Lucarini Valerio, Pavliotis Grigorios A, Zagli Niccolò
Department of Mathematics and Statistics, University of Reading, Reading, UK.
Centre for the Mathematics of Planet Earth, University of Reading, Reading, UK.
Proc Math Phys Eng Sci. 2020 Dec;476(2244):20200688. doi: 10.1098/rspa.2020.0688. Epub 2020 Dec 23.
We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.
我们研究了一个由耦合的相同主体组成的网络在热力学极限下对微扰的响应，该网络经历随机演化，一般来说，这种演化描述了非平衡条件。所有系统都被推向共同质心。我们通过平均场福克 - 普朗克方程推导了线性磁化率的克拉默斯 - 克朗尼格关系和求和规则，然后提出了与宏观情况相关的修正，该修正以自洽的方式纳入了系统间相互作用的影响。这种相互作用产生了记忆效应。我们能够推导确定特定由于系统间相互作用而发生相变的条件。这种相变存在于热力学极限中，并且与线性响应的发散相关，但对于适当定义的可观测量，其积分自相关时间并不发散。我们阐明，这种内源性相变与线性响应中可以在临界转变背景下构建的其他病态现象有着根本的不同。最后，我们展示了我们的结果如何能够阐明德赛 - 茨万齐格模型和博尼利亚 - 卡萨多 - 莫里洛模型的性质，这两个模型分别具有典型的平衡和非平衡相变。
