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 Identifies Reaction Coordinates and Explains Critical Phenomena in Noisy Interacting Systems

We consider a class of nonequilibrium systems of interacting agents with pairwise interactions and quenched disorder in the dynamics featuring, in the thermodynamic limit, phase transitions. We provide conditions on the microscopic structure of interactions among the agents that lead to a dimension reduction of the system in terms of a finite number of reaction coordinates. Such reaction coordinates prove to be proper nonequilibrium thermodynamic variables as they carry information on correlation, memory and resilience properties of the system. Phase transitions can be identified and quantitatively characterised as singularities of the complex valued susceptibility functions associated to the reaction coordinates. We provide analytical and numerical evidence of how the singularities affect the physical properties of finite size systems

Anna Franchi: l'indocile scrittura

Avanti il divorzio is undoubtedly a fundamental work for the history of women and civil rights. Anna Franchi is fighting, together with the socialists, both for freedom of choice and for the right to civil and political vote, which were denied to women. The texts also deal with the struggles for the right to work and for fair wages. Moreover, they present unpublished profiles of personalities who devoted their fight of little hope and much courage to the cause. Anna Franchi was also the first woman who believed in the movement of the Macchiaioli: she wrote the history of the artistic movement, got to know its interpreters and sought for them a market that recognised their value. The friendship with Giovanni Fattori and the correspondence with many painters are relevant: the letters reflect not only the artistic aesthetic of the Macchiaioli, but also their moods, relationships, travels, successes and many sad moments that were confided to Fattori's friend, Anna

Dimension reduction of noisy interacting systems

We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally, nonequilibrium setting. We provide a systematic dimension reduction methodology for constructing low dimensional, reduced-order dynamics based on the cumulants of the probability distribution of the infinite system. We show that the stationary properties of the reduced dynamics yield a quantitatively accurate representation of the stationary phase diagram of the system by comparing it with exact analytical results and numerical simulations. Moreover, we test the accuracy of the reduced dynamics in representing the response of the system to perturbations and show that phase transitions are associated with the breakdown of linear response operators and the emergence of poles in the susceptibility for real values of the frequency

Noise-driven neuromorphic tuned amplifier

Acknowledgments The authors acknowledge financial support from H2020-MSCA-ITN-2015 project COSMOS 642563.Peer reviewedPublisher PD

Non normal amplification of stochastic quasi-cycles

Stochastic quasi-cycles for a two species model of the excitatory-inhibitory type, arranged on a triangular loop, are studied. By increasing the strength of the inter-nodes coupling, one moves the system towards the Hopf bifurcation and the amplitude of the stochastic oscillations are consequently magnified. When the system is instead constrained to evolve on specific manifolds, selected so as to return a constant rate of deterministic damping for the perturbations, the observed amplification correlates with the degree of non normal reactivity, here quantified by the numerical abscissa. The thermodynamics of the reactive loop is also investigated and the degree of inherent reactivity shown to facilitate the out-of-equilibrium exploration of the available phase space

Treatment of Fournier's Gangrene with Combination of Vacuum-Assisted Closure Therapy, Hyperbaric Oxygen Therapy, and Protective Colostomy

Fournier's gangrene is a rare process which affects soft tissue in the genital and perirectal area. It can also progress to all different stages of sepsis, and abdominal compartment syndrome can be one of its complications. Two patients in septic shock due to Fournier gangrene were admitted to the Intensive Care Unit of Emergency Department. In both cases, infection started from the scrotum and the necrosis quickly involved genitals, perineal, and inguinal regions. Patients were treated with surgical debridement, protective colostomy, hyperbaric oxygen therapy, and broad-spectrum antibacterial chemotherapy. Vacuum-assisted closure (VAC) therapy was applied to the wound with the aim to clean, decontaminate, and avoid abdominal compartmental syndrome development. Both patients survived and were discharged from Intensive Care Unit after hyperbaric oxygen therapy cycles and abdominal closure

Unexpected Hypertensive Pneumothorax after Digestive Upper Endoscopy: A Case Report

We report an unexpected massive left pneumothorax at the end of a digestive upper endoscopy without evidences of perforation or airway over-pressure. The possible air passage through a diaphragmatic failing is discussed

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