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Register a new userComputational strategies for mapping equilibrium phase diagrams
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We survey the portfolio of computational strategies available for tackling the generic problems of phase behavior - free-energy-estimation and coexistence-curve mapping.
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A classical long-range-interacting $N$-particle system relaxes to thermal equilibrium on time scales growing with $N$; in the limit $Nto infty$ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of $N$ and brings the system towards a non-thermal quasi-stationary state. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state forever. For times smaller than the relaxation time the distribution function of the system obeys the collisionless Boltzmann equation, also known as the Vlasov equation. The Vlasov dynamics is invariant under time reversal so that it does not naturally describe a relaxational dynamics. However, as time grows the dynamics affects smaller and smaller scales in phase space, so that observables not depending upon small-scale details appear as relaxed after a short time. Herewith we present an approximation scheme able to describe violent relaxation in a one-dimensional toy-model, the Hamiltonian Mean Field (HMF). The approach described here generalizes the one proposed in G. Giachetti and L. Casetti, J. Stat. Mech.: Theory Exp. 2019, 043201 (2019), that was limited to cold initial conditions, to generic initial conditions, allowing us to to predict non-equilibrium phase diagrams that turn out to be in good agreement with those obtained from the numerical integration of the Vlasov equation.
A framework is presented for carrying out simulations of equilibrium systems in the microcanonical ensemble using annealing in an energy ceiling. The framework encompasses an equilibrium version of simulated annealing, population annealing and hybrid algorithms that interpolate between these extremes. These equilibrium, microcanonical annealing algorithms are applied to the thermal first-order transition in the 20-state, two-dimensional Potts model. All of these algorithms are observed to perform well at the first-order transition though for the system sizes studied here, equilibrium simulated annealing is most efficient.
In this work, we revisited the ZGB model in order to study the behavior of its phase diagram when two well-known random networks play the role of the catalytic surfaces: the Random Geometric Graph and the Erd{o}s-R{e}nyi network. The connectivity and, therefore, the average number of neighbors of the nodes of these networks can vary according to their control parameters, the neighborhood radius $alpha$ and the linking probability $p$, respectively. In addition, the catalytic reactions of the ZGB model are governed by the parameter $y$, the adsorption rate of carbon monoxide molecules on the catalytic surface. So, to study the phase diagrams of the model on both random networks, we carried out extensive steady-state Monte Carlo simulations in the space parameters ($y,alpha$) and ($y,p$) and showed that the continuous phase transition is greatly affected by the topological features of the networks while the discontinuous one remains present in the diagram throughout the interval of study.
We demonstrate that the occurrence of symmetry breaking phase transitions together with the emergence of a local order parameter in classical statistical physics is a consequence of the geometrical structure of probability space. To this end we investigate convex sets generated by expectation values of certain observables with respect to all possible probability distributions of classical q-state spins on a two-dimensional lattice, for several values of q. The extreme points of these sets are then given by thermal Gibbs states of the classical q-state Potts model. As symmetry breaking phase transitions and the emergence of associated order parameters are signaled by the appearance ruled surfaces on these sets, this implies that symmetry breaking is ultimately a consequence of the geometrical structure of probability space. In particular we identify the different features arising for continuous and first order phase transitions and show how to obtain critical exponents and susceptibilities from the geometrical shape of the surface set. Such convex sets thus also constitute a novel and very intuitive way of constructing phase diagrams for many body systems, as all thermodynamically relevant quantities can be very naturally read off from these sets.
We introduce a model of negotiation dynamics whose aim is that of mimicking the mechanisms leading to opinion and convention formation in a population of individuals. The negotiation process, as opposed to ``herding-like or ``bounded confidence driven processes, is based on a microscopic dynamics where memory and feedback play a central role. Our model displays a non-equilibrium phase transition from an absorbing state in which all agents reach a consensus to an active stationary state characterized either by polarization or fragmentation in clusters of agents with different opinions. We show the exystence of at least two different universality classes, one for the case with two possible opinions and one for the case with an unlimited number of opinions. The phase transition is studied analytically and numerically for various topologies of the agents interaction network. In both cases the universality classes do not seem to depend on the specific interaction topology, the only relevant feature being the total number of different opinions ever present in the system.
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