arXiv subjects

Denis Ullmo

Publications and source records attributed to Denis Ullmo.

At least 19 recordsLinked to original sources

arXiv2025

Mean-field game approach to epidemic propagation on networks

We investigate an SIR model of epidemic propagation on networks in the context of mean-field games. In a real epidemic, individuals adjust their behavior depending on the epidemic level and the impact it might have on them in the future. These individual behaviors in turn affect the epidemic dynamics. Mean-field games are a framework in which these retroaction effects can be captured. We derive dynamical equations for the epidemic quantities in terms of individual contact rates, and via mean-field approximations we obtain the Nash equilibrium associated with the minimization of a certain cost function. We first consider homogeneous networks, where all individuals have the same number of neighbors, and discuss how the individual behaviors are modified when that number is varied. We then investigate the case of a realistic heterogeneous network based on real data from a social contact network. Our results allow to assess the potential of such an approach for epidemic mitigation in real-world implementations.

physics.soc-ph↗

arXiv2024

Modelling vehicle and pedestrian collective dynamics: Challenges and advances

In our urbanised societies, the management and regulation of traffic and pedestrian flows is of considerable interest for public safety, economic development, and the conservation of the environment. However, modelling and controlling the collective dynamics of vehicles and pedestrians raises several challenges. Not only are the individual entities self-propelled and hard to describe, but their complex nonlinear physical and social interactions makes the multi-agent problem of crowd and traffic flow even more involved. In this chapter, we purport to review the suitability and limitations of classical modelling approaches through four examples of collective behaviour: stop-and-go waves in traffic flow, lane formation, long-term avoidance behaviour, and load balancing in pedestrian dynamics. While stop-and-go dynamics and lane formation can both be addressed by basic reactive models (at least to some extent), the latter two require anticipation and/or coordination at the level of the group. The results highlight the limitations of classical force-based models, but also the need for long-term anticipation mechanisms and multiscale modelling approaches. In response, we review new developments and modelling concepts.

physics.soc-ph↗

arXiv2024

Mean Field Game Approach to Non-Pharmaceutical Interventions in a Social Structure model of Epidemics

The design of coherent and efficient policies to address infectious diseases and their consequences requires to model not only epidemics dynamics, but also individual behaviors, as the latter has a strong influence on the former. In our work, we provide a theoretical model for this problem, taking into account the social structure of a population. This model is based on a Mean Field Game version of a SIR compartmental model, in which individuals are grouped by their age class and interact together in different settings. This social heterogeneity allows to reproduce realistic situations while remaining usable in practice. In our game theoretical approach, individuals can choose to limit their contacts by making a trade-off between the risks incurred by infection and the cost of being confined. The aggregation of all these individual choices and optimizations forms a Nash equilibrium through a system of coupled equations that we derive and solve numerically. The global cost born by the population within this scenario is then compared to its societal optimum counterpart (i.e. the optimal cost from the society viewpoint), and we investigate how the gap between these two costs can be partially bridged within a constrained Nash equilibrium for which a governmental institution would impose lockdowns. Finally we consider the consequences of the finiteness of the population size $N$, or of a time $T$ at which an external event would end the epidemic, and show that the variation of these parameters could lead to first order phase transitions in the choice of optimal strategies. In this paper, all the strategies considered to mitigate epidemics correspond to non-pharmaceutical interventions (NPI), and we provide here a theoretical framework within which guidelines for public policies depending on the characteristics of an epidemic and on the cost of restrictions on the society could be assessed.

physics.soc-ph↗

arXiv2024

Mean-Field Games Modeling of Anticipation in Dense Crowds

Understanding and modeling pedestrian dynamics in dense crowds is a complex yet essential aspect of crowd management and urban planning. In this work, we investigate the dynamics of a dense crowd crossed by a cylindrical intruder using a Mean-Field Game (MFG) model. By incorporating a discount factor to account for pedestrians' limited anticipation and information processing, we examine the model's ability to simulate two distinct experimental configurations: pedestrians facing the obstacle and pedestrians giving their back to the intruder. Through a comprehensive comparison with experimental data, we demonstrate that the MFG model effectively captures essential crowd behaviors, including anticipatory motion and collision avoidance.

physics.soc-ph↗

arXiv2024

Modeling of obstacle avoidance by a dense crowd as a Mean-Field Game

In this paper we use a minimal model based on Mean-Field Games (a mathematical framework apt to describe situations where a large number of agents compete strategically) to simulate the scenario where a static dense human crowd is crossed by a cylindrical intruder. After a brief explanation of the mathematics behind it, we compare our model directly against the empirical data collected during a controlled experiment replicating the aforementioned situation. We then summarize the features that make the model adhere so well to the experiment and clarify the anticipation time in this framework.

physics.soc-ph↗

arXiv2023

Epidemic models on homogeneous networks : some analytical results

The ability to actually implement epidemic models is a crucial stake for public institutions, as they may be overtaken by the increasing complexity of current models and sometimes tend to revert to less elaborate models such as the SIR. In our work, we study a simple epidemic propagation model, called SIR-$k$, which is based on a homogeneous network of degree $k$, where each individual has the same number $k$ of neighbors. This model is more refined than the basic SIR which assumes a completely homogeneous population. We show that nevertheless, analytical expressions, simpler and richer than the ones existing for the SIR model, can be derived for this SIR-$k$ model. In particular we obtain an exact implicit analytical solution for any $k$, from which quantities such as the epidemic threshold or the total number of agents infected during the epidemic can be obtained. We furthermore obtain simple exact explicit solutions for small $k$'s, and in the large $k$ limit we find a new formulation of the analytical solution of the basic SIR model, which comes with new insights.

physics.soc-ph↗

arXiv2023

Computing Quantum Mean Values in the Deep Chaotic Regime

We study the time evolution of mean values of quantum operators in a regime plagued by two difficulties: The smallness of $\hbar$ and the presence of strong and ubiquitous classical chaos. While numerics become too computationally expensive for purely quantum calculations as $\hbar \to 0$, methods that take advantage of the smallness of $\hbar$ -- that is, semiclassical methods -- suffer from both conceptual and practical difficulties in the deep chaotic regime. We implement an approach which addresses these conceptual problems, leading to a deeper understanding of the origin of the interference contributions to the operator's mean value. We show that in the deep chaotic regime our approach is capable of unprecedented accuracy, while a standard semiclassical method (the Herman-Kluk propagator) produces only numerical noise. Our work paves the way to the development and employment of more efficient and accurate methods for quantum simulations of systems with strongly chaotic classical limits.

quant-ph↗

arXiv2023

Discounted Mean-Field Game model of a dense static crowd with variable information crossed by an intruder

It was demonstrated in [Bonnemain et al., Phys. Rev. E 107, 024612 (2023)] that the anticipation pattern displayed by a dense crowd crossed by an intruder can be successfully described by a minimal Mean-Field Games model. However, experiments show that changes in the pedestrian knowledge significantly modify the dynamics of the crowd. Here, we show that the addition of a single parameter, the discount factor $\gamma$, which gives a lower weight to events distant in time, is sufficient to observe the whole variety of behaviors observed in the experiments. We present a comparison between the discounted MFG and the experimental data, also providing new analytic results and insight about how the introduction of $\gamma$ modifies the model.

physics.soc-ph↗

arXiv2022

A social structure description of epidemics propagation with the mean field game paradigm

We consider the spread of infectious diseases through a Mean Field Game version of a SIR compartmental model with social structure, in which individuals are grouped by their age class and interact together in different settings. In our game theoretical approach, individuals can choose to limit their contacts if the epidemic is too virulent, but this effort comes with a social cost. We further compare the Nash equilibrium obtained in this way with the societal optimum that would be obtained if a benevolent central planner could decide of the strategy of each individual, as well as to the more realistic situation where an approximation of this optimum is reached through social policies such as lockdown.

physics.soc-ph↗

arXiv2022

Resurgent revivals in bosonic quantum gases: a striking signature of many-body quantum interferences

Matter wave revivals depend on a delicate interplay of constructive many-body quantum interferences in the developing dynamics of an ultracold bosonic system in an optical lattice. It is shown that the interplay between weak intersite tunneling and strong onsite interactions can lead to the quantum dynamics of a density wave displaying several features not found in the mean-field limit: occupancy oscillations, resurgent revivals, and a (anti-) synchronization of revival peaks and occupancy oscillation peaks. This implies cooperative interference effects that alternate between constructive and destructive features leading to the peak revival behaviors. These many-body quantum interference phenomena create striking features in various observables, which are accessible in experimental measurements.

cond-mat.quant-gas↗

arXiv2022

Pedestrians in static crowds are not grains, but game players

The local navigation of pedestrians amid a crowd is generally believed to involve no anticipation beyond (at best) the avoidance of the most imminent collisions. We show that current models rooted in this belief fail to reproduce some key features experimentally evidenced when a dense static crowd is crossed by an intruder. We identify the missing ingredient as the pedestrians' ability to plan their motion well beyond the next interaction, whence they may accept to move towards denser regions for a short time. To account for this effect, we introduce a minimal model based on mean-field game theory, which proves remarkably successful in replicating the aforementioned observations as well as other daily-life situations involving collective behaviour in dense crowds, such as partial metro boarding. This demonstrates the ability of game approaches to capture the anticipatory effects at play in operational crowd dynamics.

physics.soc-ph↗

arXiv2020

Chaos-Assisted Long-Range Tunneling for Quantum Simulation

We present an extension of the chaos-assisted tunneling mechanism to spatially periodic lattice systems. We demonstrate that driving such lattice systems in an intermediate regime of modulation maps them onto tight-binding Hamiltonians with chaos-induced long-range hoppings $t_n \propto 1/n$ between sites at a distance $n$. We provide a numerical demonstration of the robustness of the results and derive an analytical prediction for the hopping term law. Such systems can thus be used to enlarge the scope of quantum simulations to experimentally realize long-range models of condensed matter.

quant-ph↗

arXiv2020

Lax Connection and Conserved Quantities of Quadratic Mean Field Games

Mean Field Game is a rather new field initially developed in applied mathematics and engineering in order to deal with the dynamics of a large number of controlled agents or objects in interaction. For a large class of these models, there exists a deep relationship between the associated system of equations and the non linear Schr\"odinger equation, which allows to get new insights on the structure of their solutions. In this work, we deal with related aspects of integrability for such systems, exhibiting in some cases a full hierarchy of conserved quantities, and bringing some new questions which arise in this specific context.

physics.soc-ph↗

arXiv2020

Schr\"odinger approach to Mean Field Games with negative coordination

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schr\"odinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential varies, which makes possible to get insights on the role of the forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected.

physics.soc-ph↗

arXiv2019

Semiclassical evaluation of expectation values

Semiclassical Mechanics allows for a description of quantum systems which preserves their phase information, while using only the system's classical dynamics as an input. Over the time an identification has been developed between stationary phase approximation and semiclassical mechanics. Although it is true that in most of the cases in semiclassical mechanics the significant contributions come from the neighborhood of the stationary points, there are some important exceptions to it. In this paper we address one of these exceptions, occurring in the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. For our analysis we employ and discuss two major semiclassical tools. The first one is the association of the quantum evolution of a wavefunction to the classical evolution of a Lagrangian manifold, as done by Maslov. The second one is the derivation of an expression for the semiclassical Wigner function whose properties under canonical transformation are made explicit. Using the canonical invariance of the formalism, we derive an expression for the expectation value of observables for the one-dimensional case and then generalize it to higher dimensions. We find that the expression can be written as the sum of a classical contribution which corresponds to what is referred to as the Truncated Wigner Approximation (TWA) in the cold-atoms physics context, or the Linearized Semiclassical Initial Value Representation(LSC-IVR) in chemical or molecular physics, and additional terms associated with interferences. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.

quant-ph↗

arXiv2019

Universal behavior in non stationary Mean Field Games

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled objects in interaction. Though these models are much simpler than the underlying differential games they describe in some limit, their behavior is still far from being fully understood. When the system is confined, a notion of "ergodic state" has been introduced that characterizes most of the dynamics for long optimization times. Here we consider a class of models without such an ergodic state, and show the existence of a scaling solution that plays similar role. Its universality and scaling behavior can be inferred from a mapping to an electrostatic problem.

physics.soc-ph↗

arXiv2019

Enhancement of many-body quantum interference in chaotic bosonic systems

Although highly successful, the truncated Wigner approximation (TWA) leaves out many-body quantum interference between mean-field Gross-Pitaevskii solutions as well as other quantum effects, and is therefore essentially classical. Turned around, this implies that if a system's quantum properties deviate from TWA, they must be exhibiting some quantum phenomenon, such as localization, diffraction, or tunneling. Here, we consider in detail a particular interference effect arising from discrete symmetries, which can lead to a significant enhancement of quantum observables with respect to the TWA prediction, and derive an augmented version of the TWA in order to incorporate them. Using the Bose-Hubbard model for illustration, we further show strong evidence for the presence of dynamical localization due to remaining differences between the TWA predictions and quantum results.

cond-mat.quant-gas↗

arXiv2018

Mean Field Games in the weak noise limit : A WKB approach to the Fokker-Planck equation

Motivated by the study of a Mean Field Game toy model called the "seminar problem", we consider the Fokker-Planck equation in the small noise regime for a specific drift field. This gives us the opportunity to discuss the application to diffusion problem of the WKB approach "a la Maslov", making it possible to solve directly the time dependant problem in an especially transparent way.

physics.soc-ph↗