Search

We examine two analytical characterisation of the metastable behavior of a Markov chain. The first one expressed in terms of its transition probabilities, and the second one in terms of its large deviations rate functional. Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Under a hypothesis on the jump rates, we prove the existence of times-scales $\theta^{(p)}_n$ and probability measures with disjoint supports $\pi^{(p)}_j$, $j\in S_p$, $1\le p \le q$, such that (a) $\theta^{(1)}_n \to \infty$, $\theta^{(k+1)}_n/\theta^{(k)}_n \to \infty$, (b) for all $p$, $x\in V$, $t>0$, starting from $x$, the distribution of $X^{(n)}_{t \theta^{(p)}_n}$ converges, as $n\to\infty$, to a convex combination of the probability measures $\pi^{(p)}_j$. The weights of the convex combination naturally depend on $x$ and $t$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under the same hypothesis on the jump rates and assuming, furthermore, that the process is reversible, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/\theta^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$ which take finite values only at convex combinations of the measures $\pi^{(p)}_j$: $I^{(p)}(\mu) < \infty$ if, and only if, $\mu = \sum_{j\in S_p} \omega_j\, \pi^{(p)}_j$ for some probability measure $\omega$ in $S_p$.

We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.

We consider an irreducible continuous time Markov chain on a finite state space and with time periodic jump rates and prove the joint large deviation principle for the empirical measure and flow and the joint large deviation principle for the empirical measure and current. By contraction we get the large deviation principle of three types of entropy production flow. We derive some Gallavotti-Cohen duality relations and discuss some applications., Comment: 37 pages. corrected version

In the context of driven diffusive systems, for thermodynamic transformations over a large but finite time window, we derive an expansion of the energy balance. In particular, we characterize the transformations which minimize the energy dissipation and describe the optimal correction to the quasi-static limit. Surprisingly, in the case of transformations between homogeneous equilibrium states of an ideal gas, the optimal transformation is a sequence of inhomogeneous equilibrium states., Comment: arXiv admin note: text overlap with arXiv:1404.6466

We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained in \cite{BFG}. By improving such results we also show, under additional assumptions, that the LDP holds with the strong L^1 topology on the space of currents. We deduce a general version of the Gallavotti-Cohen (GC) symmetry for the current field and show that it implies the so-called fluctuation theorem for the GC functional. We also analyze the large deviation properties of generalized empirical currents associated to a fundamental basis in the cycle space, which, as we show, are given by the first class homological coefficients in the graph underlying the Markov chain. Finally, we discuss in detail some examples., Comment: 38 pages, 2 figures

We consider the Cahn-Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity $\epsilon^{\frac 12}$, and we investigate the effect of the noise, as $\epsilon \to 0$, on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given $\gamma< \frac 23$, with probability going to one as $\epsilon \to 0$, the solution remains close to a front for times of the order of $\epsilon^{-\gamma}$, and we study the fluctuations of the front in this time scaling. They are given by a one dimensional continuous process, self similar of order $\frac 14$ and non Markovian, related to a fractional Brownian motion and for which a couple of representations are given., Comment: 33 pages

We recover the Donsker-Varadhan large deviations principle (LDP) for the empirical measure of a continuous time Markov chain on a countable (finite or infinite) state space from the joint LDP for the empirical measure and the empirical flow proved in [2]., Comment: 14 pages, no figure

In a recent work, Bodineau and Derrida analyzed the phase fluctuations in the ABC model. In particular, they computed the asymptotic variance and, on the basis of numerical simulations, they conjectured the presence of a drift, which they guessed to be an antisymmetric function of the three densities. By assuming the validity of the fluctuating hydrodynamic approximation, we prove the presence of such a drift, providing an analytical expression for it. This expression is then shown to be an antisymmetric function of the three densities. The antisymmetry of the drift can also be inferred from a symmetry property of the underlying microscopic dynamics.

Nonequilibrium stationary states of thermodynamic systems dissipate a positive amount of energy per unit of time. If we consider transformations of such states that are realized by letting the driving depend on time, the amount of energy dissipated in an unbounded time window becomes then infinite. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work performed along any given transformation. We then show that the renormalized work satisfies a Clausius inequality and prove that equality is achieved for very slow transformations, that is in the quasi static limit. We finally connect the renormalized work to the quasi potential of the macroscopic fluctuation theory, that gives the probability of fluctuations in the stationary nonequilibrium ensemble.

We consider a macroscopic system in contact with boundary reservoirs and/or under the action of an external field. We discuss the case in which the external forcing depends explicitly on time and drives the system from a nonequilibrium state to another one. In this case the amount of energy dissipated along the transformation becomes infinite when an unbounded time window is considered. Following the general proposal by Oono and Paniconi and using results of the macroscopic fluctuation theory, we give a natural definition of a renormalized work. We then discuss its thermodynamic relevance by showing that it satisfies a Clausius inequality and that quasi static transformations minimize the renormalized work. In addition, we connect the renormalized work to the quasi potential describing the fluctuations in the stationary nonequilibrium ensemble. The latter result provides a characterization of the quasi potential that does not involve rare fluctuations.

We consider finite range Gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi--invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely analytical Gibbs fields; in this context the tree decay of semi--invariants has been proven via analyticity arguments. However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called Griffiths' phase when analyticity arguments fail.

We analyze the Block Averaging Transformation applied to the two--dimensional Ising model in the uniqueness region. We discuss the Gibbs property of the renormalized measure and the convergence of renormalized potential under iteration of the map. It turns out that for any temperature $T$ higher than the critical one $T_c$ the renormalized measure is strongly Gibbsian, whereas for $TT_c$ and in a weak sense for $T

In this paper we develop a general theory which provides a unified treatment of two apparently different problems. The weak Gibbs property of measures arising from the application of Renormalization Group maps and the mixing properties of disordered lattice systems in the Griffiths' phase. We suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely, large regions are widely separated. We then show how it is possible to construct a convergent multi-scale cluster expansion.

We analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature. We prove weak Gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense. Since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem we have to use a scale-adapted expansion. Moreover, such a model below the critical temperature resembles a disordered system in presence of Griffiths' singularity. Then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached.

We consider a two-dimensional Ising model with random i.i.d. nearest-neighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a convergent cluster expansion with probability one. The associated polymers are defined on a sequence of increasing scales; in particular the convergence of the above expansion implies the infinite differentiability of the free energy but not its analyticity. The basic tools in the proof are a general theory of graded cluster expansions and a stochastic domination of the disorder.

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.

We consider the van der Waals' free energy functional, with a scaling small parameter epsilon, in the plane domain given by the first quadrant, and inhomogeneous Dirichlet boundary conditions. The boundary data are chosen in such a way that the interface between the pure phases tends to be horizontal and is pinned at some point on the y-axis which approaches zero as epsilon converges to zero. We show that there exists a critical scaling for the pinning point, such that, as the small parameter epsilon tends to zero, 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., Comment: 20 pages, 1 figure

In previous papers we have introduced a natural nonequilibrium free energy by considering the functional describing the large fluctuations of stationary nonequilibrium states. While in equilibrium this functional is always convex, in nonequilibrium this is not necessarily the case. We show that in nonequilibrium a new type of singularities can appear that are interpreted as phase transitions. In particular, this phenomenon occurs for the one-dimensional boundary driven weakly asymmetric exclusion process when the drift due to the external field is opposite to the one due to the external reservoirs, and strong enough., Comment: 10 pages, 2 figures

In this paper we present a self-contained macroscopic description of diffusive systems interacting with boundary reservoirs and under the action of external fields. The approach is based on simple postulates which are suggested by a wide class of microscopic stochastic models where they are satisfied. The description however does not refer in any way to an underlying microscopic dynamics: the only input required are transport coefficients as functions of thermodynamic variables, which are experimentally accessible. The basic postulates are local equilibrium which allows a hydrodynamic description of the evolution, the Einstein relation among the transport coefficients, and a variational principle defining the out of equilibrium free energy. Associated to the variational principle there is a Hamilton-Jacobi equation satisfied by the free energy, very useful for concrete calculations. Correlations over a macroscopic scale are, in our scheme, a generic property of nonequilibrium states. Correlation functions of any order can be calculated from the free energy functional which is generically a non local functional of thermodynamic variables. Special attention is given to the notion of equilibrium state from the standpoint of nonequilibrium., Comment: 21 pages

We investigate the joint description of the interest-rate term stuctures of Italy and an AAA-rated European country by mean of a --here proposed-- correlated CIR-like bivariate model where one of the state variables is interpreted as a benchmark risk-free rate and the other as a credit spread. The model is constructed by requiring the strict positivity of interest rates and the asymptotic decoupling of the joint distribution of the two state variables on a long time horizon. The second condition is met by imposing the reversibility of the process with respect to a product measure, the first is then implemented by using the tools of potential theory. It turns out that these conditions select a class of non-affine models, out of which we choose one that is quadratic in the two state variables both in the drift and diffusion matrix. We perform a numerical analysis of the model by investigating a cross section of the term structures comparing the results with those obtained with an uncoupled bivariate CIR model.

Experiments show that macroscopic systems in a stationary nonequilibrium state exhibit long range correlations of the local thermodynamic variables. In previous papers we proposed a Hamilton-Jacobi equation for the nonequilibrium free energy as a basic principle of nonequilibrium thermodynamics. We show here how an equation for the two point correlations can be derived from the Hamilton-Jacobi equation for arbitrary transport coefficients for dynamics with both external fields and boundary reservoirs. In contrast with fluctuating hydrodynamics, this approach can be used to derive equations for correlations of any order. Generically, the solutions of the equation for the correlation functions are non-trivial and show that long range correlations are indeed a common feature of nonequilibrium systems. Finally, we establish a criterion to determine whether the local thermodynamic variables are positively or negatively correlated in terms of properties of the transport coefficients., Comment: 4 pages

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., Comment: 36 pages

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling., Comment: 33 pages

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy., Comment: 34 pages

We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences., Comment: Extended version of the lectures given by G. Jona-Lasinio at the 9th Brazilian school of Probability, August 2005

We study current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, we prove a large deviation principle for the empirical current in the symmetric simple exclusion process with rate functional I. We then estimate the asymptotic probability of a fluctuation of the average current over a large time interval and show that the corresponding rate function can be obtained by solving a variational problem for the functional I. For the symmetric simple exclusion process the minimizer is time independent so that this variational problem can be reduced to a time independent one. On the other hand, for other models the minimizer is time dependent. This phenomenon is naturally interpreted as a dynamical phase transition., Comment: 26 pages

We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles. More precisely, we establish a large deviation principle for a space-time fluctuation $j$ of the empirical current with a rate functional $\mc I (j)$. We then estimate the probability of a fluctuation of the average current over a large time interval; this probability can be obtained by solving a variational problem for the functional $\mc I $. We discuss several possible scenarios, interpreted as dynamical phase transitions, for this variational problem. They actually occur in specific models. We finally discuss the time reversal properties of $\mc I$ and derive a fluctuation relationship akin to the Gallavotti-Cohen theorem for the entropy production., Comment: 36 Pages, No figure

We study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach to density fluctuations developed in previous articles. More precisely, we derive large deviation estimates for the space--time fluctuations of the empirical current which include the previous results. Large time asymptotic estimates for the fluctuations of the time average of the current, recently established by Bodineau and Derrida, can be derived in a more general setting. There are models where we have to modify their estimates and some explicit examples are introduced., Comment: 4 pages, LaTeX, Changed content

We generalize to non equilibrium states Onsager's minimum dissipation principle. We also interpret this principle and some previous results in terms of optimal control theory. Entropy production plays the role of the cost necessary to drive the system to a prescribed macroscopic configuration.

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., Comment: 29 pages

Stochastic lattice gases with degenerate rates, namely conservative particle systems where the exchange rates vanish for some configurations, have been introduced as simplified models for glassy dynamics. We introduce two particular models and consider them in a finite volume of size $\ell$ in contact with particle reservoirs at the boundary. We prove that, as for non--degenerate rates, the inverse of the spectral gap and the logarithmic Sobolev constant grow as $\ell^2$. It is also shown how one can obtain, via a scaling limit from the logarithmic Sobolev inequality, the exponential decay of a macroscopic entropy associated to a degenerate parabolic differential equation (porous media equation). We analyze finally the tagged particle displacement for the stationary process in infinite volume. In dimension larger than two we prove that, in the diffusive scaling limit, it converges to a Brownian motion with non--degenerate diffusion coefficient., Comment: 25 pages, 3 figures

We formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Within this theory we derive the following results: the modification of the Onsager-Machlup theory in the SNS; a general Hamilton-Jacobi equation for the macroscopic entropy; a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of systems; an H theorem for the entropy. We discuss in detail two models of stochastic boundary driven lattice gases: the zero range and the simple exclusion processes. In the first model the invariant measure is explicitly known and we verify the predictions of the general theory. For the one dimensional simple exclusion process, as recently shown by Derrida, Lebowitz, and Speer, it is possible to express the macroscopic entropy in terms of the solution of a non linear ordinary differential equation; by using the Hamilton-Jacobi equation, we obtain an logically independent derivation of this result., Comment: 34 pages

In this paper we formulate a dynamical fluctuation theory for stationary non equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic regime and is verified explicitly in stochastic models of interacting particles. In our theory a crucial role is played by the time reversed dynamics. Our results include the modification of the Onsager-Machlup theory in the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a non equilibrium, non linear fluctuation dissipation relation valid for a wide class of systems.

In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite range lattice gases, when suitable strong mixing conditions are satisfied. Using block decimation procedure, cluster expansion (like in [HK]) and detailed comparison between statistical ensembles, we are able to prove Gibbsianess and convergence to a trivial (i.e. Gaussian and product) fixed point. Our results apply to 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field., Comment: Plain TeX, 68 pages, 10 PostScript figures

We consider the scaling limits for a one-dimensional random growth model, the weakly asymmetric single step Solid-on-Solid process. We show that the fluctuation field, if considered in an appropriate (long) space-time scale, solves the Kardar-Parisi-Zhang equation., Comment: 4 pages, RevTeX (twocolumn), no figures