arXiv subjects

Davide Gabrielli

Publications and source records attributed to Davide Gabrielli.

At least 37 records · Page 2Linked to original sources

arXiv2021

Concurrent Donsker-Varadhan and hydrodynamical large deviations

We consider the weakly asymmetric exclusion process on the $d$-dimensional torus. We prove a large deviations principle for the time averaged empirical density and current in the joint limit in which both the time interval and the number of particles diverge. This result is obtained both by analyzing the variational convergence, as the number of particles diverges, of the Donsker-Varadhan functional for the empirical process and by considering the large time behavior of the hydrodynamical rate function. The large deviations asymptotic of the time averaged current is then deduced by contraction principle. The structure of the minimizers of this variational problem corresponds to the possible occurrence of dynamical phase transitions.

math.PR↗

arXiv2020

Dynamical Phase Transitions for Flows on Finite Graphs

We study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.

cond-mat.stat-mech↗

arXiv2020

Box-ball system: soliton and tree decomposition of excursions

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma in 1990. Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari, Nguyen, Rolla and Wang 2018 proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors have proposed a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In this paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall 2005. A ball configuration distributed as independent Bernoulli variables of parameter $λ<1/2$ is in correspondence with a simple random walk with negative drift $2λ-1$ and infinitely many excursions over the local minima. In this case the authors have proven that the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables. We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.

math.PR↗

arXiv2020

Remarks on the interpolation method

We discuss a generalization of the conditions of validity of the interpolation method for the density of quenched free energy of mean field spin glasses. The condition is written just in terms of the $L^2$ metric structure of the Gaussian random variables. As an example of application we deduce the existence of the thermodynamic limit for a GREM model with infinite branches for which the classic conditions of validity fail.

math-ph↗

arXiv2020

A combinatorial representation for the invariant measure of diffusion processes on metric graphs

We give a generalization to a continuous setting of the classic Markov chain tree Theorem. In particular, we consider an irreducible diffusion process on a metric graph. The unique invariant measure has an atomic component on the vertices and an absolutely continuous part on the edges. We show that the corresponding density at $x$ can be represented by a normalized superposition of the weights associated to metric arborescences oriented toward the point $x$. The weight of each oriented metric arborescence is obtained by the exponential of integrals of the form $\int\frac{b}{σ^2}$ along the oriented edges time a weight for each node determined by the local orientation of the arborescence around the node time the inverse of the diffusion coefficient at $x$. The metric arborescences are obtained cutting the original metric graph along some edges.

math.PR↗

arXiv2019

BBS invariant measures with independent soliton components

The Box-Ball System (BBS) is a one-dimensional cellular automaton in $\{0,1\}^\Z$ introduced by Takahashi and Satsuma \cite{TS}, who also identified conserved sequences called \emph{solitons}. Integers are called boxes and a ball configuration indicates the boxes occupied by balls. For each integer $k\ge1$, a $k$-soliton consists of $k$ boxes occupied by balls and $k$ empty boxes (not necessarily consecutive). Ferrari, Nguyen, Rolla and Wang \cite{FNRW} define the $k$-slots of a configuration as the places where $k$-solitons can be inserted. Labeling the $k$-slots with integer numbers, they define the $k$-component of a configuration as the array $\{ζ_k(j)\}_{j\in \mathbb Z}$ of elements of $\Z_{\ge0}$ giving the number $ζ_k(j)$ of $k$-solitons appended to $k$-slot $j\in \mathbb Z$. They also show that if the Palm transform of a translation invariant distribution $μ$ has independent soliton components, then $μ$ is invariant for the automaton. We show that for each $λ\in[0,1/2)$ the Palm transform of a product Bernoulli measure with parameter $λ$ has independent soliton components and that its $k$-component is a product measure of geometric random variables with parameter $1-q_k(λ)$, an explicit function of $λ$. The construction is used to describe a large family of invariant measures with independent components under the Palm transformation, including Markov measures.

math.PR↗

arXiv2018

Bounds on current fluctuations in periodically driven systems

Small nonequilibrium systems in contact with a heat bath can be analyzed with the framework of stochastic thermodynamics. In such systems, fluctuations, which are not negligible, follow universal relations such as the fluctuation theorem. More recently, it has been found that, for nonequilibrium stationary states, the full spectrum of fluctuations of any thermodynamic current is bounded by the average rate of entropy production and the average current. However, this bound does not apply to periodically driven systems, such as heat engines driven by periodic variation of the temperature and artificial molecular pumps driven by an external protocol. We obtain a universal bound on current fluctuations for periodically driven systems. This bound is a generalization of the known bound for stationary states. In general, the average rate that bounds fluctuations in periodically driven systems is different from the rate of entropy production. We also obtain a local bound on fluctuations that leads to a trade-off relation between speed and precision in periodically driven systems, which constitutes a generalization to periodically driven systems of the so called thermodynamic uncertainty relation. From a technical perspective, our results are obtained with the use of a recently developed theory for 2.5 large deviations for Markov jump processes with time-periodic transition rates.

cond-mat.stat-mech↗

arXiv2018

Fick and Fokker--Planck diffusion law in inhomogeneous media

We discuss diffusion of particles in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un--oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker--Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart of reversibility is encoded into a tensor metrics and a positive function. The Fick's law with inhomogeneous diffusion matrix is obtained in the case when the spatial inhomogeneity is associated exclusively with the edge weights. We discuss also some related properties of the systems like a non-homogeneous Einstein relation and the possibility of uphill diffusion.

cond-mat.stat-mech↗

arXiv2018

Gradient structure and transport coefficients for strong particles

We introduce and study a simple and natural class of solvable stochastic lattice gases. This is the class of \emph{Strong Particles}. The name is due to the fact that when they try to jump to an occupied site they succeed pushing away a pile of particles. For this class of models we explicitly compute the transport coefficients. We also discuss some generalizations and the relations with other classes of solvable models.

cond-mat.stat-mech↗

arXiv2017

Gibbsian stationary non equilibrium states

We study the structure of stationary non equilibrium states for interacting particle systems from a microscopic viewpoint. In particular we discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. Since divergence free flows are characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. According to this, for example, the instantaneous current of any interacting particle system on a finite torus can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components are associated to functions on the configuration space. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field and we use this decomposition to construct models having a fixed invariant measure.

cond-mat.stat-mech↗

arXiv2017

Stochastic monotonicity from an Eulerian viewpoint

Stochastic monotonicity is a well known partial order relation between probability measures defined on the same partially ordered set. Strassen Theorem establishes equivalence between stochastic monotonicity and the existence of a coupling compatible with respect to the partial order. We consider the case of a countable set and introduce the class of \emph{finitely decomposable flows} on a directed acyclic graph associated to the partial order. We show that a probability measure stochastically dominates another probability measure if and only if there exists a finitely decomposable flow having divergence given by the difference of the two measures. We illustrate the result with some examples. In fluid theory the Lagrangian description follows the trajectories of the particles while the Eulerian one observes the local flow. A coupling gives a Lagrangian description of the transference plan of mass while a flow gives an Eulerian one

math.PR↗

arXiv2017

The energy of the alphabet model

We call \emph{Alphabet model} a generalization to N types of particles of the classic ABC model. We have particles of different types stochastically evolving on a one dimensional lattice with an exchange dynamics. The rates of exchange are local but under suitable conditions the dynamics is reversible with a Gibbsian like invariant measure with long range interactions. We discuss geometrically the conditions of reversibility on a ring that correspond to a gradient condition on the graph of configurations or equivalently to a divergence free condition on a graph structure associated to the types of particles. We show that much of the information on the interactions between particles can be encoded in associated \emph{Tournaments} that are a special class of oriented directed graphs. In particular we show that the interactions of reversible models are corresponding to strongly connected tournaments. The possible minimizers of the energies are in correspondence with the Hamiltonian cycles of the tournaments. We can then determine how many and which are the possible minimizers of the energy looking at the structure of the associated tournament. As a byproduct we obtain a probabilistic proof of a classic Theorem of Camion \cite{Camion} on the existence of Hamiltonian cycles for strongly connected tournaments. Using these results we obtain in the case of an equal number of k types of particles new representations of the Hamiltonians in terms of translation invariant $k$-body long range interactions. We show that when $k=3,4$ the minimizer of the energy is always unique up to translations. Starting from the case $k=5$ it is possible to have more than one minimizer. In particular it is possible to have minimizers for which particles of the same type are not joined together in single clusters.

cond-mat.stat-mech↗

arXiv2016

Totally asymmetric limit for models of heat conduction

We consider one dimensional weakly asymmetric boundary driven models of heat conduction. In the cases of a constant diffusion coefficient and of a quadratic mobility we compute the quasi-potential that is a non local functional obtained by the solution of a variational problem. This is done using the dynamic variational approach of the macroscopic fluctuation theory \cite{MFT}. The case of a concave mobility corresponds essentially to the exclusion model that has been discussed in \cite{Lag,CPAM,BGLa,ED}. We consider here the convex case that includes for example the Kipnis-Marchioro-Presutti (KMP) model and its dual (KMPd) \cite{KMP}. This extends to the weakly asymmetric regime the computations in \cite{BGL}. We consider then, both microscopically and macroscopically, the limit of large external fields. Microscopically we discuss some possible totally asymmetric limits of the KMP model. In one case the totally asymmetric dynamics has a product invariant measure. Another possible limit dynamics has instead a non trivial invariant measure for which we give a duality representation. Macroscopically we show that the quasi-potentials of KMP and KMPd, that for any fixed external field are non local, become local in the limit. Moreover the dependence on one of the external reservoirs disappears. For models having strictly positive quadratic mobilities we obtain instead in the limit a non local functional having a structure similar to the one of the boundary driven asymmetric exclusion process.

cond-mat.stat-mech↗

arXiv2015

Macroscopic fluctuation theory

Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. We give a detailed review of this theory including its main predictions and most relevant applications.

cond-mat.stat-mech↗

arXiv2015

Large deviations of the empirical flow for continuous time Markov chains

We consider a continuous time Markov chain on a countable state space and prove a joint large deviation principle for the empirical measure and the empirical flow, which accounts for the total number of jumps between pairs of states. We give a direct proof using tilting and an indirect one by contraction from the empirical process.

math.PR↗

arXiv2013

Large deviation principles for nongradient weakly asymmetric stochastic lattice gases

We consider a lattice gas on the discrete d-dimensional torus $(\mathbb{Z}/N\mathbb{Z})^d$ 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.

math-ph↗

arXiv2012

Which random walks are cyclic?

A cyclic random walk is a random walk whose transition probabilities/rates can be written as a superposition of the empirical measures of a family of finite cycles. This identifies a convex set of models. We discuss the problem of characterization of cyclic random walks in some special cases showing that it is related to several remarkable and classical results. In particular we introduce the notion of balanced measure and show that a translation invariant random walk on $\mathbb Z^d$ is cyclic if and only if its transition probability is balanced. The characterization of the extremal elements is obtained using the Carathéodory's Theorem of convex analysis. We then show that a random walk on a finite set is cyclic if and only if at every vertex the outgoing flux of the transition graph is equal to the ingoing flux. The extremal elements are characterized by the Birkhoff-Von-Neumann Theorem. Finally we consider the discrete torus and discuss when the cyclic decomposition can be done using only homotopically trivial cycles or elementary cycles associated to edges and two dimensional faces. While in one dimension this is equivalent to require some geometric properties of a discrete vector field associated to the transition rates, in two dimension this is not the case. In particular we give a simple characterization of the polyhedron of the rates admitting a cyclic decomposition with elementary cycles. The proof is based on a discrete Hodge decomposition, elementary homological algebra and the Helly's Theorem of convex analysis. Finally we discuss a natural discretization procedure of smooth divergence free continuous vector fields and an application to random walks in random environments.

math.PR↗

arXiv2010

Action functional and quasi-potential for the Burgers equation in a bounded interval

Consider the viscous Burgers equation $u_t + f(u)_x = ε\, u_{xx}$ on the interval $[0,1]$ with the inhomogeneous Dirichlet boundary conditions $u(t,0) = ρ_0$, $u(t,1) = ρ_1$. The flux $f$ is the function $f(u)= u(1-u)$, $ε>0$ is the viscosity, and the boundary data satisfy $0<ρ_0<ρ_1<1$. We examine the quasi-potential corresponding to an action functional, arising from non-equilibrium statistical mechanical models, associated to the above equation. We provide a static variational formula for the quasi-potential and characterize the optimal paths for the dynamical problem. In contrast with previous cases, for small enough viscosity, the variational problem defining the quasi potential admits more than one minimizer. This phenomenon is interpreted as a non-equilibrium phase transition and corresponds to points where the super-differential of the quasi-potential is not a singleton.

math.AP↗