arXiv subjects

Francesca R. Nardi

Publications and source records attributed to Francesca R. Nardi.

25 records · Page 2Linked to original sources

arXiv2012

Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We introduce the notion of relaxation height in a general energy landscape and we prove sufficient conditions which are valid even in presence of multiple metastable states. We show how these results can be used to approach the problem of multiple metastable states via the use of the modern theories of metastability. We finally apply these general results to the Blume--Capel model for a particular choice of the parameters ensuring the existence of two multiple, and not degenerate in energy, metastable states.

math-ph↗

arXiv2012

Entropy-driven cutoff phenomena

In this paper we present, in the context of Diaconis' paradigm, a general method to detect the cutoff phenomenon. We use this method to prove cutoff in a variety of models, some already known and others not yet appeared in literature, including a chain which is non-reversible w.r.t. its stationary measure. All the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon. In the case of birth- and-death chains this mechanism is fairly well understood; our work is an effort to generalize this picture to more general systems, such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one-dimensional problem. In those situations the drift may be looked for by means of a suitable partitioning of the state space into classes; using a statistical mechanics language it is then possible to set up a kind of energy-entropy competition between the weight and the size of the classes. Under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease.

math-ph↗

arXiv2011

Kawasaki dynamics with two types of particles: stable/metastable configurations and communication heights

This is the second in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and are interested in the transition time to the full box. This transition is triggered by a critical droplet appearing somewhere in the box. In the first paper we identified the parameter range for which the system is metastable, showed that the first entrance distribution on the set of critical droplets is uniform, computed the expected transition time up to and including a multiplicative factor of order one, and proved that the nucleation time divided by its expectation is exponentially distributed, all in the limit of low temperature. These results were proved under three hypotheses, and involve three model-dependent quantities: the energy, the shape and the number of critical droplets. In this second paper we prove the first and the second hypothesis and identify the energy of critical droplets. The paper deals with understanding the geometric properties of subcritical, critical and supercritical droplets, which are crucial in determining the metastable behavior of the system. The geometry turns out to be considerably more complex than for Kawasaki dynamics with one type of particle, for which an extensive literature exists. The main motivation behind our work is to understand metastability of multi- type particle systems.

math.PR↗

arXiv2009

Metastability for a stochastic dynamics with a parallel heat bath updating rule

We consider the problem of metastability for a stochastic dynamics with a parallel updating rule with single spin rates equal to those of the heat bath for the Ising nearest neighbors interaction. We study the exit from the metastable phase, we describe the typical exit path and evaluate the exit time. We prove that the phenomenology of metastability is different from the one observed in the case of the serial implementation of the heat bath dynamics. In particular we prove that an intermediate chessboard phase appears during the excursion from the minus metastable phase toward the plus stable phase.

cond-mat.stat-mech↗

arXiv2009

Competitive nucleation in metastable systems

Metastability is observed when a physical system is close to a first order phase transition. In this paper the metastable behavior of a two state reversible probabilistic cellular automaton with self-interaction is discussed. Depending on the self-interaction, competing metastable states arise and a behavior very similar to that of the three state Blume-Capel spin model is found.

cond-mat.stat-mech↗

arXiv2009

Metastability for reversible probabilistic cellular automata with self--interaction

The problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the Freidlin--Wentzel regime, namely, finite volume, small magnetic field, and small temperature. The model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. %The characterization of the metastable behavior %of a system in the context of parallel dynamics is a very difficult task, %since all the jumps in the configuration space are allowed. Our strategy is based on recent powerful approaches, not needing a complete description of the fixed points of the dynamics, but relying on few model dependent results. We compute the exit time, in the sense of logarithmic equivalence, and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state. We need to supply two model dependent inputs: (1) the communication energy, that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one; (2) a recurrence property stating that for any configuration different from the metastable state there exists a path, starting from such a configuration and reaching a lower energy state, such that its maximal energy is lower than the communication energy.

cond-mat.stat-mech↗

arXiv2009

Competitive nucleation in reversible Probabilistic Cellular Automata

The problem of competitive nucleation in the framework of Probabilistic Cellular Automata is studied from the dynamical point of view. The dependence of the metastability scenario on the self--interaction is discussed. An intermediate metastable phase, made of two flip--flopping chessboard configurations, shows up depending on the ratio between the magnetic field and the self--interaction. A behavior similar to the one of the stochastic Blume--Capel model with Glauber dynamics is found.

cond-mat.stat-mech↗