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

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

Advanced accident research system based on a medical and engineering data in the metropolitan area of Florence

BACKGROUND: In the metropolitan area of Florence, 62% of major traumas involve powered two wheeler rider and pillion passengers, 10% cyclists, and 7% pedestrians. The urban and extra-urban areas are the most dangerous for the vulnerable road user. In-depth investigations are needed for assessing detailed information on road accidents. This type of study has been very limited in time frame in Italy, and completely absent in the Tuscan region. Consequently a study called “In-depth Study of road Accident in FlorencE” (In-SAFE) has been initiated. METHODS: A network between the Department of Mechanics and Industrial Technologies (University of Florence) and the Intensive Care Unit of the Emergency Department (Careggi Teaching Hospital, Florence) was created with the aim of collecting information about the road accidents. The data collected includes: on-scene data, data coming from examination of the vehicles, kinematics and dynamic crash data, injuries, treatment, and injury mechanisms. Each injury is codified thorough the AIS score, localized by a three-dimensional human body model based on computer tomography slices, and the main scores are calculated. We then associate each injury with its cause and crash technical parameters. Finally, all the information is collected in the In-SAFE database. RESULTS: Patient mean age at the time of the accident was 34.6 years, and 80% were males. The ISS mean is 24.2 (SD 8.7) and the NISS mean is 33.6 (SD 10.5). The main road accident configurations are the “car-to-PTW” (25%) and “pedestrian run over” (17,9%). For the former, the main collision configuration is “head-on crash” (57%). Cyclists and PTW riders-and-pillions-passengers suffer serious injuries (AIS3+) mainly to the head and the thorax. The head (56.4%) and the lower extremities (12.7%) are the most frequently injured pedestrian body regions. CONCLUSIONS: The aim of the project is to create an in-depth road accident study with special focus on the correlation between technical parameters and injuries. An in-depth investigation team was setup and is currently active in the metropolitan area of Florence. Twenty-eight serious road accidents involving twenty-nine ICU patients are studied. PTW users, cyclist and pedestrians are the most frequently involved in metropolitan accidents

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 low-dimensional dynamics returns the correct diagnostic properties since it produces a quantitatively accurate representation of the stationary phase diagram of the system that we compare with exact analytical results and numerical simulations. Moreover, we prove that the reduced order dynamics yields also the prognostic, i.e., time-dependent properties, as it provides the correct response of the system to external perturbations. On the one hand, this validates the use of our complexity reduction methodology since it retains information not only of the invariant measure of the system but also of the transition probabilities and time-dependent correlation properties of the stochastic dynamics. On the other hand, the breakdown of linear response properties is a key signature of the occurence of a phase transition. We show that the reduced response operators capture the correct diverging resonant behavior by quantitatively assessing the singular nature of the susceptibility of the system and the appearance of a pole for real values of frequencies. Hence, this methodology can be interpreted as a low-dimensional, reduced order approach to the investigation and detection of critical phenomena in high-dimensional interacting systems in settings where order parameters are not known