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Register a new userFluctuations in interacting particle systems with memory
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Authors
Rosemary J. Harris
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We consider the effects of long-range temporal correlations in many-particle systems, focusing particularly on fluctuations about the typical behaviour. For a specific class of memory dependence we discuss the modification of the large deviation principle describing the probability of rare currents and show how superdiffusive behaviour can emerge. We illustrate the general framework with detailed calculations for a memory-dependent version of the totally asymmetric simple exclusion process as well as indicating connections to other recent work.
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We propose a method to calculate the large deviations of current fluctuations in a class of stochastic particle systems with history-dependent rates. Long-range temporal correlations are seen to alter the speed of the large deviation function in analogy with long-range spatial correlations in equilibrium systems. We give some illuminating examples and discuss the applicability of the Gallavotti-Cohen fluctuation theorem.
Driven particles in presence of crowded environment, obstacles or kinetic constraints often exhibit negative differential mobility (NDM) due to their decreased dynamical activity. We propose a new mechanism for complex many-particle systems where slowing down of certain {it non-driven} degrees of freedom by the external field can give rise to NDM. This phenomenon, resulting from inter-particle interactions, is illustrated in a pedagogical example of two interacting random walkers, one of which is biased by an external field while the same field only slows down the other keeping it unbiased. We also introduce and solve exactly the steady state of several driven diffusive systems, including a two species exclusion model, asymmetric misanthrope and zero-range processes, to show explicitly that this mechanism indeed leads to NDM.
This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in stationary states of open systems. The outcome of this approach is a theory connecting the non equilibrium thermodynamics to the transport coefficients via a variational principle. This leads ultimately to a functional derivative equation of Hamilton-Jacobi type for the non equilibrium free energy in which local thermodynamic variables are the independent arguments. In the first part of the paper we give a detailed introduction to the microscopic dynamics considered, while the second part, devoted to the macroscopic properties, illustrates many consequences of the Hamilton-Jacobi equation. In both parts several novelties are included.
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We consider the asymptotic behavior of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein-Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot-Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier-Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein-Uhlenbeck process. The method relies on the martingale approach and the Donsker-Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of large deviation principle.
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