On Viscosity and Fluctuation-Dissipation in Exclusion Processes (original) (raw)
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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.
Large deviations for the boundary driven symmetric simple exclusion process
2003
The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in non-equilibrium, namely for non reversible systems. In this paper we consider a simple example of a non-equilibrium 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.
From diffusive to fractional behavior in a boundary driven exclusion process
2020
The purpose of this article is to study the hydrodynamic limit of the symmetric exclusion process with long jumps and in contact with infinitely extended reservoirs for a particular critical regime. The jumps are given in terms of a transition probability that can have finite or infinite variance and the hydrodynamic equation is a diffusive equation, in the former case, or a fractional equation, in the latter case. In this work we treat the critical case, that is, when the variance is infinite but of logarithm order wrt the system size. This is the case in which there is a transition from diffusive to super-diffusive behavior.
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We prove that the equilibrium fluctuations of the symmetric simple exclusion process in contact with slow boundaries is given by an Ornstein-Uhlenbeck process with Dirichlet, Robin or Neumann boundary conditions depending on the range of the parameter that rules the slowness of the boundaries.
arXiv: Mathematical Physics, 2020
We provide a complete description of the equilibrium fluctuations for diffusive symmetric exclusion processes with long jumps in contact with infinitely extended reservoirs and prove that they behave as generalized Ornstein-Uhlenbeck processes with various boundary conditions, depending mainly on the strength of the reservoirs. On the way, we also give a general statement about uniqueness of the Ornstein-Uhlenbeck process originated by the microscopic dynamics of the underlying interacting particle systems and adapt it to our study.
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Probability Theory and Related Fields, 2007
We examine the long-range exclusion process introduced by Spitzer and studied by Liggett and answer some of the open questions raised by Liggett. In particular, we show the existence of equilibria corresponding to bounded dual harmonic functions and that the process can have right-discontinuous paths at strictly positive times. We also show that "explosions" when they occur, do so at fixed times determined by the initial configuration. Finally, we give an example for which the configuration with all sites occupied is not stable although the rate at which particles arrive at any given site for that configuration is infinite.
A generalized fluctuation-dissipation theorem for the one-dimensional diffusion process
Communications in Mathematical Physics, 1985
The [α, β, y]-Langevin equation describes the time evolution of a real stationary process with T-positivity (reflection positivity) originating in the axiomatic quantum field theory. For this [α,/?,y]-Langevin equation a generalized fluctuation-dissipation theorem is proved. We shall obtain, as its application, a generalized fluctuation-dissipation theorem for the onedimensional non-linear diffusion process, which presents one solution of Ryogo Kubo's problem in physics.