Non-Local Scattering Kernel and the Hydrodynamic Limit

In this paper we study the interaction of a fluid with a wall in the framework of the kinetic theory. We consider the possibility that the fluid molecules can penetrate the wall to be reflected by the inner layers of the wall. This results in a scattering kernel which is a non-local generalization of the classical Maxwell scattering kernel. The proposed scattering kernel satisfies a global mass conservation law and a generalized reciprocity relation. We study the hydrodynamic limit performing a Knudsen layer analysis, and derive a new class of (weakly) nonlocal boundary conditions to be imposed to the Navier–Stokes equations.     

Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exist a family of solutions to the Boltzmann equation globally in time for small Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number ε tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.     

Hydrodynamic Limit for an Arc Discharge at Atmospheric Pressure

In this paper we study a partially ionized plasma that corresponds to an arc discharge at atmospheric pressure. We derive an inviscid hydrodynamic/diffusion limit, characterized by two temperatures, from a system of Boltzmann type transport equations modelling that plasma problem. The original property of this system is that impact ionization is a leading order collisional process. As a consequence, the density of electrons is given in terms of the density of the other species (and its temperature) via a Saha law.     

Hydrodynamic Limit of Coagulation-Fragmentation Type Models of k-Nary Interacting Particles

Hydrodynamic limit of general k-nary mass exchange processes with discrete mass distribution is described by a system of kinetic equations that generalize classical Smoluchovski's coagulation equations and many other models that are intensively studied in the current mathematical and physical literature. Existence and uniqueness theorems for these equations are proved. At last, for k-nary mass exchange processes with k>2 an alternative nondeterministic measure-valued limit (diffusion approximation) is discussed.     

Hydrodynamic Limit for an Evolutional Model of Two-Dimensional Young Diagrams

We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.     

Hydrodynamic Limit for a Zero-Range Process in the Sierpinski Gasket

We consider a system of random walks on graph approximations of the Sierpinski gasket, coupled with a zero-range interaction. We prove that the hydrodynamic limit of this system is given by a nonlinear heat equation on the Sierpinski gasket.     

The Hydrodynamic Limit for Local Mean-Field Dynamics with Unbounded Spins

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean-field interactions. We prove convergence of the empirical measure to the solution of a McKean–Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of Gärtner, Comets and others for bounded spins or strict mean-field interactions.     

Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term plus independent Brownian motions: is the sum of pair potentials, V(r)+ ^d J(r); the second term has the form of a Kac potential with inverse range . Using diffusive hydrodynamic scaling (spatial scale ^–1, temporal scale ^–2) we obtain, in the limit 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.     

Hydrodynamic Limit of Boundary Driven Exclusion Processes with Nonreversible Boundary Dynamics

Using duality techniques, we derive the hydrodynamic limit for one-dimensional, boundary-driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary, for which the classical entropy method fails.     

Hydrodynamic Limit for Weakly Asymmetric Simple Exclusion Processes in Crystal Lattices

We investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices. We construct a suitable scaling limit by using a discrete harmonic map. As we shall observe, the quasi-linear parabolic equation in the limit is defined on a flat torus and depends on both the local structure of the crystal lattice and the discrete harmonic map. We formulate the local ergodic theorem on the crystal lattice by introducing the notion of local function bundle, which is a family of local functions on the configuration space. The ideas and methods are taken from the discrete geometric analysis to these problems. Results we obtain are extensions of ones by Kipnis, Olla and Varadhan to crystal lattices.     

Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles

Two-species condensing zero range processes (ZRPs) are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub-critical densities. We prove the hydrodynamic limit of nearest neighbour mean zero two-species condensing ZRP with bounded local jump rate for sub-critical initial profiles, i.e., for initial profiles whose image is contained in the region of sub-critical densities. The proof is based on H.T. Yau’s relative entropy method, which relies on the existence of sufficiently regular solutions to the hydrodynamic equation. In the particular case of the species-blind ZRP, we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times.     

Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold

Journal of Statistical Physics - We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these...     

Perturbation of Singular Equilibria of Hyperbolic Two-Component Systems: A Universal Hydrodynamic Limit

We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under the Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws:with where is a convex compact polygon in ². The system is typically strictly hyperbolic in the interior of with possible non-hyperbolic degeneracies on the boundary . We consider the case of an isolated singular (i.e. non-hyperbolic) point on the interior of one of the edges of , call it (₀,u₀). We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities (₀,u₀) of the conserved quantities. We prove that for a very rich class of systems, under a proper hydrodynamic limit the propagation of these small perturbations are universally driven by the two-by-two systemwhere the parameter is the only trace of the microscopic structure.The proof relies on the relative entropy method and thus, it is valid only in the regime of smooth solutions of the pde. But there are essential new elements: in order to control the fluctuations of the terms with Poissonian (rather than Gaussian) decay coming from the low density approximations we have to apply refined pde estimates. In particular Lax entropies of these pde systems play a not merely technical key role in the main part of the proof.     

Onsager Relations and Eulerian Hydrodynamic Limit for Systems with Several Conservation Laws

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsager's reciprocity relations must hold. We check validity of these identities whenever a stationary measure with product structure exists. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena. The Onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel. The fact that equilibrium thermodynamic entropy is Lax entropy for the arising Euler equations was noticed earlier in the context of Hamiltonian systems with weak noise, see ref. 7.     

The Hydrodynamic Limit of a Deterministic Particle System with Conservation of Mass and Momentum

We study the hydrodynamic limit of a deterministic one-dimensional particle system with nearest neighbour interaction and an additional regularizing force. Under its evolution mass and momentum are conserved. In the limit with Euler scaling their macroscopic distributions are shown to be governed by the compressible Navier–Stokes equations with a density dependent viscosity.