Fernando Albuquerque Oliveira

In this review, we scrutinize historical and modern results on the linear response of dynamical systems to external perturbations with a particular emphasis on the celebrated relationship between fluctuations and dissipation expressed by the fluctuation–dissipation theorem (FDT). The conceptual foundation of FDT originates from the definition of the equilibrium state and Onsager’s regression hypothesis. Over time, the fluctuation–dissipation relation has been vividly investigated also in systems far from equilibrium, which often exhibit wild fluctuations in measured parameters. In this review, we recall the major formulations of the FDT, including those proposed by Langevin, Onsager and Kubo. We discuss the role of fluctuations in a broad class of growth and diffusion phenomena and examine the violation of the FDT resulting from a transition from Euclidean to fractal geometry. Finally, we highlight possible generalizations of the FDT formalism and discuss situations where the relation breaks down and is no longer applicable.

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 develop a hypothesis that the dynamics of equilibrium systems at criticality have their dynamics constricted to a fractal subspace. We relate the correlation fractal dimension associated with this subspace to the Fisher critical exponent controlling the singularity associated with the correlation function. This fractal subspace is different from that associated with the order parameter. We propose a relation between the correlation fractal dimension and the order parameter fractal dimension. The fractal subspace we identify has as a defining property that the correlation function is restored at the critical point by restricting the dynamics this way. We determine the correlation fractal dimension of the two-dimensional Ising model and validate it by computer simulations. We discuss growth models briefly in this context.

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.