Small noise asymptotic of the Gallavotti-Cohen functional for diffusion processes
Abstract.
We consider, for a diffusion process in $\bb R^n$, the Gallavotti-Cohen functional, defined as the empirical power dissipated in a time interval by the non-conservative part of the drift. We prove a large deviation principle in the limit in which the noise vanishes and the time interval diverges. The corresponding rate functional, which satisfies the fluctuation theorem, is expressed in terms of a variational problem on the classical Freidlin-Wentzell functional.
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Abstract.
Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced by Bertini et al., we analyze a Lyapunov functional for such equation which gives the large deviations asymptotics of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functional, whose minimizer is given by the unique stationary solution, as the length of the interval diverges. We focus on boundary data corresponding to a standing wave solution to the Burgers equation in the whole line. In this case, the limiting functional has in fact a one-parameter family of minimizers and we analyze the so-called development by Γ-convergence; this amounts to compute the sharp asymptotic cost corresponding to a given shift of the stationary solution.
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Large deviation principles for non gradient weakly asymmetric stochastic lattice gases
Abstract.
We consider a lattice gas on the discrete $d$-dimensional torus $(\bb Z/N \bb Z)^d$ with a generic translation invariant, finite range interaction satisfying a uniform strong mixing condition. The lattice gas performs a Kawasaki dynamics in the presence of a weak external field $E/N$. We show that, under diffusive rescaling, the hydrodynamic behavior of the lattice gas is described by a nonlinear driven diffusion equation. We then prove the associated dynamical large deviation principle. Under suitable assumptions on the external field (e.g. $E$ constant), we finally analyze the variational problem defining the quasi-potential and characterize the optimal exit trajectory. From these results we deduce the asymptotic behavior of the stationary measures of the stochastic lattice gas, which are not explicitly known. In particular, when the external field $E$ is constant, we prove a stationary large deviation principle for the empirical density and show that the rate function does not depend on $E$.
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Boundary effects in the gradient theory of phase transitions
Abstract.
We consider the van der Waals' free energy functional, with scaling parameter $\epsilon$, in the plane domain $\mathbb{R}_+\times \mathbb{R}_+$, with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries $\mathbb{R}_+ \times\{0\}$ and $\mathbb{R}_+\times\{\infty\}$, and free boundary conditions on $\{\infty\}\times\mathbb{R}_+$. Finally, the datum on $\{0\}\times \mathbb{R}_+$ is chosen in such a way that the interface between the pure phases is pinned at some point $(0,y)$. We show that there exists a critical scaling, $y=y_\epsilon$, such that, as $\epsilon\to 0$, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations.
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On the dynamical behavior of the ABC model
Abstract.
We consider the ABC dynamics, with equal density of the three species, on the discrete ring with $N$ sites. In this case, the process is reversible with respect to a Gibbs measure with a mean field interaction that undergoes a second order phase transition. We analyze the relaxation time of the dynamics and show that at high temperature it grows at most as $N^2$ while it grows at least as $N^3$ at low temperature.
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