Difusão singular em um sistema confinado

Abstract

Patterns of scale invariance, associated with power laws, are often found in nature, for instance, in the fluctuations of prices of items in stock markets and in the energy spectrum of turbulent systems. These two systems and many others that exhibit scale invariance present some common properties: they are comprised of several elements that interact in a non-linear way, are not in equilibrium, and exhibit self-organization. Scale invariance is also found in the correlations observed in the critical state of systems that present phase transitions. The concept of self-organized criticality suggests that the properties of invariance spontaneously arise in complex systems. Several models exhibit properties of self-organized critically, including invasion percolation, sand-piles and the trough model, however it is not clear what are the necessary ingredients for criticality to arise. It is known that this property appears in some non-linear diffusive systems. In this work, we introduce a confining potential in a one-dimensional diffusion model with a singular non-linearity on diffusion coefficient, and analyze how this affects in the steady state of the system. We then derive a diffusion equation and obtain a solution for stationary density profile. Our analytical solution is in good agreement with the numerical results. We also present a statistical study of the distribution of avalanches sizes in this model, and obtain profiles following power laws, what is not usually observed in other one-dimensional systems. We also investigated how these profiles vary when the confinement increases, and using finite size scaling we found a universal curve for the distribution of avalanche sizes. Our results show that the action of confinement in a one-dimensional system can yield scale invariance.