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.     

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.     

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.     

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” $$varLambda _N=[1,N]cap mathbb {N}$$...     

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 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.     

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.     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Strong Asymmetric Limit of the Quasi-Potential of the Boundary Driven Weakly Asymmetric Exclusion Process

We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. In the case in which the bulk asymmetry is in the same direction as the drift due to the boundary reservoirs, we prove that the quasi-potential can be expressed in terms of the solution to a one-dimensional boundary value problem which has been introduced by Enaud and Derrida [16]. We consider the strong asymmetric limit of the quasi-potential and recover the functional derived by Derrida, Lebowitz, and Speer [15] for the asymmetric exclusion process.     

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.     

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     

Complex Networks from Chaotic Time Series on Riemannian Manifold

Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities.In this work we propose a reliable method for constructing complex networks from chaotic time series.We first estimate the covariance matrices,then a geodesic-based distance between the covariance matrices is introduced.Consequently the network can be constructed on a Riemannian manifold where the nodes and edges correspond to the covariance matrix and geodesic-based distance,respectively.The proposed method provides us with an intrinsic geometry viewpoint to understand the time series.