sobolev constant grow as $\ell^2$. It is also shown how one can obtain, via a scaling limit from the logarithmic Sobolev inequality, the exponential decay of a macroscopic entropy associated to a degenerate parabolic differential equation (porous media equation). We analyze finally the tagged particle displacement for the stationary process in infinite volume. In dimension larger than two we prove that, in the diffusive scaling limit, it converges to a Brownian motion with non--degenerate diffusion coefficient.

Quasi-Stationary measures for conservative dynamics in the infinite lattice

We study quasi-stationary measures for conservative particle systems in the in finite lattice. Existence of quasi-stationary measures is established for a fairly general class of reversible systems. For the special cases of a system of independent random walks and the symmetric simple exclusion process, it is shown that qualitative features of quasi-stationary measures change drastically with dimension.

On extremal measures for conservative particle systems

It is well known that the exclusion, zero-range and misanthrope particle systems possess families of invariant measures due to the mass conservation property. Although these families have been classified a great deal, a full characterization of their extreme points is not available. In this article, we consider an approach to the study of this classification. One of the results in this note is that the zero-range product invariant measures, ∏ i∈ S μ α(·) , for an infinite countable set S, under mild conditions, are identified as extremal for α(·)∈ H ZR where μ α( i) ( k)= Z( α( i)) −1 α( i) k / g(1)⋯ g( k) with g and Z the rate function and normalization respectively, and H ZR is the set of invariant measures for the transition probability p.