Marcio Santos

Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing insights into causality, structure, and hidden patterns within complex systems. In physical systems with many degrees of freedom, such as gases, liquids, and solids, a statistical analysis of these correlations is essential. For a field Ψ(x,t) that depends on spatial position x and time t, it is often necessary to understand the correlation with itself at another position and time Ψ(x_{0},t_{0}). This specific function is called the autocorrelation function. In this context, the autocorrelation function for order-parameter fluctuations, introduced by Fisher [J. Math. Phys. 5, 944 (1964)0022-248810.1063/1.1704197], provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent η is introduced to correct the spatial behavior of the correlation function at T=T_{c}. In recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)2470-004510.1103/PhysRevE.110.L062107] demonstrated that at T_{c} a fractal analysis is necessary for a complete description of the correlation function. In this study, we investigate the fundamental physics and mathematics underlying phase transitions, emphasizing the deep interplay between scaling behavior, critical exponents, and fractal geometry. In particular, we show that the application of modern fractional differentials allows us to write an equation for the correlation function that recovers the correct exponents below the upper critical dimension. We obtain the exact expression for the Fisher exponent η. Furthermore, we examine the Rushbrooke scaling relation, which has been questioned in certain magnetic systems, and, drawing on results from the Ising model, we confirm that both our relations and the Rushbrooke scaling law hold even when d is not an integer.

We investigated the dynamical behavior of the Ziff, Gulari, and Barshad model considering the adsorption rates of the carbon monoxide and oxygen molecules on a catalytic surface as a periodic function of time. For a two-dimensional square lattice and using the Monte Carlo simulation method, we found the stationary states of the model for various values of the amplitude and period of oscillation, taking into account different periodic expressions for the adsorption rates. Depending on the values of the parameters of the model, as well as of the form of the periodic adsorption rates, the stationary states can be active or inactive. For each periodic adsorption rate used in this work we determined a phase diagram in the plane amplitude versus period of oscillation. These phase diagrams can exhibit continuous and/or discontinuous lines separating the active phase from the inactive ones, and a nonexpected new discontinuous transition line separating the two absorbing states. Moreover, we also observed that the area of the active phase shows a strong dependence on the way how the periodic adsorption rates are applied on the system, and in specific situations the phase poisoned by oxygen atoms can even disappear.

In this work we consider the nonequilibrium mechanical and magnetic work performed on a one-dimensional compressible Ising model. In the harmonic approximation we easily integrate the mechanical degrees of freedom of the model, and the resulting effective Hamiltonian depends on two external parameters, the magnetic field and the force applied along the chain. As the model is exactly soluble in one dimension we can determine the free energy difference between two arbitrary thermodynamic states of the system. We show the validity of the Jarzynski equality, which relates the free energy difference between two thermodynamic states of the system and the average work performed by external agents in a finite time, through nonequilibrium paths between the same thermodynamic states. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field and the mechanical force applied to the system.

In this paper we determine the nonequilibrium magnetic work performed on a Ising model and relate it to the fluctuation theorem derived some years ago by Jarzynski. The basic idea behind this theorem is the relationship connecting the free energy difference between two thermodynamic states of a system and the average work performed by an external agent, in a finite time, through nonequilibrium paths between the same thermodynamic states. We test the validity of this theorem by considering the one-dimensional Ising model where the free energy is exactly determined as a function of temperature and magnetic field. We have found that the Jarzynski theorem remains valid for all the values of the rate of variation of the magnetic field applied to the system. We have also determined the probability distribution function for the work performed on the system for the forward and reverse processes and verified that predictions based on the Crooks relation are equally correct. We also propose a method to calculate the lag between the current state of the system and that of the equilibrium based on macroscopic variables. We have shown that the lag increases with the sweeping rate of the field at its final value for the reverse process, while it decreases in the case of the forward process. The lag increases linearly with the size of the chain and with a slope decreasing with the inverse of the rate of variation of the field.

The stationary states of the prisoner's dilemma model are studied on a square lattice taking into account the role of a noise parameter in the decision-making process. Only first neighboring players—defectors and cooperators—are considered in each step of the game. Through Monte Carlo simulations we determined the phase diagrams of the model in the plane noise versus the temptation to defect for a large range of values of the noise parameter. We observed three phases: cooperators and defectors absorbing phases, and a coexistence phase between them. The phase transitions as well as the critical exponents associated with them were determined using both static and dynamical scaling laws.

We study the prisoner's dilemma model with a noisy imitation evolutionary dynamics on directed out-homogeneous and uncorrelated directed random networks. A heterogeneous pair mean-field approximation is presented showing good agreement with Monte Carlo simulations in the limit of weak selection (high noise) where we obtain analytical predictions for the critical temptations. We discuss the phase diagram as a function of temptation, intensity of noise and coordination number of the networks and we consider both the model with and without self-interaction. We compare our results with available results for non-directed lattices and networks.

We study the prisoner's dilemma model with a noisy imitation evolutionary dynamics on directed out-homogeneous and uncorrelated directed random networks. An heterogeneous pair mean-field approximation is presented showing good agreement with Monte Carlo simulations in the limit of weak selection (high noise) where we obtain analytical predictions for the critical temptations. We discuss the phase diagram as a function of temptation, intensity of noise and coordination number of the networks and we consider both the model with and without self-interaction. We compare our results with available results for non-directed lattices and networks.