Mário J. de Oliveira

Stochastic dynamics and irreversibility

Random Variables.- Sequence of Independent Variables.- Langevin equation.- Fokker-Planck Equation I.- Fokker-Planck Equation II.- Markov Chains.- Master Equation I.- Master Equation II.- Phase Transitions and Criticality.- Reactive Systems.- Glauber Model.- Systems with Inversion Symmetry.- Systems with Absorbing States.- Population Dynamics.- Probabilistic Cellular automata.- Reaction-Diffusion Processes.- Random Sequential Adsoprtion.- Percolation.

Susceptible-infected-recovered and susceptible-exposed-infected models

Two stochastic epidemic lattice models, the susceptible-infected-recovered and the susceptible-exposed-infected models, are studied on a Cayley tree of coordination number k. The spreading of the disease in the former is found to occur when the infection probability b is larger than bc = k/2(k − 1). In the latter, which is equivalent to a dynamic site percolation model, the spreading occurs when the infection probability p is greater than pc = 1/(k − 1). We set up and solve the time evolution equations for both models and determine the final and time-dependent properties, including the epidemic curve. We show that the two models are closely related by revealing that their relevant properties are exactly mapped into each other when p = b/[k − (k − 1)b]. These include the cluster size distribution and the density of individuals of each type, quantities that have been determined in closed forms.

Irreversible models with Boltzmann–Gibbs probability distribution and entropy production

We analyze irreversible interacting spin models evolving according to a master equation with spin flip transition rates that do not obey detailed balance but obey global balance with a Boltzmann–Gibbs probability distribution. Spin flip transition rates with up–down symmetry are obtained for a linear chain, a square lattice, and a cubic lattice with a stationary state corresponding to the Ising model with nearest neighbor interactions. We show that these irreversible dynamics describes the contact of the system with particle reservoirs that cause a flux of particles through the system. Using a microscopic definition, we determine the entropy production rate of these irreversible models and show that it can be written as a macroscopic bilinear form in the forces and fluxes. Exact expressions for this property are obtained for the linear chain and the square lattice. In this last case the entropy production rate displays a singularity at the phase transition point of the same type as the entropy itself.

Entropy production in linear Langevin systems

We study the entropy production rate in systems described by linear Langevin equations, containing mixed even and odd variables under time reversal. Exact formulas are derived for several important quantities in terms only of the means and covariances of the random variables in question. These include the total rate of change of the entropy, the entropy production rate, the entropy flux rate and the three components of the entropy production. All equations are cast in a way suitable for large-scale analysis of linear Langevin systems. Our results are also applied to different types of electrical circuits, which suitably illustrate the most relevant aspects of the problem.

Critical discontinuous phase transition in the threshold contact process

We analyze a threshold contact process on a square lattice in which particles are created on empty sites with at least two neighboring particles and are annihilated spontaneously. We show by means of Monte Carlo simulations that the process undergoes a discontinuous phase transition at a definite value of the annihilation parameter, in accordance with the Gibbs phase rule, and that the discontinuous transition exhibits critical behavior. The simulations were performed by using boundary conditions in which the sites of the border of the lattice are permanently occupied by particles.

Crystal vs. glass formation in lattice models with many coexisting ordered phases

We present here new evidence that after a quench the planar Potts model on the square lattice relaxes towards a glassy state if the number of states q is larger than four. By extrapolating the finite size data we compute the average energy of these states for the infinite system with periodic boundary conditions, and find that it is comparable with that previously found using fixed boundary conditions. We also report preliminary results on the behaviour of these states in the presence of thermal fluctuations.

Canonical and microcanonical Monte Carlo simulations of lattice-gas mixtures.

We propose strict canonical and microcanonical Monte Carlo algorithms for an arbitrary lattice-gas binary mixture. We deduce formulas that allow us to obtain field quantities over the ensembles in which their conjugate extensive quantities are conserved. As an example, we have considered a lattice-gas mixture that is equivalent to the spin-1 Blume-Emery-Griffiths model [Phys. Rev. A 4, 1071 (1971)]. For a finite system and near a phase coexistence, the field as a function of its extensive conjugate shows a loop that disappears in the thermodynamic limit giving rise to the usual tie line. The first-order phase transition was determined by the use of three criteria.

Glassy behaviour in short-range lattice models without quenched disorder

Abstract We investigate the quenching process in lattice systems with short-range interaction and several crystalline states as ground states. We consider in particular the following systems on a square lattice: hard-core particles with r ground states; and the q-state planar Potts model. The system is initially in a homogeneous disordered phase and relaxes towards a new equilibrium state as soon as the temperature is rapidly lowered. Its evolution can be described numerically by a stochastic process such as the Metropolis algorithm and it is known that for r or q ⩾ d + 1 the final equilibrium state may be polycrystalline, that is not made of a uniform phase. We find that, in addition, r g and q g exist such that for r > r g or q > q g the system evolves towards a glassy state, that is a state in which the ratio of the interaction energy between the different crystalline phases to the total energy of the system never vanishes; moreover we find indications that r g = q g. We infer that, for the Potts model, q = q g corresponds to the crossing from second order to discontinuous transition in the phase diagram of the system.

The chimical potential as an ensemble average

Abstract We show that the chemical potential of a fluid is related to the canonical ensemble average of exp{ ψ kT } where ψ is the energy of interaction between one particle and the others. A similar result holds for the case of a lattice gas model in an ensemble where the number of occupied sites is kept fixed.

Order-Disorder Transition

A crystalline solid has as a fundamental property the ordered structure of its atoms. This structure consists of an array of sites forming a regular three-dimensional periodic lattice. The ordered structure, strictly speaking, does not mean that the atoms are located exactly on the sites of the lattice as they are in constant motion due to thermal agitation. In fact, the average positions of the atoms are the points in space that should be considered as the sites of the ordered lattice. Only for simplicity we say that atoms themselves form the crystal lattice.