Gibbsian Stationary Non-equilibrium States

We study the structure of stationary non-equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated with functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Particle Model for the Reservoirs in the Simple Symmetric Exclusion Process

Journal of Statistical Physics - In this paper, we will study the long time behavior of the simple symmetric exclusion process in the “channel” $$varLambda _N=[1,N]cap mathbb {N}$$...     

Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process

The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in nonequilibrium, namely for nonreversible systems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.     

Stationary States of Boundary Driven Exclusion Processes with Nonreversible Boundary Dynamics

We prove a law of large numbers for the empirical density of one-dimensional, boundary driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary. The proofs rely on duality techniques.     

Flux Fluctuations in the One Dimensional Nearest Neighbors Symmetric Simple Exclusion Process

Let J(t) be the the integrated flux of particles in the symmetric simple exclusion process starting with the product invariant measure ν _ρ with density ρ. We compute its rescaled asymptotic variance: $$\mathop {\lim }\limits_{t \to \infty } t^{ - 1/2} \mathbb{V}J(t) = \sqrt {2/\pi } (1 - \rho )\rho$$ Furthermore we show that t ^?1/4 J(t) converges weakly to a centered normal random variable with this variance. From these results we compute the asymptotic variance of a tagged particle in the nearest neighbor case and show the corresponding central limit theorem.     

Symmetric Simple Exclusion Process:¶Regularity of the Self-Diffusion Coefficient

We prove that the self-diffusion coefficient of a tagged particle in the symmetric exclusion process in Z ^d , which is in equilibrium at density α, is of class C ^∞ as a function of α in the closed interval [0,1]. The proof provides also a recursive method to compute the Taylor expansion at the boundaries. Received: 6 December 2000 / Accepted: 6 April 2001     

Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition

For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current Q _t during time t through the origin when, in the initial condition, the sites are occupied with density ρ _a on the negative axis and with density ρ _b on the positive axis. All the cumulants of Q _t grow like . In the range where , the decay exp [−Q _t ³/t] of the distribution of Q _t is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities derived recently by Tracy and Widom for exclusion processes on the infinite line. We acknowledge the support of the French Ministry of Education through the ANR BLAN07-2184264 grant.     

Convergence Speed for Simple Symmetric Exclusion: An Explicit Calculation

For the infinite-volume simple symmetric nearest-neighbor exclusion process in one dimension, we investigate the speed of convergence to equilibrium from a particular initial distribution. We use duality to reduce the analysis to that of the two-particle process, which we further reduce to a random walk reflecting rightward at zero, whose generator is self-adjoint on l ²(Z). We obtain the spectral representation of the generator and use asymptotic analysis to show that convergence is slow.     

Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0<ρ<1, is stationary in space and time. Let N_t(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For we prove that the fluctuations of N_t(j) for large t are of order t^1/3 and we determine the limiting distribution function , which is a generalization of the GUE Tracy-Widom distribution. The family of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at through the asymptotics of a Fredholm determinant. is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle. An erratum to this article can be found at     

Large Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process

We consider an open one dimensional lattice gas on sites i=1,..., N, with particles jumping independently with rate 1 to neighboring interior empty sites, the simple symmetric exclusion process. The particle fluxes at the left and right boundaries, corresponding to exchanges with reservoirs at different chemical potentials, create a stationary nonequilibrium state (SNS) with a steady flux of particles through the system. The mean density profile in this state, which is linear, describes the typical behavior of a macroscopic system, i.e., this profile occurs with probability 1 when N. The probability of microscopic configurations corresponding to some other profile (x), x=i/N, has the asymptotic form exp[–N ({})]; is the large deviation functional. In contrast to equilibrium systems, for which _eq({}) is just the integral of the appropriately normalized local free energy density, the we find here for the nonequilibrium system is a nonlocal function of . This gives rise to the long range correlations in the SNS predicted by fluctuating hydrodynamics and suggests similar non-local behavior of in general SNS, where the long range correlations have been observed experimentally.     

Current Fluctuations in the One-Dimensional Symmetric Exclusion Process with Open Boundaries

We calculate the first four cumulants of the integrated current of the one-dimensional symmetric simple exclusion process of N sites with open boundary conditions. For large system size N, the generating function of the integrated current depends on the densities ρ _a and ρ _b of the two reservoirs and on the fugacity z, the parameter conjugated to the integrated current, through a single parameter. Based on our expressions for these first four cumulants, we make a conjecture which leads to a prediction for all the higher cumulants. In the case ρ _a =1 and ρ _b =0, our conjecture gives the same universal distribution as the one obtained by Lee, Levitov, and Yakovets for one-dimensional quantum conductors in the metallic regime.     

Finite Size Corrections to the Large Deviation Function of the Density in the One Dimensional Symmetric Simple Exclusion Process

The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.     

Vortices in the Two-Dimensional Simple Exclusion Process

We show that the fluctuations of the partial current in two dimensional diffusive systems are dominated by vortices leading to a different scaling from the one predicted by the hydrodynamic large deviation theory. This is supported by exact computations of the variance of partial current fluctuations for the symmetric simple exclusion process on general graphs. On a two-dimensional torus, our exact expressions are compared to the results of numerical simulations. They confirm the logarithmic dependence on the system size of the fluctuations of the partial flux. The impact of the vortices on the validity of the fluctuation relation for partial currents is also discussed in an Appendix.     

Current Reservoirs in the Simple Exclusion Process

We consider the symmetric simple exclusion process in the interval [−N,N] with additional birth and death processes respectively on (N−K,N], K>0, and [−N,−N+K). The exclusion is speeded up by a factor N ², births and deaths by a factor N. Assuming propagation of chaos (a property proved in a companion paper, De Masi et al., ) we prove convergence in the limit N→∞ to the linear heat equation with Dirichlet condition on the boundaries; the boundary conditions however are not known a priori, they are obtained by solving a non-linear equation. The model simulates mass transport with current reservoirs at the boundaries and the Fourier law is proved to hold.     

On the Asymmetric Simple Exclusion Process with Multiple Species

In previous work the authors, using the Bethe Ansatz, found for the N-particle asymmetric simple exclusion process on the integers a formula—a sum of multiple integrals—for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations hold.     

On the Convergence of Entropy for Stationary Exclusion Processes with Open Boundaries

We prove that, for a wide class of stochastic lattice gases in contact with reservoirs, despite long-range correlations, the leading-order term of the Gibbs–Shannon entropy in the nonequilibrium stationary state is given by the local equilibrium entropy.     

Rotating,Stationary, Axially Symmetric Spacetimes with Collisionless Matter

The existence of stationary solutions to the Einstein–Vlasov system which are axially symmetric and have non-zero total angular momentum is shown. This provides mathematical models for rotating, general relativistic and asymptotically flat non-vacuum spacetimes. If angular momentum is allowed to be non-zero, the system of equations to solve contains one semilinear elliptic equation which is singular on the axis of rotation. This can be handled very efficiently by recasting the equation as one for an axisymmetric unknown on ${\mathbb{R}^5}$ .     

Gyroscopic Precession and Inertial Forces in Axially Symmetric Stationary Spacetimes

We study the phenomenon of gyroscopic precession and the analogues of inertial forces within the framework of general relativity. Covariant connections between the two are established for circular orbits in stationary spacetimes with axial symmetry. Specializing to static spacetimes, we prove that gyroscopic precession and centrifugal force both reverse at the photon orbits. Simultaneous non-reversal of these in the case of stationary spacetimes is discussed. Further insight is gained in the case of static spacetime by considering the phenomena in a spacetime conformal to the original one. Gravi-electric and gravi-magnetic fields are studied and their relation to inertial forces is established.     

Formulas and Asymptotics for the Asymmetric Simple Exclusion Process

This is an expanded version of a series of lectures delivered by the second author in June, 2009. It describes the results of three of the authors’ papers on the asymmetric simple exclusion process, from the derivation of exact formulas for configuration probabilities, through Fredholm determinant representation, to asymptotics with step initial condition establishing KPZ universality. Although complete proofs are in general not given, at least the main elements of them are.