Derivation and analytical solutions of a non-linear diffusion equation applied to non-constant heat conductivity and ionic diffusion in glasses.

This paper derives a non-linear diffusion equation discussing two possible applications: the ionic diffusion in glasses and temperature-dependent conductivity in semiconductors. The first equation is a logarithmic diffusion derived formally from the continuity of ion concentration, but the latter is a more phenomenological example. A power-law ansatz with time-dependent parameters maximizes a non-standard entropy and gives a set of coupled motion equations we can solve analytically. These results build the general solution to the non-linear diffusion equation.

Analyzing the 2019 Chilean social outbreak: Modelling Latin American economies.

In this work, we propose a quantitative model for the 2019 Chilean protests. We utilize public data for the consumer price index, the gross domestic product, and the employee and per capita income distributions as inputs for a nonlinear diffusion-reaction equation, the solutions to which provide an in-depth analysis of the population dynamics. Specifically, the per capita income distribution stands out as a solution to the extended Fisher-Kolmogorov equation. According to our results, the concavity of employee income distribution is a decisive input parameter and, in contrast to the distributions typically observed for Chile and other countries in Latin America, should ideally be non-negative. Based on the results of our model, we advocate for the implementation of social policies designed to stimulate social mobility by broadening the distribution of higher salaries.

Solving Equations of Motion by Using Monte Carlo Metropolis: Novel Method Via Random Paths Sampling and the Maximum Caliber Principle.

A permanent challenge in physics and other disciplines is to solve Euler-Lagrange equations. Thereby, a beneficial investigation is to continue searching for new procedures to perform this task. A novel Monte Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the path space with a probability functional obtained by using the maximum caliber principle. Free particle and harmonic oscillator problems are numerically implemented by sampling the path space for a given action by using Monte Carlo simulations. The average path converges to the solution of the equation of motion from classical mechanics, analogously as a canonical system is sampled for a given energy by computing the average state, finding the least energy state. Thus, this procedure can be general enough to solve other differential equations in physics and a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.

Discontinuous spirals of stability in an optically injected semiconductor laser.

We report a new kind of discontinuous spiral with stable periodic orbits in the parameter space of an optically injected semiconductor laser model, which is a combination of the intercalation of fish-like and cuspidal-like structures (the two normal forms of complex cubic dynamics). The spiral has a tridimensional structure that rolls up in at least three directions. A turn of approximately 2π radians along the spiral and toward the center increases the number of peaks in the laser intensity by one, which does not occur when traversing the discontinuities. We show that as we vary the linewidth enhancement factor (α), discontinuities are created (destroyed) through disaggregation (collapses) from (into) the so-called shrimp-like structures. Future experimental verification and applications, as well as theoretical studies to explain its origin and relation with homoclinic spirals that exist in its neighborhood, are needed.

Tricorn-like structures in an optically injected semiconductor laser.

This study reports the existence of tricorn-like structures of stable periodic orbits in the parameter plane of an optically injected semiconductor laser model (a continuous-time dynamical system). These tricorns appear inside tongue-like structures that are created through simple Shi'lnikov bifurcations. As the linewidth enhancement factor-α of the laser increases, these tongues invade the laser locking zone and extends over the zone of stable period-1 orbits. This invasion provokes a rich overlap dynamics of the parameter planes that produces an abundant multistability. As α increases, the tricorn exhibits a phenomenon of codimension-3 rotating in the clockwise and counterclockwise directions in the plane of the injected field rate K vs its detuning ω. We hope that the numerical evidence of the tricorns presented herein motivates the study of mathematical conditions for their genesis. We also encourage the experimental verification of these tricorn-like structures because our results also open new possibilities for optical switching between several different laser outputs in the neighborhood of these structures.

Publisher's Note: Dynamics and thermodynamics of systems with long-range dipole-type interactions [Phys. Rev. E 95, 022110 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.95.022110.

Dynamics and thermodynamics of systems with long-range dipole-type interactions.

A Hamiltonian mean field model, where the potential is inspired by dipole-dipole interactions, is proposed to characterize the behavior of systems with long-range interactions. The dynamics of the system remains in quasistationary states before arriving at equilibrium. The equilibrium is analytically derived from the canonical ensemble and coincides with that obtained from molecular dynamics simulations (microcanonical ensemble) at only long time scales. The dynamics of the system is characterized by the behavior of the mean value of the kinetic energy. The significance of the results, compared to others in the recent literature, is that two plateaus sequentially emerge in the evolution of the model under the special considerations of the initial conditions and systems of finite size. The first plateau decays to a different second one before the system reaches equilibrium, but the dynamics of the system is expected to have only one plateau when the thermodynamics limit is reached because the difference between them tends to disappear as N tends to infinity. Hence, the first plateau is a type of quasistationary state the lifetime of which depends on a power law of N and the second seems to be a true quasistationary state as reported in the literature. We characterize the general behavior of the model according to its dynamics and thermodynamics.