Mean-field Interaction Research Articles

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

Propagation of chaos for stochastic particle systems with singular mean-field interaction of Lq−Lp type

Propagation of chaos for stochastic particle systems with singular mean-field interaction of Lq−Lp type

Asymptotic behavior of stochastic currents under large deviation scaling with mean field interaction and vanishing noise

Asymptotic behavior of stochastic currents under large deviation scaling with mean field interaction and vanishing noise

Effective potentials in a rotating spin-orbit-coupled spin-1 spinor condensate

We theoretically study the stationary-state vortex lattice configurations of rotating spin-orbit (SO)- and coherently-coupled spin-1 Bose–Einstein condensates (BECs) trapped in quasi-two-dimensional harmonic potentials. The combined effects of rotation, SO and coherent couplings are analyzed systematically from the single-particle perspective. Through the single-particle Hamiltonian, which is exactly solvable for one-dimensional coupling, we illustrate that a boson in these rotating SO- and coherently-coupled condensates are subjected to effective toroidal, symmetric double-well, or asymmetric double-well potentials under specific coupling and rotation strengths. In the presence of mean-field interactions, using the coupled Gross–Pitaevskii formalism at moderate to high rotation frequencies, the analytically obtained effective potential minima and the numerically obtained coarse-grained density maxima position are in excellent agreement. On rapid rotation, we further find that the spin-expectation per particle of an antiferromagnetic spin-1 BEC approaches unity indicating a similarity in the response with ferromagnetic SO-coupled condensates.

Extreme events in a complex network: Interplay between degree distribution and repulsive interaction.

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.

Weak and strong error analysis for mean-field rank-based particle approximations of one-dimensional viscous scalar conservation laws

In this paper, we analyse the rate of convergence of a system of N interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikov (Ann. Probab. 46 (2018) 1042–1069) to check trajectorial propagation of chaos with optimal rate N−1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy (Math. Comp. 73 (2004) 777–812) to check the convergence in L1(R) with rate O(1N+h) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one-dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as O(1N+h). We provide numerical results which confirm our theoretical estimates.

Graphon particle system: Uniform-in-time concentration bounds

In this paper, we consider graphon particle systems with heterogeneous mean-field type interactions and the associated finite particle approximations. Under suitable growth (resp. convexity) assumptions, we obtain uniform-in-time concentration estimates, over finite (resp. infinite) time horizon, for the Wasserstein distance between the empirical measure and its limit, extending the work of Bolley–Guillin–Villani (2007).

Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the point vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher-dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding “microscopic” concavity result at finite N, shown by leveraging an unexpected link to Kähler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions, strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application, it is shown that the critical inverse negative temperatures—in the macroscopic as well as the microscopic setting—coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way, we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.

A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria

We analyze a class of nonlinear partial differential equations (PDEs) defined on R d × P 2 ( R d ) , \mathbb {R}^d \times \mathcal {P}_2(\mathbb {R}^d), where P 2 ( R d ) \mathcal {P}_2(\mathbb {R}^d) is the Wasserstein space of probability measures on R d \mathbb {R}^d with a finite second-order moment. We show that such equations admit a classical solutions for sufficiently small time intervals. Under additional constraints, we prove that their solution can be extended to arbitrary large intervals. These nonlinear PDEs arise in the recent developments in the theory of large population stochastic control. More precisely they are the so-called master equations corresponding to asymptotic equilibria for a large population of controlled players with mean-field interaction and subject to minimization constraints. The results in the paper are deduced by exploiting this connection. In particular, we study the differentiability with respect to the initial condition of the flow generated by a forward-backward stochastic system of McKean-Vlasov type. As a byproduct, we prove that the decoupling field generated by the forward-backward system is a classical solution of the corresponding master equation. Finally, we give several applications to mean-field games and to the control of McKean-Vlasov diffusion processes.

Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

AbstractWe use probabilistic methods to study properties of mean-field models, which arise as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle ‘jumps forward’ at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles’ distribution when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of—and attraction to—mean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

Increased exchange interaction range by Fe substitution in Ni3-xFexAl: Evidence from critical behavior

The critical behavior of Fe substituted intermetallic compounds Ni3-xFexAl (x = 0, 0.18, and 0.36) has been investigated around the paramagnetic-ferromagnetic (PM-FM) phase transition based on the static magnetization data. Reliable critical exponents (β, γ, and δ) and transition temperature TC are determined for the compounds, by using Modified Arrott plot, Kouvel-Fisher plot, and critical isotherm plot. The self-consistence and reliability of the obtained critical exponents are further verified by the Widom scaling relation δ = 1 + γ/β and scaling equations of the state M(H,ε) = εβf±(H/εβ+γ). It is found that, Fe substitution for Ni dramatically increases TC from 44.44 K for x = 0 to 301.20 K for x = 0.36. The critical exponents for the three compounds are all found to follow mean-field model. The exchange interaction J(r) is found to decay asr-4.657,r-4.434, and r-4.257 for x=0, 0.18 and 0.36, respectively. The results suggest that, a long range mean-field spin interaction dominates in Ni3-xFexAl (0≤x≤0.36) series. Furthermore, Fe substitution to Ni site is found to increase the spin interaction distance to longer range.

Superposition and mimicking theorems for conditional McKean–Vlasov equations

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs are governed by non-linear stochastic partial differential equations (SPDEs) of the second order, whereas their laws satisfy Fokker–Planck equations on the space of probability measures. Our paper establishes two superposition principles: The first asserts that any solution of the SPDE can be lifted to a solution of the conditional McKean–Vlasov SDE, and the second guarantees that any solution of the Fokker–Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Itô process can be emulated by those of a solution to a conditional McKean–Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean–Vlasov dynamics.

Phase Behaviors of a Binary Blend of Oppositely Charged Polyelectrolytes: A Weak Segregation Approach

In our quest for a better understanding of phase behaviors regarding polyelectrolyte systems, we develop a Landau free energy for a stoichiometric blend of weakly charged polyanions and polycations. Its free energy functional is first formulated as that for Gaussian chains subject to effective interaction fields. The local interaction part, which necessitates excluded volume and charged chain connectivity, is given by a locally equilibrated excess equation of state for the charged hard-sphere chain blend with mean-field dispersion interactions. The long-ranged part of Coulomb interactions is included through an electric potential field generated inside. The resultant functional is then expanded as a series in two proper order parameters to yield the desired Landau free energy. Unfavorable energetics makes the blend phase segregated but only on a nanoscopic scale due to global electroneutrality. The interplay of self- and mutual dispersion interactions produces a variety of phase behaviors ranging from conventional order–disorder-type transitions to complicated ordering transitions in carefully selected blends. Critical compositions are revealed to depend on asymmetry in chain sizes and disparity in self-dispersion interactions between the constituents. Furthermore, it is argued that nanostructured materials with drastically different responses to pressure can be designed by blending polyelectrolytes.

An Asymptotic Method for Predicting Risks in Problems of Stochastic Monitoring and Control

To ensure the stabilization of the equilibrium state in a nonlinear system in the presence of noise, it is not enough to solve the local stabilization problem, it is also necessary to ensure continuous monitoring of a possible transition to a critical state leading to system failure. To organize such monitoring, we use the large deviations principle applied to dynamical systems with small perturbations. For the purposes of monitoring, the optimal path that we call the A-profile is important. We use the A-profile to build a situational forecast in the risk control problem for a multi-agent system. In addition to the nonlinear mechanism of internal stabilization of the level h for each of the agents, there are forces of mean field interaction between the agents. The weak limit in this model with the number of agents tending to infinity is described by the FokerPlanck-Kolmogorov equation, but the use of approximation up to O(h2) leads to a finite-dimensional Wentzel-Freidlin scheme. According to the scheme, we obtain an explicit A-profile as a solution of the degenerate Abel equation of the second kind. At the same time, the approximation in h makes it possible to develop a method of successive approximations for the A-profile. In this paper, the A-profile is synthesized as a solution of the optimal control problem, where the state-dependent Riccati equation method and the method of the approximating sequence of Riccati equations are used. In the article, these methods are applied and compared within the framework of the risk control problem.

Electronic and structural properties of RbCeX2 (X2: O2, S2, SeS, Se2, TeSe, Te2)

Triangular lattice delafossite compounds built from magnetic lanthanide ions are a topic of recent interest due to their frustrated magnetism and realization of quantum disordered magnetic ground states. Here we report the evolution of the structure and electronic ground states of $\mathrm{RbCe}{X}_{2}$ compounds, built from a triangular lattice of ${\mathrm{Ce}}^{3+}$ ions, upon varying their anion character $({X}_{2}={\mathrm{O}}_{2}$, ${\mathrm{S}}_{2}, \mathrm{SeS}, {\mathrm{Se}}_{2}, \mathrm{TeSe}, {\mathrm{Te}}_{2})$. This includes the discovery of a new member of this series, ${\mathrm{RbCeO}}_{2}$, that potentially realizes a quantum disordered ground state analogous to ${\mathrm{NaYbO}}_{2}$. Magnetization and susceptibility measurements reveal that all compounds manifest mean-field antiferromagnetic interactions and, with the exception of the oxide, possess signatures of magnetic correlations onset below 1 K. The crystalline electric field level scheme is explored via neutron scattering and ab initio calculations in order to model the intramultiplet splitting of the $J=5/2$ multiplet. In addition to the two excited doublets expected within the $J=5/2$ manifold, we observe one extra local mode present across the sample series. This added mode shifts downward in energy with increasing anion mass and decreasing crystal field strength, suggesting a long-lived anomalous mode endemic to anion motion about the ${\mathrm{Ce}}^{3+}$ sites.

Conditional propagation of chaos in a spatial stochastic epidemic model with common noise

We study a stochastic spatial epidemic model where the N individuals carry two features: a position and an infection state, interact and move in $${\mathbb {R}}^d$$ . In this Markovian model, the evolution of infection states are described with the help of the Poisson Point Processes , whereas the displacement of individuals are driven by mean field interactions, a (state dependence) diffusion and also a common noise, so that the spatial dynamic is a random process. We prove that when the number N of individual goes to infinity, the conditional propagation of chaos holds : conditionally to the common noise, the individuals are asymptotically independent and the stochastic dynamic converges to a “random” nonlinear McKean-Vlasov process. As a consequence, the associated empirical measure converges to a measure, which is solution of a stochastic mean-field PDE driven by the common noise.

Quantum Droplet in Lower Dimensions

The study of Bose–Einstein condensation (BEC) in lower dimensions plays an important role in understanding the fundamentals of many-body physics as they can be treated theoretically with relative ease and can be verified experimentally. Recently, observation of a liquid-like state in a BEC mixture has been reported along with a theoretical prescription for its observation in the lower dimension. This observation is unique and has serious ramifications in our prevailing conception of the liquid state, which has a deep influence on the van der Waals theory. In explaining the self-bound nature of this state, quantum fluctuation and its fine balance with mean-field (MF) interaction turn out to be playing a key role. Though the experiments are performed predominantly in three dimensions, theoretical studies extend to the lower dimensions. In this brief review, we plan to summarize the recent theoretical advances in droplet research in the lower dimension and elaborate on the description of our contributions. We will mainly focus on analytical results related to this self-bound state in a one-dimension and quasi one-dimension environment. We aim to cover a few results from the family of cnoidal solutions to droplet solutions with smooth transitions between each other, finishing it by carrying a modest discussion on the supersolid phase.

Self-Consistent Transfer Operators: Invariant Measures, Convergence to Equilibrium, Linear Response and Control of the Statistical Properties

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented in the paper are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We apply the general statements to examples of different nature: coupled continuous maps, coupled expanding maps, coupled systems with additive noise, systems made of different maps coupled by a mean field interaction and other examples of self consistent transfer operators not coming from coupled maps. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.

Chiral matter-wave soliton in a Bose-Einstein condensate under density-dependent gauge potential

We theoretically investigate the chiral matter-wave soliton in a Bose-Einstein Condensate (BEC) under a density-dependent gauge potential. By means of the reductive perturbation method, the dark and bright solitons excited upon the background of the system are obtained. The physical mechanism for generating the chirality of the nonlinear excitation is revealed. The coupling of the nonlinear excitation and background of the system leads to a complex competition relationship between the current nonlinearity induced by the gauge potential and the mean-field atomic interaction, which results in rich chiral solitons. Generally, the amplitude and width of the chiral soliton depend on the direction of motion. Interestingly, when the current nonlinearity dominates over the mean-field atomic interaction, unidirectional chiral bright and dark solitons are obtained. The theoretical results are confirmed by the numerical simulations of the full nonlinear GP equation describing the system.

Klein–Gordon equation with mean field interaction. Orbital and asymptotic stability of solitary waves * *Research supported by the Austrian Science Fund (FWF) under Grant No. P 34177.

We investigate orbital and asymptotic stability of solitary wave solutions to the U(1)-invariant nonlinear Klein–Gordon equation with mean field interaction.