Response theory and critical phenomena for noisy interacting systems

In this thesis we investigate critical phenomena for ensembles of identical interacting agents, namely weakly interacting diffusions. These interacting systems undergo two qualitatively different scenarios of criticality, critical transitions and phase transitions. The former situation conforms to the classical tipping point phenomenology that is observed in finite dimensional systems and originates from a setting where negative feedbacks that stabilise the system progressively loose their efficiency, resulting in amplified fluctuations and correlation properties of the system. On the other hand, \textit{phase transitions} stem from the complex interplay between the agents' own dynamics, the coupling among them and the noise, leading to macroscopic emergent behaviour of the system, and are only observed in the thermodynamic limit. Classically, \textit{phase transitions} are investigated with the use of suitable macroscopic variables, called order parameters, acting as effective reaction coordinates that capture the relevant features of the macroscopic dynamics. However, identifying an order parameter is not always possible. In this thesis we adopt a complementary point of view, based on Linear Response theory, to investigate such critical phenomena. We are able to identify the conditions leading either to a critical transition or a phase transition in terms of spectral properties of suitable response operators. We associate critical phenomena to settings where the response of the system breaks down. In particular, we are able to characterise these critical scenarios as settings where the complex valued susceptibility of the system develops a non analytical behaviour for real values of frequencies, resulting in a macroscopic resonance of the system. We provide multiple paradigmatic examples of equilibrium and nonequilibrium phase transitions where we are able to prove mathematically and numerically the clear signature of a singular behaviour of the susceptibility at the phase transition as the thermodynamic limit is reached. Being associated to spectral properties of suitable operators describing either correlation or response properties, these resonant phenomena do not depend on the specific details of the applied forcing nor on the observable under investigation, allowing one to bypass the problem of the identification of the order parameter for the system.Open Acces

Response theory and phase transitions for the thermodynamic limit of interacting identical systems

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