Hydrodynamics of stationary non-equilibrium states for some stochastic lattice gas models

We consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior, which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. The unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non-reversible. We study the structure of the stationary measure in the hydrodynamic limit, as the microscopic lattice size goes to infinity. In particular, we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by Fick's law.Dedicated to Res Jost and Arthur WightmanSupported in part by NSF Grants DMR 89-18903 and INT 8521407. H.S. also supported by the Deutsche Forschungsgemeinschaft     

Lattice gas models in contact with stochastic reservoirs: Local equilibrium and relaxation to the steady state

Extending the results of a previous work, we consider a class of discrete lattice gas models in a finite interval whose bulk dynamics consists of stochastic exchanges which conserve the particle number, and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials. We establish here the local equilibrium structure of the stationary measures for these models. Further, we prove as a law of large numbers that the time-dependent empirical density field converges to a deterministic limit process which is the solution of the initial-boundary value problem for a nonlinear diffusion equation.Supported in part by NSF Grants DMR89-18903 and INT85-21407. G.E. and H.S. also supported by the Deutsche Forschungsgemeinschaft     

Condensation in the Inclusion Process and Related Models

We study condensation in several particle systems related to the inclusion process. For an asymmetric one-dimensional version with closed boundary conditions and drift to the right, we show that all but a finite number of particles condense on the right-most site. This is extended to a general result for independent random variables with different tails, where condensation occurs for the index (site) with the heaviest tail, generalizing also previous results for zero-range processes. For inclusion processes with homogeneous stationary measures we establish condensation in the limit of vanishing diffusion strength in the dynamics, and give several details about how the limit is approached for finite and infinite systems. Finally, we consider a continuous model dual to the inclusion process, the so-called Brownian energy process, and prove similar condensation results.     

Spatial Moments for the Nonstationary One-Velocity Problem of Transport Theory with Isotropic Scattering. 2. Plane Instantaneous Source

The spatial distribution of the density of particles emitted by a plane infinite isotropic source with a unit surface particle density is reconstructed for the nonstationary one-velocity problem of transport theory by the method of polynomial expansions with the use of Legendre and Hermite polynomials. The diffusion approximation is examined and the boundaries of the spatiotemporal region in which this approximation is valid are estimated.     

Lattice Gas Model in Random Medium and Open Boundaries: Hydrodynamic and Relaxation to the Steady State

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions d≥3, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a quasilinear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation.     

Directed Transport of Brownian Particles in a Periodic Channel

The transport of Brownian particles in the infinite channel within an external force along the axis of the channel has been studied. In this paper, we study the transport of Brownian particle in the infinite channel within an external force along the axis of the channel and an external force in the transversal direction. In this more sophisticated situation, some property is similar to the simple situation, but some interesting property also appears.     

Directed Transport of Brownian Particles in a Periodic Channel

The transport of Brownian particles in the infinite channel within an external force along the axis of the channel has been studied. In this paper, we study the transport of Brownian particle in the infinite channel within an external force along the axis of the channel and an external force in the transversal direction. In this more sophisticated situation, some property is similar to the simple situation, but some interesting property also appears.     

Equilibrium Fluctuations for Interacting Ornstein-Uhlenbeck Particles

 We study the hydrodynamic density fluctuations of an infinite system of interacting particles on ℝ ^d . The particles interact between them through a two body superstable potential, and with a surrounding fluid in equilibrium through a random viscous force of Ornstein-Uhlenbeck type. The stationary initial distribution is the Gibbs measure associated with the potential and with a given temperature and fugacity. We prove that the time-dependent density fluctuation field converges in law, under diffusive scaling of space and time, to the solution of a linear stochastic partial differential equation driven by white noise. Received: 10 July 2001 / Accepted: 9 September 2002 Published online: 8 January 2003 RID="*" ID="*" We thank J. Fritz for fruitful discussions, in particular about the existence of the infinite dynamics. A special thanks to L. Bertini for help in the proof of the spectral gap estimate (cf. Appendix B). Communicated by H. Spohn     

Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.     

Hydrodynamic Limit for a Boundary Driven Stochastic Lattice Gas Model with Many Conserved Quantities

We prove the hydrodynamic limit for a particle system in which particles may have different velocities. We assume that we have two infinite reservoirs of particles at the boundary: this is the so-called boundary driven process. The dynamics we considered consists of a weakly asymmetric simple exclusion process with collision among particles having different velocities.     

Open systems of splitting particles

Systems with an infinite variety of types of splitting particles are investigated. It is shown that if there is a stationary source of particles but no sink, a steady state with finite density of each species is nevertheless possible due to the infinite number of degrees of freedom. It is demonstrated that the limiting (steady) state is independent of the initial state of the system. Typical features of the steady state, which do not depend on the particle splitting law, are shown.     

Dynamical Large Deviations for a Boundary Driven Stochastic Lattice Gas Model with Many Conserved Quantities

We prove the dynamical large deviations for a particle system in which particles may have different velocities. We assume that we have two infinite reservoirs of particles at the boundary: this is the so-called boundary driven process. The dynamics we considered consists of a weakly asymmetric simple exclusion process with collision among particles having different velocities.     

Population growth in random media. I. Variational formula and phase diagram

We consider an infinite system of particles on the integer lattice Z that: (1) migrate to the right with a random delay, (2) branch along the way according to a random law depending on their position (random medium). In Part I, the first part of a two-part presentation, the initial configuration has one particle at each site. The long-time limit exponential growth rate of the expected number of particles at site 0 (local particle density) does not depend on the realization of the random medium, but only on the law. It is computed in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In earlier work the exponential growth rate of the Cesaro limit of the number of particles per site (global particle density) was studied and a different variational formula was found, but with similar structure, solution, and phases. Combination of the two results reveals an intermediate phase where the population globally survives but locally becomes extinct (i.e., dies out on any fixed finite set of sites).     

Entropy of Non-equilibrium Stationary Measures of Boundary Driven TASEP

We examine the entropy of non-equilibrium stationary states of boundary driven totally asymmetric simple exclusion processes. As a consequence, we obtain that the Gibbs–Shannon entropy of the non equilibrium stationary state converges to the Gibbs–Shannon entropy of the local equilibrium state. Moreover, we prove that its fluctuations are Gaussian, except when the mean displacement of particles produced by the bulk dynamics agrees with the particle flux induced by the density reservoirs in the maximal phase regime.     

Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J _ν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.     

Biased Brownian motion in extremely corrugated tubes

Biased Brownian motion of point-size particles in a three-dimensional tube with varying cross-section is investigated. In the fashion of our recent work, Martens et al. [Phys. Rev. E 83, 051135 (2011)] we employ an asymptotic analysis to the stationary probability density in a geometric parameter of the tube geometry. We demonstrate that the leading order term is equivalent to the Fick-Jacobs approximation. Expression for the higher order corrections to the probability density is derived. Using this expansion orders, we obtain that in the diffusion dominated regime the average particle current equals the zeroth order Fick-Jacobs result corrected by a factor including the corrugation of the tube geometry. In particular, we demonstrate that this estimate is more accurate for extremely corrugated geometries compared with the common applied method using a spatially-dependent diffusion coefficient D(x, f) which substitutes the constant diffusion coefficient in the common Fick-Jacobs equation. The analytic findings are corroborated with the finite element calculation of a sinusoidal-shaped tube.     

Trapping without traps by correlated random walks

Correlated random walk of particles in the infinite cluster of percolating lattices in two dimensions is investigated. For infinitely strong forward correlations (no change of direction except at the boundaries) trapping of the particles in small regions of the infinite cluster is observed.     

Scaling limit for a mechanical system of interacting particles

A system of a large number of classical particles moving on a onedimensional segment with virtually reflecting boundaries is studied. The particles interact with one another through repulsive pair-potential forces and are subjected to resistance proportional to their velocities. Because of the latter it is only the number of particles that is conserved under the evolution of the system. It is proved that in the hydrodynamic limit of diffusion type scaling the normalized counting measure of particle locations converges and its limiting density is governed by a non-linear diffusion equation which in typical cases is of porous media equation type.     

Large deviation function for the current in the open asymmetric simple exclusion process

We consider the one-dimensional asymmetric exclusion process with particle injection and extraction at two boundaries. The model is known to exhibit four distinct phases in its stationary state. We analyze the current statistics at the first site in the low and high density phases. In the limit of infinite system size, we conjecture an exact expression for the current large deviation function.     

Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.Supported by NSF Grant MCS-8301364, ONR Contract N00014-81-K-0012 and a Fellowship from John S. Guggenheim Memorial Foundation