arXiv subjects

Cristian Giardina

Publications and source records attributed to Cristian Giardina.

13 recordsLinked to original sources

arXiv2023

Uphill in reaction-diffusion multi-species interacting particles systems

We study reaction-diffusion processes with multi-species of particles and hard-core interaction. We add boundary driving to the system by means of external reservoirs which inject and remove particles, thus creating stationary currents. We consider the condition that the time evolution of the average occupation evolves as the discretized version of a system of coupled diffusive equations with linear reactions. In particular, we identify a specific one-parameter family of such linear reaction-diffusion systems where the hydrodynamic limit behaviour can be obtained by means of a dual process. We show that partial uphill diffusion is possible for the discrete particle systems on the lattice, whereas it is lost in the hydrodynamic limit.

math-ph↗

arXiv2020

Duality and hidden equilibrium in transport models

A large family of diffusive models of transport that has been considered in the past years admits a transformation into the same model in contact with an equilibrium bath. This mapping holds at the full dynamical level, and is independent of dimension or topology. It provides a good opportunity to discuss questions of time reversal in out of equilibrium contexts. In particular, thanks to the mapping one may define the free-energy in the non-equilibrium states very naturally as the (usual) free energy of the mapped system.

cond-mat.stat-mech↗

arXiv2017

Exact formulas for two interacting particles and applications in particle systems with duality

We consider two particles performing continuous-time nearest neighbor random walk on $\mathbb Z$ and interacting with each other when they are at neighboring positions. Typical examples are two particles in the partial exclusion process or in the inclusion process. We provide an exact formula for the Laplace-Fourier transform of the transition probabilities of the two-particle dynamics. From this we derive a general scaling limit result, which shows that the possible scaling limits are coalescing Brownian motions, reflected Brownian motions, and sticky Brownian motions. In particle systems with duality, the solution of the dynamics of two dual particles provides relevant information. We apply the exact formula to the the symmetric inclusion process, that is self-dual, in the condensation regime. We thus obtain two results. First, by computing the time-dependent covariance of the particle occupation number at two lattice sites we characterize the time-dependent coarsening in infinite volume when the process is started from a homogeneous product measure. Second, we identify the limiting variance of the density field in the diffusive scaling limit, relating it to the local time of sticky Brownian motion.

math.PR↗

arXiv2014

Energy-exchange stochastic models for non-equilibrium

Non-equilibrium steady states are subject to intense investigations but still poorly understood. For instance, the derivation of Fourier law in Hamiltonian systems is a problem that still poses several obstacles. In order to investigate non-equilibrium systems, stochastic models of energy-exchange have been introduced and they have been used to identify universal properties of non-equilibrium. In these notes, after a brief review of the problem of anomalous transport in 1-dimensional Hamiltonian systems, some boundary-driven interacting random systems are considered and the "duality approach" to their rigorous mathematical treatment is reviewed. Duality theory, of which a brief introduction is given, is a powerful technique to deal with Markov processes and interacting particle systems. The content of these notes is mainly based on the papers [10, 11, 12].

math.PR↗

arXiv2011

Simulating rare events in dynamical processes

Atypical, rare trajectories of dynamical systems are important: they are often the paths for chemical reactions, the haven of (relative) stability of planetary systems, the rogue waves that are detected in oil platforms, the structures that are responsible for intermittency in a turbulent liquid, the active regions that allow a supercooled liquid to flow... Simulating them in an efficient, accelerated way, is in fact quite simple. In this paper we review a computational technique to study such rare events in both stochastic and Hamiltonian systems. The method is based on the evolution of a family of copies of the system which are replicated or killed in such a way as to favor the realization of the atypical trajectories. We illustrate this with various examples.

cond-mat.stat-mech↗

arXiv2009

Interaction-Flip Identities in Spin Glasses

We study the properties of fluctuation for the free energies and internal energies of two spin glass systems that differ for having some set of interactions flipped. We show that their difference has a variance that grows like the volume of the flipped region. Using a new interpolation method, which extends to the entire circle the standard interpolation technique, we show by integration by parts that the bound imply new overlap identities for the equilibrium state. As a side result the case of the non-interacting random field is analyzed and the triviality of its overlap distribution proved.

math-ph↗

arXiv2009

On the structure of correlations in the three dimensional spin glasses

We investigate the low temperature phase of three-dimensional Edwards-Anderson model with Bernoulli random couplings. We show that at a fixed value $Q$ of the overlap the model fulfills the clustering property: the connected correlation functions between two local overlaps decay as a power whose exponent is independent of $Q$ for all $0\le |Q| < q_{EA}$. Our findings are in agreement with the RSB theory and show that the overlap is a good order parameter.

cond-mat.dis-nn↗

arXiv2008

Thinking transport as a twist

The determination of the conductivity of a deterministic or stochastic classical system coupled to reservoirs at its ends can in general be mapped onto the problem of computing the stiffness (the `energy' cost of twisting the boundaries) of a quantum-like operator. The nature of the coupling to the reservoirs determines the details of the mechanical coupling of the torque at the ends.

cond-mat.stat-mech↗

arXiv2008

Duality and hidden symmetries in interacting particle systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the "hidden" symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.

math-ph↗

arXiv2008

Spin Glass Identities and the Nishimori Line

For a general spin glass model with asymmetric couplings we prove a family of identities involving expectations of generalized overlaps and magnetizations in the quenched state. Those identities holds pointwise in the Nishimori line and are reached at the rate of the inverse volume while, in the general case, they can be proved in integral average.

cond-mat.dis-nn↗

arXiv2006

Overlap Equivalence in the Edwards-Anderson Model

We study the relative fluctuations of the link overlap and the square standard overlap in the three dimensional Gaussian Edwards-Anderson model with zero external field. We first analyze the correlation coefficient and find that the two quantities are uncorrelated above the critical temperature. Below the critical temperature we find that the link overlap has vanishing fluctuations for fixed values of the square standard overlap and large volumes. Our data show that the conditional variance scales to zero in the thermodynamic limit. This implies that, if one of the two random variables tends to a trivial one (i.e. delta-like distributed), then also the other does and, by consequence, the TNT picture should be dismissed. We identify the functional relation among the two variables using the method of the least squares which turns out to be a monotonically increasing function. Our results show that the two overlaps are completely equivalent in the description of the low temperature phase of the Edwards-Anderson model.

cond-mat.dis-nn↗

arXiv2005

Relative entropy and waiting times for continuous-time Markov processes

For discrete-time stochastic processes, there is a close connection between return/waiting times and entropy. Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one does need a reference measure and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of suitable waiting-times and their fluctuation properties (central limit theorem and large deviation principle).

math.PR↗

arXiv2005

Factorization Properties in the 3D Edwards-Anderson Model

Starting from the study of a linear combination of multi-overlaps which can be rigorously shown to vanish for large systems we numerically analyze the factorization properties of the link-overlaps multi-distribution for the 3D Gaussian Edward-Anderson spin-glass model. We find evidence of a pure factorization law for the multi-correlation functions. For instance the quantity [ - ]/ tends to zero at increasing volumes. We also perform the same analysis for the standard overlap for which instead the lack of factorization persists increasing the size of the system. The necessity of a better understanding of the mutual relation between the two overlaps is pointed out.

cond-mat.dis-nn↗