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Milton Jara

Publications and source records attributed to Milton Jara.

At least 55 records · Page 3Linked to original sources
arXiv2015

Thermal conductivity in harmonic lattices with random collisions

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.

cond-mat.stat-mech↗
arXiv2015

Abrupt convergence for stochastic small perturbations of one dimensional dynamical systems

We study the cut-off phenomenon for a family of stochastic small perturbations of a one dimensional dynamical system. We will focus in a semi-flow of a deterministic differential equation which is perturbed by adding to the dynamics a white noise of small variance. Under suitable hypothesis on the potential we will prove that the family of perturbed stochastic differential equations present a profile cut-off phenomenon with respect to the total variation distance. We also prove a local cut-off phenomenon in a neighborhood of the local minima (metastable states) of multi-well potential.

math.PR↗
arXiv2015

Superdiffusion of energy in a chain of harmonic oscillators with noise

We consider a one dimensional infinite chain of har- monic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to a fractional diffusion. For a pinned system we prove that energy evolves diffusively, generalizing some of the results of [4].

cond-mat.stat-mech↗
arXiv2015

A stochastic Burgers equation from a class of microscopic interactions

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^{-γ})$ for $1/2<γ\leq1$, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein-Uhlenbeck process. However, at the critical weak asymmetry when $γ=1/2$, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp "Boltzmann-Gibbs" estimate which improves on earlier bounds.

math.PR↗
arXiv2014

From normal diffusion to superdiffusion of energy in the evanescent flip noise limit

We consider a harmonic chain perturbed by an energy conserving noise depending on a parameter $γ$. When $γ$ is of order one, the energy diffuses according to the standard heat equation after a space-time diffusive scaling. On the other hand, when $γ=0$, the energy superdiffuses according to a 3/4 fractional heat equation after a subdiffusive space-time scaling. In this paper, we study the existence of a crossover between these two regimes as a function of $γ$.

math.PR↗
arXiv2014

The Einstein relation for the KPZ equation

We compute the non-universal constants in the KPZ equation in one dimension, in terms of the thermodynamical quantities associated to the underlying microscopic dynamics. In particular, we derive the second-order Einstein relation $λ= \frac{1}{2}\frac{d^2}{dρ^2} χ(ρ) D(ρ)$ for the transport coefficient $λ$ of the KPZ equation, in terms of the conserved quantity $ρ$, the diffusion coefficient $D$ and the static compressibility of the system $χ$.

math.PR↗
arXiv2014

Symmetric exclusion as a random environment: hydrodynamic limits

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.

math.PR↗
arXiv2014

Quadratic Fluctuations of the Simple Exclusion Process

We introduce a two-dimensional, distribution-valued field which we call the quadratic field associated to the one-dimensional Ornstein-Uhlenbeck process. We show that the stationary quadratic fluctuations of the simple exclusion process, when rescaled in the diffusive scaling, converge to this quadratic field. We show that this quadratic field evaluated at the diagonal corresponds to the Wick-renormalized square of the Ornstein-Uhlenbeck process, and we use this new representation in order to prove some small and large-time properties of it.

math.PR↗
arXiv2013

Nonlinear fluctuations of weakly asymmetric interacting particle systems

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.

math.PR↗
arXiv2012

Scaling limits of additive functionals of interacting particle systems

Using the renormalization method introduced in \cite{GJ}, we prove what we call the {\em local} Boltzmann-Gibbs principle for conservative, stationary interacting particle systems in dimension $d=1$. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by-product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation.

math.PR↗
arXiv2012

Crossover to the KPZ equation

We characterize the crossover regime to the KPZ equation for a class of one-dimensional weakly asymmetric exclusion processes. The crossover depends on the strength asymmetry $an^{2-γ}$ ($a,γ>0$) and it occurs at $γ=1/2$. We show that the density field is a solution of an Ornstein-Uhlenbeck equation if $γ\in(1/2,1]$, while for $γ=1/2$ it is an energy solution of the KPZ equation. The corresponding crossover for the current of particles is readily obtained.

math.PR↗
arXiv2010

Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear rate zero-range processes

Nonequilibrium fluctuations of a tagged, or distinguished particle in a class of one dimensional mean-zero zero-range systems with sublinear, increasing rates are derived. In Jara-Landim-Sethuraman (2009), processes with at least linear rates are considered. A different approach to establish a main "local replacement" limit is required for sublinear rate systems, given that their mixing properties are much different. The method discussed also allows to capture the fluctuations of a "second-class" particle in unit rate, symmetric zero-range models.

math.PR↗
arXiv2010

Non-markovian limits of additive functionals of Markov processes

In this paper we consider an additive functional of an observable $V(x)$ of a Markov jump process. We assume that the law of the expected jump time $t(x)$ under the invariant probability measure $π$ of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled limit of the functional is a Mittag-Leffler proces, provided that $Ψ(x):=V(x)t(x)$ is square integrable w.r.t. $π$. When the law of $Ψ(x)$ belongs to a domain of attraction of a stable law the resulting process can be described by a composition of a stable process and the inverse of a subordinator and these processes are not necessarily independent. On the other hand when the singularities of $Ψ(x)$ and $t(x)$ do not overlap with large probability the law of the resulting process has some scaling invariance property. We provide an application of the results to a process that arises in quantum transport theory.

math.PR↗
arXiv2010

Universality of KPZ equation

We introduce the notion of energy solutions of the KPZ equation. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, weakly asymmetric, conservative particle systems with respect to the stationary states are given by energy solutions of the KPZ equation. As a consequence, we prove that the Cole-Hofp solutions are also energy solutions of the KPZ equation.

math.PR↗
arXiv2009

Limit theorems for additive functionals of a Markov chain

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $π$. Let $Ψ$ a function on the state space of the chain, with $α$-tails with respect to $π$, $α\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/α}\sum_n^N Ψ(X_n)$ to a $α$-stable law. ``Martingale approximation'' approach and ``coupling'' approach give two different sets of conditions. We extend these results to continuous time Markov jump processes $X_t$, whose skeleton chain satisfies our assumptions. If waiting time between jumps has finite expectation, we prove convergence of $N^{-1/α}\int_0^{Nt} V(X_s) ds$ to a stable process. In the case of waiting times with infinite average, we prove convergence to a Mittag-Leffler process.

math.PR↗
arXiv2009

Hydrodynamic limit of the exclusion process in inhomogeneous media

We obtain the hydrodynamic limit of a simple exclusion process in an inhomogeneous environment of divergence form. Our main assumption is a suitable version of Gamma-convergence for the environment. In this way we obtain an unified approach to recent works on the field.

math.PR↗