Self-organized quantization and oscillations on continuous fixed-energy sandpiles
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912129127284736 |
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| author | Niehues, Jakob Jensen, Gorm Gruner Haerter, Jan O. |
| author_facet | Niehues, Jakob Jensen, Gorm Gruner Haerter, Jan O. |
| contents | Atmospheric self-organization and activator-inhibitor dynamics in biology provide examples of checkerboard-like spatio-temporal 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 non-negative real number, its energy. Upon each timestep all sites with energy exceeding a unit threshold re-distribute 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 re-distribution 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 checkerboard-like 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 $\partial_μσ$ - 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04470 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Self-organized quantization and oscillations on continuous fixed-energy sandpiles Niehues, Jakob Jensen, Gorm Gruner Haerter, Jan O. Statistical Mechanics Adaptation and Self-Organizing Systems Pattern Formation and Solitons Computational Physics Atmospheric self-organization and activator-inhibitor dynamics in biology provide examples of checkerboard-like spatio-temporal 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 non-negative real number, its energy. Upon each timestep all sites with energy exceeding a unit threshold re-distribute 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 re-distribution 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 checkerboard-like 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 $\partial_μσ$ - 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. |
| title | Self-organized quantization and oscillations on continuous fixed-energy sandpiles |
| topic | Statistical Mechanics Adaptation and Self-Organizing Systems Pattern Formation and Solitons Computational Physics |
| url | https://arxiv.org/abs/2111.04470 |