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 Schrödinger 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 vary, 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.
Generalized hydrodynamics of the KdV soliton gas Bonnemain, Thibault; Doyon, Benjamin; El, Gennady
Journal of physics. A, Mathematical and theoretical,
09/2022, Letnik:
55, Številka:
37
Journal Article
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 “à la Maslov (Maslov and Fedoriuk, 1981)”, making it possible to solve directly the time dependent problem in an especially transparent way.
•The Fokker-Planck equation is studied in the weak noise limit.•A time-dependent WKB approximation “à la Maslov” is developed.•A general scheme as well as a formal derivation are provided.•A mean field games toy model, the “seminar problem”, is solved as an example.
Generalised hydrodynamics (GHD) is a recent and powerful framework to study many-body integrable systems, quantum or classical, out of equilibrium. It has been applied to several models, from the ...delta Bose gas to the XXZ spin chain, the KdV soliton gas and many more. Yet it has only been applied to (1+1)-dimensional systems and generalisation to higher dimensions of space is non-trivial. We study the Boussinesq equation which, while generally considered to be less physically relevant than the KdV equation, is interesting as a stationary reduction of the (boosted) Kadomtsev-Petviashvili (KP) equation, a prototypical and universal example of a nonlinear integrable PDE in (2+1) dimensions. We follow a heuristic approach inspired by the Thermodynamic Bethe Ansatz in order to construct the GHD of the Boussinesq soliton gas. Such approach allows for a statistical mechanics interpretation of the Boussinesq soliton gas that comes naturally with the GHD picture. This is to be seen as a first step in the construction of the KP soliton gas, yielding insight on some classes of solutions from which we may be able to build an intuition on how to devise a more general theory. This also offers another perspective on the construction of anisotropic bidirectional soliton gases previously introduced phenomenologically by Congy et al (2021).
•Mean Field Games describe the dynamics of many controlled objects in interaction.•MFG systems consist in two coupled PDEs, one forward and one backward in time.•We consider MFG systems which ...exhibits a yet unexplained t2/3 scaling behavior.•In a weak noise limit, 1D MFG can be mapped onto a 3D electrostatic problem.•The stable t2/3 asymptotics follows from an electric monopole approximation.
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 a similar role. Its universality and scaling behavior can be inferred from a mapping to an electrostatic problem.
Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid ...density or quasi-particle density: the GHD equation. We consider a new, general form of the GHD equation: we allow for spatially extended interaction kernels, generalising previous constructions. We show that the GHD equation, in our general form and hence also in its conventional form, is Hamiltonian. This holds also including force terms representing inhomogeneous external potentials coupled to conserved densities. To this end, we introduce a new Poisson bracket on functionals of the fluid density, which is seen as our dynamical field variable. The total energy is the Hamiltonian whose flow under this Poisson bracket generates the GHD equation. The fluid density depends on two (real and spectral) variables so the GHD equation can be seen as a \(2+1\)-dimensional classical field theory. We further show the system admits an infinite set of conserved quantities that are in involution for our Poisson bracket, hinting at integrability of this field theory.
We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for ...many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg-de Vries (KdV) equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons' statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.
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.
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.
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