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.

中文翻译：

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相互作用的相同系统的热力学极限的响应理论和相变

我们研究了在经历随机演化的耦合相同代理网络的热力学极限中对扰动的响应，该网络通常描述非平衡条件。所有系统都被推向共同的质心。我们为通过平均场 Fokker-Planck 方程获得的线性磁化率推导出 Kramers-Kronig 关系和求和规则，然后提出与宏观情况相关的修正，它以自洽的方式结合了系统之间相互作用的影响。这种相互作用会产生记忆效应。我们能够推导出确定相变发生的条件，特别是由于系统到系统的相互作用。这种相变存在于热力学极限中，并与线性响应的发散相关，但不伴随适当定义的可观察量的积分自相关时间的发散。我们澄清，这种内源性相变与线性响应中的其他病理有根本不同，可以在关键转变的背景下构建。最后，我们展示了我们的结果如何阐明 Desai-Zwanzig 模型和 Bonilla-Casado-Morillo 模型的特性，它们分别具有典型的平衡和非平衡相变。我们澄清，这种内源性相变与线性响应中的其他病理有根本不同，可以在关键转变的背景下构建。最后，我们展示了我们的结果如何阐明 Desai-Zwanzig 模型和 Bonilla-Casado-Morillo 模型的特性，它们分别具有典型的平衡和非平衡相变。我们澄清，这种内源性相变与线性响应中的其他病理有根本不同，可以在关键转变的背景下构建。最后，我们展示了我们的结果如何阐明 Desai-Zwanzig 模型和 Bonilla-Casado-Morillo 模型的特性，它们分别具有典型的平衡和非平衡相变。