Self-organized quantization and oscillations on continuous fixed-energy sandpiles.

Abstract

Atmospheric self-organization and activator-inhibitor dynamics in biology provide examples of checkerboardlike spatiotemporal organization. We study a simple model for local activation-inhibition processes. Our model, first introduced in the context of atmospheric moisture dynamics, is a continuous-energy and non-Abelian version of the fixed-energy sandpile model. Each lattice site is populated by a nonnegative real number, its energy. Upon each timestep all sites with energy exceeding a unit threshold redistribute their energy at equal parts to their nearest neighbors. The limit cycle dynamics gives rise to a complex phase diagram in dependence on the mean energy μ: For low μ, all dynamics ceases after few redistribution events. For large μ, the dynamics is well-described as a diffusion process, where the order parameter, spatial variance σ, is removed. States at intermediate μ are dominated by checkerboardlike period-two phases which are however interspersed by much more complex phases of far longer periods. Phases are separated by discontinuous jumps in σ or ∂_{μ}σ-akin to first- and higher-order phase transitions. Overall, the energy landscape is dominated by few energy levels which occur as sharp spikes in the single-site density of states and are robust to noise.