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 ∂μσ—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.
Niehues J., Jensen G.G. and Haerter J.O. (2022) Self-organized quantization and oscillations on continuous fixed-energy sandpiles. Physical review. E 105(3). 10.1103/PhysRevE.105.034314
@article{Niehues2022,
Title = {Self-organized quantization and oscillations on continuous fixed-energy sandpiles},
Author = {Niehues, Jakob and Jensen, Gorm Gruner and Haerter, Jan O.},
Editor = {},
Journal = {Physical review. E},
Year = {2022},
Volume = {105},
Doi = {10.1103/PhysRevE.105.034314},
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 ∂μσ—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.},
}
TY - JOUR
AU - Niehues, Jakob
AU - Jensen, Gorm Gruner
AU - Haerter, Jan O.
TI - Self-organized quantization and oscillations on continuous fixed-energy sandpiles
T2 - Physical review. E
PY - 2022
VL - 105
DO - 10.1103/PhysRevE.105.034314
AB - 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 ∂μσ—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.
ER -
