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Cedric Bernardin

Publications and source records attributed to Cedric Bernardin.

At least 19 recordsLinked to original sources
arXiv2026

Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics

We investigate a boundary-driven Ginzburg-Landau dynamics with long-range interactions. In the hydrodynamic limit, the macroscopic evolution is governed by a fractional heat equation with Dirichlet boundary conditions, while the corresponding stationary profile is characterized by a fractional Laplace equation. We establish a dynamical large deviations principle for the empirical measure and derive the associated stationary large deviations principle for the non-equilibrium steady state, which can be computed semi-explicitly. We further show that the stationary rate function coincides with the quasi-potential associated with the dynamical large deviations functional.

math.PR↗
arXiv2025

Spikes in Poissonian quantum trajectories

We consider the dynamics of a continuously monitored qubit in the limit of strong measurement rate where the quantum trajectory is described by a stochastic master equation with Poisson noise. Such limits are expected to give rise to quantum jumps between the pointer states associated with the non-demolition measurement. A surprising discovery in earlier work [Tilloy et al., Phys. Rev. A 92, 052111 (2015)] on quantum trajectories with Brownian noise was the phenomena of spikes observed in between the quantum jumps. Here, we show that spikes are observed also for Poisson noise. We consider three cases where the non-demolition is broken by adding, to the basic strong measurement dynamics, either unitary evolution or thermal noise or additional measurements. We present a complete analysis of the spike and jump statistics for all three cases using the fact that the dynamics effectively corresponds to that of stochastic resetting. We provide numerical results to support our analytic results.

quant-ph↗
arXiv2021

Equilibrium fluctuations for diffusive symmetric exclusion with long jumps and infinitely extended reservoirs

We provide a complete description of the equilibrium fluctuations for diffusive symmetric exclusion processes with long jumps in contact with infinitely extended reservoirs and prove that they behave as generalized Ornstein-Uhlenbeck processes with various boundary conditions, depending mainly on the strength of the reservoirs. On the way, we also give a general statement about uniqueness of the Ornstein-Uhlenbeck process originated by the microscopic dynamics of the underlying interacting particle systems and adapt it to our study.

math-ph↗
arXiv2020

Quantum Dynamics under continuous projective measurements: non-Hermitian description and the continuous space limit

The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schrodinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented.

quant-ph↗
arXiv2019

A Microscopic Derivation Of Coupled Spde's With A Kpz Flavor

We consider an interacting particles system composed of a Hamiltonian part and perturbed by a conservative stochastic noise so that the full system conserves two quantities: energy and volume. The Hamiltonian part is regulated by a scaling parameter vanishing in the limit. We study the form of the fluctuations of these quantities at equilibrium and derive coupled stochastic partial differential equations with a KPZ flavor.

math-ph↗
arXiv2019

Gamma Convergence Approach For The Large Deviations Of The Density In Systems Of Interacting Diffusion Processes

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on , the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle (LDP) with a rate functional. We study the $Γ$-convergence of as $\rightarrow$ 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory (MFT) for diffusive systems.

math.AP↗
arXiv2019

Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes

We consider the fluctuation fields of multi-species weakly-asymmetric zero-range interacting particle systems in one dimension, where the mass density of each species is conserved. Although such fields have been studied in systems with a single species, the multi-species setting is much less understood. Among other results, we show that, when the system starts from stationary states, with a particular property, the scaling limits of the multi-species fluctuation fields, seen in a characteristic traveling frame, solve a coupled Burgers SPDE, which is a formal spatial gradient of a coupled KPZ equation.

math.PR↗
arXiv2018

A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $α$ at the left of the system and $β$ at the right of the system. The strength of the reservoirs is ruled by $κ$N --$θ$ > 0. Here N is the size of the system, $κ$ > 0 and $θ$ $\in$. Our results are valid for $θ$ $\le$ 0. For $θ$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $κ$ and with Dirichlet boundary conditions. Their solutions also depend on $κ$. For $θ$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $θ$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $θ$ = 0 when we send the parameter $κ$ to zero. Indeed, we conjecture that the limiting profile when $κ$ $\rightarrow$ 0 is the one that we should obtain when taking small values of $θ$ > 0.

math-ph↗
arXiv2017

Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps

We consider an exclusion process with long jumps in the box $Λ\_N=\{1, \ldots,N-1\}$, for $N \ge 2$, in contact with infinitely extended reservoirs on its left and on its right. The jump rate is described by a transition probability $p(\cdot)$ which is symmetric, with infinite support but with finite variance. The reservoirs add or remove particles with rate proportional to $κN^{-θ}$, where $κ>0$ and $θ\in\mathbb R$. If $θ>0$ (resp. $θ<0$) the reservoirs add and fastly remove (resp. slowly remove) particles in the bulk. According to the value of $θ$ we prove that the time evolution of the spatial density of particles is described by some reaction-diffusion equations with various boundary conditions.

math.PR↗
arXiv2017

Density large deviations for multidimensional stochastic hyperbolic conservation laws

We investigate the density large deviation function for a multidimensional conservation law in the vanishing viscosity limit, when the probability concentrates on weak solutions of a hyperbolic conservation law conservation law. When the conductivity and dif-fusivity matrices are proportional, i.e. an Einstein-like relation is satisfied, the problem has been solved in [4]. When this proportionality does not hold, we compute explicitly the large deviation function for a step-like density profile, and we show that the associated optimal current has a non trivial structure. We also derive a lower bound for the large deviation function, valid for a general weak solution, and leave the general large deviation function upper bound as a conjecture.

cond-mat.stat-mech↗
arXiv2016

Weakly harmonic oscillators perturbed by a conservative noise

We consider a chain of weakly harmonic coupled oscillators perturbed by a conservative noise. We show that by tuning accordingly the coupling constant energy can diffuse like a Brownian motion or superdiffuse like a maximally 3/2-stable asymmetric L{é}vy process. For a critical value of the coupling, the energy diffusion is described by a family of L{é}vy processes which interpolates between these two processes.

math.PR↗
arXiv2015

Diffusion of energy in chains of oscillators with bulk noise

These notes are based on a mini-course given during the conference Particle systems and PDE's - II which held at the Center of Mathematics of the University of Minho in December 2013. We discuss the problem of normal and anomalous diffusion of energy in systems of coupled oscillators perturbed by a stochastic noise conserving energy.

math.PR↗
arXiv2014

Green-Kubo formula for weakly coupled system with dynamical noise

We study the Green-Kubo (GK) formula $κ(\varepsilon, ξ)$ for the heat conductivity of an infinite chain of $d$-dimensional finite systems (cells) coupled by a smooth nearest neighbour potential $\varepsilon V$. The uncoupled systems evolve according to Hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength $ξ$. Noting that $κ(\varepsilon, ξ)$ exists and is finite whenever $ξ> 0$, we are interested in what happens when the strength of the noise $ξ\to 0$. For this, we start in this work by formally expanding $κ(\varepsilon, ξ)$ in a power series in $\varepsilon$, $κ(\varepsilon, ξ) = \varepsilon^2 \sum_{n\ge 2} \varepsilon^{n-2} κ_n (ξ)$ and investigating the (formal) equations satisfied by $κ_n (ξ$. We show in particular that $κ_2 (ξ)$ is well defined when no pinning potential is present, and coincides formally with the heat conductivity obtained in the weak coupling (van Hove) limit, where time is rescaled as $\varepsilon^{-2}t$, for the cases where the latter has been established \cite{LO, DL}. For one-dimensional systems, we investigate $κ_2 (ξ)$ as $ξ\to 0$ in three cases: the disordered harmonic chain, the rotor chain and a chain of strongly anharmonic oscillators. Moreover, we formally identify $κ_2 (ξ)$ with the conductivity obtained by having the chain between two reservoirs at temperature $T$ and $T+δT$, in the limit $δT\to 0$, $N \to \infty$, $\varepsilon \to 0$.

cond-mat.stat-mech↗
arXiv2013

A one-dimensional coagulation-fragmentation process with a dynamical phase transition

We introduce a reversible Markovian coagulation-fragmentation process on the set of partitions of $\{1,\ldots,L\}$ into disjoint intervals. Each interval can either split or merge with one of its two neighbors. The invariant measure can be seen as the Gibbs measure for a homogeneous pinning model \cite{cf:GBbook}. Depending on a parameter $λ$, the typical configuration can be either dominated by a single big interval (delocalized phase), or be composed of many intervals of order $1$ (localized phase), or the interval length can have a power law distribution (critical regime). In the three cases, the time required to approach equilibrium (in total variation) scales very differently with $L$. In the localized phase, when the initial condition is a single interval of size $L$, the equilibration mechanism is due to the propagation of two "fragmentation fronts" which start from the two boundaries and proceed by power-law jumps.

math.PR↗
arXiv2013

Harmonic Systems With Bulk Noises

We consider a harmonic chain in contact with thermal reservoirs at different temperatures and subject to bulk noises of different types: velocity flips or self-consistent reservoirs. While both systems have the same covariances in the nonequilibrium stationary state (NESS) the measures are very different. We study hydrodynamical scaling, large deviations, fluctuations, and long range correlations in both systems. Some of our results extend to higher dimensions.

cond-mat.stat-mech↗
arXiv2013

Homogenization results for a linear dynamics in random Glauber type environment

We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green-Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved.

cond-mat.stat-mech↗