Consistent Mean Field Approximations Research Articles

Robust mean field social control problems with applications in analysis of opinion dynamics

This paper investigates the social optimality of linear quadratic mean field control systems with unmodelled dynamics. The objective of agents is to optimise the social cost, which is the sum of costs of all agents. By variational analysis and direct decoupling methods, the social optimal control problem is analysed, and two equivalent auxiliary robust optimal control problems are obtained for a representative agent. By solving the auxiliary problem with consistent mean field approximations, a set of decentralised strategies is designed, and its asymptotic social optimality is further proved. Next, the results are applied into the study of opinion dynamics in social networks. The evolution of opinions is analysed over finite and infinite horizons, respectively. All opinions are shown to reach agreement with the average opinion in a probabilistic sense. Finally, local interactions among multiple sub-populations are examined via graphon theory.

Many-body perturbation theory for the superconducting quantum dot: Fundamental role of the magnetic field

We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated $0$-phase. It controls the critical behavior of the $0-\pi$ quantum transition, lifts the degeneracy in the $\pi$-phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in $0$- and $\pi$-phases. While the magnetic susceptibility vanishes in the $0$-phase it becomes of the Curie type and diverges in the $\pi$-phase at zero temperature.

Social optima in leader-follower mean field linear quadratic control

This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially, the auxiliary control problems with consistent mean field approximations, we can obtain a set of decentralized social optimality strategy with help of a class of forward-backward consistency systems. The relevant Stackelberg equilibrium is further proved under some proper conditions.

Social Optima in Robust Mean Field LQG Control: From Finite to Infinite Horizon

This article studies social optimal control of mean field linear-quadratic-Gaussian models with uncertainty. Specially, the uncertainty is represented by an uncertain drift, which is common for all agents. A robust optimization approach is applied by assuming all agents treat the uncertain drift as an adversarial player. In our model, both dynamics and costs of agents are coupled by mean field terms, and both finite- and infinite-time horizon cases are considered. By examining social functional variation and exploiting person-by-person optimality principle, we construct an auxiliary control problem for the generic agent via a class of forward-backward stochastic differential equation system. By solving the auxiliary problem and constructing consistent mean field approximation, a set of decentralized control strategies is designed and shown to be asymptotically optimal.

Type-II t−J model in superconducting nickelate Nd1−xSrxNiO2

The recent observation of superconductivity at relatively high temperatures in hole doped NdNiO$_2$ has generated considerable interest, particularly due to its similarity with the infinite layer cuprates. Building on the observation that the Ni$^{2+}$ ions resulting from hole doping are commonly found in the spin-triplet state, we introduce and study a variant of the $t-J$ model in which the holes carry S=1. We name this new model the Type II $t-J$ model. We find two distinct mechanisms for $d$ wave superconductivity. In both scenarios the pairing is driven by the spin coupling $J$. However, coherence is gained in distinct ways in these two scenarios. In the first case, the spin-one holes condense leading to a $d$ wave superconductor along with spin-symmetry breaking. Different orders including spin-nematic orders are possible. This scenario is captured by a spin one slave boson theory. In the second scenario, a coherent and symmetric $d$ wave superconductor is achieved from "Kondo resonance": spin one holes contribute two electrons to form large Fermi surface together with the spin 1/2 singly occupied sites. The large Fermi surface then undergoes $d$ wave pairing because of spin coupling $J$, similar to heavy fermion superconductor. We propose a three-fermion parton theory to treat these two different scenarios in one unified framework and calculate its doping phase diagram within a self consistent mean field approximation. Our study shows that a combination of "cuprate physics" and "heavy fermion physics" may emerge in the type II $t-J$ model.

Linear quadratic mean field games with a major player: The multi-scale approach

This paper considers linear quadratic (LQ) mean field games with a major player and analyzes an asymptotic solvability problem. It starts with a large-scale system of coupled dynamic programming equations and applies a re-scaling technique introduced in Huang and Zhou (2018) and Huang and Zhou (2020) to derive a set of Riccati equations in lower dimensions, the solvability of which determines the necessary and sufficient condition for asymptotic solvability. We next derive the mean field limit of the strategies and the value functions. Finally, we show that the two decentralized strategies can be interpreted as the best responses of a major player and a representative minor player embedded in an infinite population, which have the property of consistent mean field approximations.

Mean field production output control with sticky prices: Nash and social solutions

This paper presents an application of mean field control to dynamic production optimization with sticky prices and adjustment costs. Both noncooperative and cooperative solutions are considered. By solving auxiliary limiting optimal control problems subject to consistent mean field approximations, two sets of decentralized strategies are obtained and further shown to asymptotically attain Nash equilibria and social optima, respectively. A numerical example is given to compare market prices, firms’ outputs and costs under the two solution frameworks.

Robust Mean Field Linear-Quadratic-Gaussian Games with Unknown $L^2$-Disturbance

This paper considers a class of mean field linear-quadratic-Gaussian (LQG) games with model uncertainty. The drift term in the dynamics of the agents contains a common unknown function. We take a robust optimization approach where a representative agent in the limiting model views the drift uncertainty as an adversarial player. By including the mean field dynamics in an augmented state space, we solve two optimal control problems sequentially, which combined with consistent mean field approximations provides a solution to the robust game. A set of decentralized control strategies is derived by use of forward-backward stochastic differential equations (FBSDE) and shown to be a robust epsilon-Nash equilibrium.

Self-repelling polymer chains in finite volume with Dirichlet boundary conditions: crossover from the dilute to the dense phase

Extending the results for periodic boundary conditions obtained in a previous paper we analyse a single polymer chain in a good solvent contained in a finite box with Dirichlet boundary conditions. We develop a consistent mean-field approximation, apply it to different geometries and discuss in detail the chain length distribution for the `field theoretic ensemble' of chains characterized by a chemical segment potential . We show how the different regimes of the dilute and dense limits (where stands for the critical chemical potential) smoothly evolve from one another.

A perturbative treatment of fluctuating polarizability in classical fluids

The effects of fluctuating polarizability are accounted for by means of a perturbative approach closely by connected to a previous treatment of polarizabilities in the mean spherical approximation. Monte Carlo reaction field calculations are performed for a generalized Stockmayer model for ammonia with effective dipole moment, and results are in agreement with those obtained from the self consistent mean field approximation, but computed dielectric constants show significant deviations when compared with results from molecular dynamics for a model with static polarizability.

Can excitons produce superdiamagnetic currents ?

It is shown that Bose condensation of excitons cannot produce superdiamagnetic currents. The latter vanish within a consistent mean field approximation when both normal and anomalous pairings are retained.