Space-time fractional porous media equation: Application on modeling of S&P500 price return

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Sandpiles subjected to sinusoidal drive

This paper considers a sandpile model subjected to a sinusoidal external drive with the period T. We develop a theoretical model for the Green's function in a large T limit, which predicts that the avalanches are anisotropic and elongated in the oscillation direction. We track the problem numerically and show that the system additionally shows a regime where the avalanches are elongated in the perpendicular direction with respect to the oscillations. We find a crossover between these two regimes. The power spectrum of avalanche size and the grains wasted from the parallel and perpendicular directions are studied. These functions show power-law behavior in terms of the frequency with exponents, which run with T.

Statistical dynamics of driven systems of confined interacting particles in the overdamped-motion regime

Systems consisting of confined, interacting particles doing overdamped motion admit an effective description in terms of nonlinear Fokker-Planck equations. The behavior of these systems is closely related to the _S power-law entropies and can be interpreted in terms of the _S-based thermostatistics. The connection between overdamped systems and the _S measures provides valuable insights on diverse physical problems, such as the dynamics of interacting vortices in type-II superconductors. The _S-thermostatistical approach to the study of many-body systems described by nonlinear Fokker-Planck equations has been intensively explored in recent years, but most of these efforts were restricted to systems affected by time-independent external potentials. Here, we extend this treatment to systems evolving under time-dependent external forces. We establish a lower bound on the work done by these forces when they drive the system during a transformation. The bound is expressed in terms of a free energy based on the _S entropy and is satisfied even if the driving forces are not derivable from a potential function. It constitutes a generalization, for systems governed by nonlinear Fokker-Planck equations involving general time-dependent external forces, of the H-theorem satisfied by these systems when the external forces arise from a time-independent potential.