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Condensation in large random Lotka-Volterra systems

大型随机 Lotka-Volterra 系统中的凝结

基本信息

批准号：
317605153
负责人：
Professor Dr. Andreas Engel
金额：
--
依托单位：
Institut für Physik (IFP)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2021-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/317605153?language=en
关键词：
Condensation large random Lotka Volterra

项目摘要

Condensation is a general phenomenon in science to be found in large sets of interacting systems each of which may exist in different states. Condensation is said to occur if a finite fraction of these systems choose to be in the same state. Examples reach from Bose-Einstein condensation in atomic gases to the emergence of biological species. Whereas in equilibrium situations the condensed state is often unique non-equilibrium steady states typically show many different condensation patterns and the question of the number and selection of condensed states becomes relevant. A suitable mathematical framework to investigate these questions is the so-called replicator equation. It describes the time evolution of the fraction of systems in the various states. Being introduced originally in the field of evolutionary game theory this equation has been recently shown to model a variety of situations in which condensation occurs. Its main ingredient is an interaction matrix containing the rates of possible stochastic transitions between different states. Realistic situations are often characterized by a large number of states and complex interactions. To understand typical features of condensation under such conditions it is reasonable to assume that the interaction matrix is drawn from a random matrix ensemble. Such an approach will be fruitful if crucial properties of the condensation process are independent of the concrete realization of the interaction matrix and depend only on the statistical properties of the whole ensemble. Several interesting results along those lines of reasoning have been found recently in numerical simulations. In the present project we want to employ methods from the statistical mechanics of disordered systems to complement and extend these numerical findings by analytical results on the distribution of the number of condensates, its dependence on the connectivity of the underlying interaction network, and the stability of the selected pattern of condensed states.

凝聚是科学中的一种普遍现象，存在于大量相互作用的系统中，每个系统都可能以不同的状态存在。如果这些系统中有有限的一部分选择处于相同的状态，那么就说发生了凝聚。例子从原子气体中的玻色-爱因斯坦凝聚到生物物种的出现。而在平衡状态下，凝聚态往往是唯一的非平衡稳态通常显示出许多不同的凝聚模式和凝聚态的数量和选择的问题变得相关。研究这些问题的一个合适的数学框架是所谓的复制因子方程。它描述了处于不同状态的系统的分数的时间演化。被引入最初在进化博弈论领域，这个方程最近已被证明是模拟各种情况下，冷凝发生。它的主要成分是一个相互作用矩阵，其中包含不同状态之间可能的随机转换率。现实情况的特点往往是大量的状态和复杂的相互作用。为了理解在这种条件下凝结的典型特征，可以合理地假设相互作用矩阵是从随机矩阵系综中得出的。如果凝聚过程的关键性质独立于相互作用矩阵的具体实现，并且仅依赖于整个系综的统计性质，则这种方法将是富有成效的。最近在数值模拟中发现了一些有趣的结果。在本项目中，我们希望采用无序系统统计力学的方法，通过对凝聚体数量分布、其对底层相互作用网络连接性的依赖性以及所选模式的稳定性的分析结果来补充和扩展这些数值发现。凝聚态模式。