arXiv subjects

Errico Presutti

Publications and source records attributed to Errico Presutti.

At least 19 recordsLinked to original sources

arXiv2022

Scaling limit of a generalized contact process

We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice $\mathbb{Z}^d$. The states at each lattice site can take values in $0,\ldots,k$. These can be interpreted as neuronal membrane potential, with the state $k$ corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state $k$ corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from $0$ to $k$. The rate at which it climbs is determined by other neurons at state $k$, coupled to it through a Kac-type potential, of range $γ^{-1}$. The hydrodynamic equations are obtained in the limit $γ\rightarrow 0$. Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered.

math.PR↗

arXiv2021

Reservoirs, Fick law and the Darken effect

We study the stationary measures of Ginzburg-Landau (GL) stochastic processes which describe the magnetization flux induced by the interaction with reservoirs. To privilege simplicity to generality we restrict to quadratic Hamiltonians where almost explicit formulas can be derived. We discuss the case where reservoirs are represented by boundary generators (mathematical reservoirs) and compare with more physical reservoirs made by large-infinite systems. We prove the validity of the Fick law away from the boundaries. %and the existence of boundary layers for the mathematical reservoirs. We also obtain in the context of the GL models a mathematical proof of the Darken effect which shows uphill diffusion of carbon in specimen partly doped with the addition of Si.

math-ph↗

arXiv2019

A system of interacting neurons with short term synaptic facilitation

In this paper we present a simple microscopic stochastic model describing short term plasticity within a large homogeneous network of interacting neurons. Each neuron is represented by its membrane potential and by the residual calcium concentration within the cell at a given time. Neurons spike at a rate depending on their membrane potential. When spiking, the residual calcium concentration of the spiking neuron increases by one unit. Moreover, an additional amount of potential is given to all other neurons in the system. This amount depends linearly on the current residual calcium concentration within the cell of the spiking neuron. In between successive spikes, the potentials and the residual calcium concentrations of each neuron decrease at a constant rate. We show that in this framework, short time memory can be described as the tendency of the system to keep track of an initial stimulus by staying within a certain region of the space of configurations during a short but macroscopic amount of time before finally being kicked out of this region and relaxing to equilibrium. The main technical tool is a rigorous justification of the passage to a large population limit system and a thorough study of the limit equation.

math.PR↗

arXiv2019

Stationary states in infinite volume with non zero current

We study the Ginzburg-Landau stochastic models in infinite domains with some special geometry and prove that without the help of external forces there are stationary measures with non zero current in three or more dimensions.

cond-mat.stat-mech↗

arXiv2019

A note on Fick's law with phase transitions

We characterize the non equilibrium stationary states in two classes of systems where phase transitions are present. We prove that the interface in the limit is a plane which separates the two phases.

cond-mat.stat-mech↗

arXiv2019

Fast-reaction limit for Glauber-Kawasaki dynamics with two components

We consider the Kawasaki dynamics of two types of particles under a killing effect on a $d$-dimensional square lattice. Particles move with possibly different jump rates depending on their types. The killing effect acts when particles of different types meet at the same site. We show the existence of a limit under the diffusive space-time scaling and suitably growing killing rate: segregation of distinct types of particles does occur, and the evolution of the interface between the two distinct species is governed by the two-phase Stefan problem. We apply the relative entropy method and combine it with some PDE techniques.

math.PR↗

arXiv2018

Renewal properties of the $d=1$ Ising model

We consider the $d=1$ Ising model with Kac potentials at inverse temperature $β>1$ where mean field predicts a phase transition with two possible equilibrium magnetization $\pm m_β$, $m_β>0$. We show that when the Kac scaling parameter $γ$ is sufficiently small typical spin configurations are described (via a coarse graining) by an infinite sequence of successive plus and minus intervals where the empirical magnetization is "close" to $m_β$ and respectively $-m_β$. We prove that the corresponding marginal of the unique DLR measure is a renewal process.

math-ph↗

arXiv2017

Hydrodynamics of the $N$-BBM process

The Branching Brownian Motions (BBM) are particles performing independent Brownian motions in $\mathbb R$ and each particle at rate 1 creates a new particle at her current position; the newborn particle increments and branchings are independent of the other particles. The $N$-BBM starts with $N$ particles and at each branching time, the leftmost particle is removed so that the total number of particles is $N$ for all times. The $N$-BBM was proposed by Maillard and belongs to a family of processes introduced by Brunet and Derrida. We fix a density $ρ$ with a left boundary $L=\sup\{r\in\mathbb R: \int_r^\infty ρ(x)dx=1\}>-\infty$ and let the initial particle positions be iid continuous random variables with density $ρ$. We show that the empirical measure associated to the particle positions at a fixed time $t$ converges to an absolutely continuous measure with density $ψ(\cdot,t)$, as $N\to\infty$. The limit $ψ$ is solution of a free boundary problem (FBP) when this solution exists. The existence of solutions for finite time-intervals has been recently proved by Lee.

math.PR↗

arXiv2017

Microscopic models for uphill diffusion

We study a system of particles which jump on the sites of the interval $[1,L]$ of $\mathbb Z$. The density at the boundaries is kept fixed to simulate the action of mass reservoirs. The evolution depends on two parameters $λ'\ge 0$ and $λ"\ge 0$ which are the strength of an external potential and respectively of an attractive potential among the particles. When $λ'=λ"= 0$ the system behaves diffusively and the density profile of the final stationary state is linear, Fick's law is satisfied. When $λ'> 0$ and $λ"= 0$ the system models the diffusion of carbon in the presence of silicon as in the Darken experiment: the final state of the system is in qualitative agreement with the experimental one and uphill diffusion is present at the weld. Finally if $λ'=0$ and $λ">0$ is suitably large, the system simulates a vapor-liquid phase transition and we have a surprising phenomenon. Namely when the densities in the reservoirs correspond respectively to metastable vapor and metastable liquid we find a final stationary current which goes uphill from the reservoir with smaller density (vapor) to that with larger density (liquid). Our results are mainly numerical, we have convincing theoretical explanations yet we miss a complete mathematical proof.

cond-mat.stat-mech↗

arXiv2017

Particle models with self sustained current

We present some computer simulations run on a stochastic CA (cellular automaton). The CA simulates a gas of particles which are in a channel, the interval $[1,L]$ in $\mathbb Z$, but also in "reservoirs" $\mathcal R_1$ and $\mathcal R_2$. The evolution in the channel simulates a lattice gas with Kawasaki dynamics with attractive Kac interactions, the temperature is chosen smaller than the mean field critical one. There are also exchanges of particles between the channel and the reservoirs and among reservoirs. When the rate of exchanges among reservoirs is in a suitable interval the CA reaches an apparently stationary state with a non zero current, for different choices of the initial condition the current changes sign. We have a quite satisfactory theory of the phenomenon but we miss a full mathematical proof.

cond-mat.stat-mech↗

arXiv2016

Free boundary problems in PDEs and particle systems

In this volume a theory for models of transport in the presence of a free boundary is developed. Macroscopic laws of transport are described by PDEs. When the system is open, there are several mechanisms to couple the system with the external forces. Here a class of systems where the interaction with the exterior takes place in correspondence of a free boundary is considered. Both continuous and discrete models sharing the same structure are analyzed. In Part I a free boundary problem related to the Stefan Problem is worked out in all details. For this model a new notion of relaxed solution is proposed for which global existence and uniqueness is proven. It is also shown that this is the hydrodynamic limit of the empirical mass density of the associated particle system. In Part II several other models are discussed. The expectation is that the results proved for the basic model extend to these other cases. All the models discussed in this volume have an interest in problems arising in several research fields such as heat conduction, queuing theory, propagation of fire, interface dynamics, population dynamics, evolution of biological systems with selection mechanisms.In general researchers interested in the relations between PDEs and stochastic processes can find in this volume an extension of this correspondence to modern mathematical physics.

math.PR↗

arXiv2016

Latent heat and the Fourier law

We present computer simulations run with a stochastic cellular automaton which describes $d=1$ particle systems connected to reservoirs which keep two different densities at the endpoints. We fix the parameters so that there is a phase transition (of the van der Waals type) and observe that if the densities at the boundaries are metastable then, after a transient, the system reaches an apparently stationary regime where the current flows from the reservoir with smaller density to the one with larger density.

cond-mat.stat-mech↗

arXiv2015

Extinction time for a random walk in a random environment

We consider a random walk with death in $[-N,N]$ moving in a time dependent environment. The environment is a system of particles which describes a current flux from $N$ to $-N$. Its evolution is influenced by the presence of the random walk and in turn it affects the jump rates of the random walk in a neighborhood of the endpoints, determining also the rate for the random walk to die. We prove an upper bound (uniform in $N$) for the survival probability up to time $t$ which goes as $c\exp\{-bN^{-2}t\}$, with $c$ and $b$ positive constants.

math.PR↗

arXiv2015

Exponential rate of convergence in current reservoirs

In this paper, we consider a family of interacting particle systems on $[-N,N]$ that arises as a natural model for current reservoirs and Fick's law. We study the exponential rate of convergence to the stationary measure, which we prove to be of the order $N^{-2}$.

math.PR↗

arXiv2015

Highly anisotropic scaling limits

We consider a highly anisotropic $d=2$ Ising spin model whose precise definition can be found at the beginning of Section 2. In this model the spins on a same horizontal line (layer) interact via a $d=1$ Kac potential while the vertical interaction is between nearest neighbors, both interactions being ferromagnetic. The temperature is set equal to 1 which is the mean field critical value, so that the mean field limit for the Kac potential alone does not have a spontaneous magnetization. We compute the phase diagram of the full system in the Lebowitz-Penrose limit showing that due to the vertical interaction it has a spontaneous magnetization. The result is not covered by the Lebowitz-Penrose theory because our Kac potential has support on regions of positive codimension.

math-ph↗

arXiv2015

Layered systems at the mean field critical temperature

We consider the Ising model on $\mathbb Z\times \mathbb Z$ where on each horizontal line $\{(x,i), x\in \mathbb Z\}$, the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)\sim γJ(γ(x-y))$ at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength $ε$ and prove that for every $ε>0$ the systems exhibits phase transition provided $γ>0$ is small enough.

math.PR↗

arXiv2014

Phase transitions in layered systems

We consider the Ising model on the two-dimensional square lattice where on each horizontal line, called "layer", the interaction is given by a ferromagnetic Kac potential with coupling strength $J_γ(x,y)=γJ(γ(x-y))$, where $J(\cdot)$ is smooth and has compact support; we then add a nearest neighbor ferromagnetic vertical interaction of strength $γ^{A}$ (where $A\ge 2$ is fixed) and prove that for any $β$ (inverse temperature) larger than the mean field critical value there is a phase transition for all $γ$ small enough.

math.PR↗

arXiv2014

Hydrodynamic limit for interacting neurons

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential 1/N . This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

math.PR↗