Mean-field Equations Research Articles

Dynamical mean-field-driven spinor-condensate physics beyond the single-mode approximation

$^{23}\mathrm{Na}$ spin-1 Bose-Einstein condensates are used to experimentally demonstrate that mean-field physics beyond the single-mode approximation can be relevant during the nonequilibrium dynamics. The experimentally observed spin oscillation dynamics and associated dynamical spatial structure formation confirm theoretical predictions that are derived by solving a set of coupled mean-field Gross-Pitaevskii equations [J. Jie et al., Phys. Rev. A 102, 023324 (2020)]. The experiments rely on microwave dressing of the $f=1$ hyperfine states, where $f$ denotes the total angular momentum of the $^{23}\mathrm{Na}$ atom. The fact that physics beyond the single-mode approximation at the mean-field level, i.e., spatial mean-field dynamics that distinguishes the spatial density profiles associated with different Zeeman levels, can, in certain parameter regimes, have a pronounced effect on the dynamics when the spin healing length is comparable to or larger than the size of the Bose-Einstein condensate has implications for using Bose-Einstein condensates as models for quantum phase transitions and spin squeezing studies as well as for nonlinear SU(1,1) interferometers.

Projective embedding of dynamical systems: Uniform mean field equations

We study embeddings of continuous dynamical systems in larger dimensions via projector operators. We call this technique PEDS, projective embedding of dynamical systems, as the stable fixed point of the original system dynamics are recovered via projection from the higher dimensional space. In this paper we provide a general definition and prove that for a particular type of rank-1 projector operator, the uniform mean field projector, the equations of motion become a mean field approximation of the dynamical system. While in general the embedding depends on a specified variable ordering, the same is not true for the uniform mean field projector. We prove a variety of results on the relationship between the spectrum of the Jacobian for fixed points in the original and in the embedded system. Direct applications of PEDS can be non-convex optimization and machine learning.

Analysis of Mean-Field Approximation for Deffuant Opinion Dynamics on Networks

Mean-field equations have been developed recently to approximate the dynamics of the Deffuant model of opinion formation. These equations can describe both fully mixed populations and the case where individuals interact only along edges of a network. In each case, interactions only occur between individuals whose opinions differ by less than a given parameter, called the confidence bound. The size of the confidence bound parameter is known to strongly affect both the dynamics and the number and location of opinion clusters. In this work, we carry out a mathematical analysis of the mean-field equations to investigate the role of the confidence bound and boundaries on these important observables of the model. We consider the limit in which the confidence bound interval is small, and we identify the key mechanisms driving opinion evolution. We show that linear stability analysis can predict the number and location of opinion clusters. Comparison with numerical simulations of the model illustrates that the early-time dynamics and the final cluster locations can be accurately approximated for networks composed of two degree classes, as well as for the case of a fully mixed population.

A revisit of the vapor–liquid interface of the Lennard-Jones fluid

In this work a mean-field crossover generated by the Maxwell construction is defined as the dividing interface for the vapor–liquid interface area. An accurate and physically preferable density-profile equation is used to correlate the local density and to predict various interfacial properties. By using the density gradient theory and a mean-field equation of state for the Lennard-Jones fluid, we are able to extensively explore the interface system in terms of the Gibbs free energy, the Helmholtz free energy and heat capacity. The peak of the intrinsic free energy at the interface reveals the most unstable state and suggests that the interface area is thus divided into vapor- enriched and liquid- enriched sub-regions. The local maxima of the intrinsic isobaric heat capacity echo such a dividing, which is in accordance with the Widom line in the supercritical region. Consequently the mean-field dividing interface is the natural extension of the Widom line into the coexistence region. Hence the entire phase space is coherently divided into liquid-like and gas-like regions in all three (temperature–pressure-volume) planes. Finally, the modeling of adsorption on the interface is briefly discussed.

Uniform stability and emergent dynamics of particle and kinetic Lohe matrix models

We study asymptotic behaviors of the second-order Lohe matrix model on the unitary group. For the emergent behaviors, we show that a solution to the Lohe matrix model with identical Hamiltonians always converges to equilibrium using a gradient flow approach. Moreover, we establish the uniform-in-time stability with respect to initial data using the complete aggregation estimate, which can be applied to obtain the corresponding mean-field kinetic equation for the one-oscillator distribution function on the phase space. In fact, we rigorously show that Wasserstein-2 distance between two solutions in different regimes converges to zero in any finite time interval, when the number of oscillators tends to infinity. For the asymptotic emergent dynamics of the kinetic equation, we use an energy estimate and temporal evolution for the diameter to provide sufficient frameworks leading to the complete and practical aggregations for identical and non-identical Hamiltonians, respectively.

Revisiting the Mechanisms of Charge Transport in Solutions of Redox-Active Molecules Using Computer Simulations: When and Why Do Analytical Theories Fail?

Understanding charge transport is essential for the development of energy-storage applications. This work introduces a new theoretical methodology to model diffusive charge transport in solutions of redox-active molecules by combining Langevin dynamics for the spatial degrees of freedom and a master-equation formalism to describe the electron-hopping events between redox molecules. The model is used to analyze the effects of the concentration of the redox molecules and the strength of the intermolecular interactions on the charge-transport mechanism. In the past, the rate of charge transport has been modeled with the analytical Dahms-Ruff equation; however, this is a mean-field equation, whose range of validity has not been tested with less approximate theories. We show that the Dahms-Ruff equation fails to quantitatively predict the diffusion coefficient for charge transport for large concentrations of the redox species and high bimolecular electron-transfer rates, i.e., the most relevant conditions for energy-storage applications. Under these conditions, the diffusion coefficient for charge transport obtained from simulations is larger than that predicted from the Dahm-Ruff equation because of the formation of transient clusters of redox molecules. Also, intermolecular interactions, which are not taken into account by the Dahms-Ruff equation, play a central role in the charge transport of redox species. We show that the apparent diffusion coefficient experiences a maximum with respect to the strength of the intermolecular attractions. This maximum is traced back to the formation of clusters and their two opposite effects on the diffusion coefficient: electron hopping is fast within a cluster but inefficient between neighboring clusters.

Renormalization group flow to effective quantum mechanics in the IR in an emergent dual holographic description for spontaneous chiral symmetry breaking

Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative way, we construct an effective holographic dual description with an emergent extradimension identified with an RG scale. Taking the large$-N$ limit, we obtain an equation of motion of an order-parameter field, here the chiral condensate for our explicit demonstration. In particular, an intertwined structure manifests between the first-order RG flow equations of renormalized coupling functions and the second-order differential equation of the order-parameter field, thus referred to as a nonperturbative RG-improved mean-field theory. Assuming translational symmetry as a vacuum state, we solve these nonlinear coupled mean-field equations based on a matching method between UV- and IR-regional solutions. As a result, we find an RG flow from a weakly-coupled chiral-symmetric UV fixed point to a strongly-correlated chiral-symmetry broken IR fixed point, where the renormalized velocity of Dirac fermions vanishes most rapidly and effective quantum mechanics appears at IR. Furthermore, we translate these RG flows of coupling functions into those of emergent metric tensors and extract out geometrical properties of the emergent holographic spacetime constructed from the UV- and IR-regional solutions. Surprisingly, we obtain the volume law of entanglement entropy in the Ryu-Takayanagi formula, which implies appearance of a black hole type solution in the limit of infinite cutoff even at zero temperature. We critically discuss our field theoretic interpretation for this solution in terms of potentially gapless multi-particle excitation spectra.

Learning Swarm Interaction Dynamics From Density Evolution

In this article, we consider the problem of understanding the coordinated movements of biological or artificial swarms. In this regard, we propose a learning scheme to estimate the coordination laws of the interacting agents from observations of the swarm's density over time. We describe the dynamics of the swarm based on pairwise interactions according to a Cucker–Smale flocking model, and express the swarm's density evolution as the solution to a system of mean-field hydrodynamic equations. We propose a new family of parametric functions to model the pairwise interactions, which allows for the mean-field macroscopic system of integro-differential equations to be efficiently solved as an augmented system of partial differential equations. Finally, we incorporate the augmented system in an iterative optimization scheme to learn the dynamics of the interacting agents from observations of the swarm's density evolution over time. The results of this work can offer an alternative approach to study how animal flocks coordinate, create new control schemes for large networked systems, and serve as a central part of defense mechanisms against adversarial drone attacks.

Next generation neural population models

Low-dimensional neural mass models are often invoked to model the coarse-grained activity of large populations of neurons and synapses and have been used to help understand the coordination of large scale brain rhythms. However, they are phenomenological in nature and, although motivated by neurobiological considerations, the absence of a direct link to an underlying biophysical reality is a weakness that means they may not be best suited to capturing some of the rich behaviors seen in real neuronal tissue. In this perspective article I discuss a simple spiking neuron network model that has recently been shown to admit to an exact mean-field description for synaptic interactions. This has many of the features of a neural mass model coupled to an additional dynamical equation that describes the evolution of population synchrony. Thisnext generationneural mass model is ideally suited to understanding the patterns of brain activity that are ubiquitously seen in neuroimaging recordings. Here I review the mean-field equations, the way in which population synchrony, firing rate, and average voltage are intertwined, together with their application in large scale brain modeling. As well as natural extensions of this new approach to modeling the dynamics of neuronal populations I discuss some of the open mathematical challenges in developing a statistical neurodynamics that can generalize the one discussed here.

Excitonic Insulators and Superfluidity in Two-Dimensional Bilayers without External Fields.

We explore a new platform for realizing excitonic insulators, namely van der Waals (vdW) bilayers comprising two-dimensional Janus materials. In previous studies, type II heterobilayers have been brought to the excitonic insulating regime by tuning the band alignment using external gates. In contrast, the Janus bilayers presented here represent intrinsic excitonic insulators. We first conduct ab initio calculations to obtain the quasiparticle band structures, screened Coulomb interaction, and interlayer exciton binding energies of the bilayers. These ab initio-derived quantities are then used to construct a BCS-like Hamiltonian of the exciton condensate. By solving the mean-field gap equation, we identify 16 vdW Janus bilayers with insulating ground states and superfluid properties. Our calculations expose a new class of advanced materials that are likely to exhibit novel excitonic phases at low temperatures and highlight the subtle competition between interlayer hybridization, spin-orbit coupling, and dielectric screening that governs their properties.

Macroscopic dynamics of neural networks with heterogeneous spiking thresholds.

Mean-field theory links the physiological properties of individual neurons to the emergent dynamics of neural population activity. These models provide an essential tool for studying brain function at different scales; however, for their application to neural populations on large scale, they need to account for differences between distinct neuron types. The Izhikevich single neuron model can account for a broad range of different neuron types and spiking patterns, thus rendering it an optimal candidate for a mean-field theoretic treatment of brain dynamics in heterogeneous networks. Here we derive the mean-field equationsfor networks of all-to-all coupled Izhikevich neurons with heterogeneous spiking thresholds. Using methods from bifurcation theory, we examine the conditions under which the mean-field theory accurately predicts the dynamics of the Izhikevich neuron network. To this end, we focus on three important features of the Izhikevich model that are subject here to simplifying assumptions: (i) spike-frequency adaptation, (ii) the spike reset conditions, and (iii) the distribution of single-cell spike thresholds across neurons. Our results indicate that, while the mean-field model is not an exact model of the Izhikevich network dynamics, it faithfully captures its different dynamic regimes and phase transitions. We thus present a mean-field model that can represent different neuron types and spiking dynamics. The model comprises biophysical state variables and parameters, incorporates realistic spike resetting conditions, and accounts for heterogeneity in neural spiking thresholds. These features allow for a broad applicability of the model as well as for a direct comparison to experimental data.

On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure

On the existence of solutions to a general mean field equation of nonlinear diffusion with the Newtonian potential pressure

Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning.

We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler-Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to "physics-informed" gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner.

Stability and optimal control for delayed rumor-spreading model with nonlinear incidence over heterogeneous networks

On the multilingual online social networks of global information sharing, the wanton spread of rumors has an enormous negative impact on people’s lives. Thus, it is essential to explore the rumor-spreading rules in multilingual environment and formulate corresponding control strategies to reduce the harm caused by rumor propagation. In this paper, considering the multilingual environment and intervention mechanism in the rumor-spreading process, an improved ignorants–spreaders-1–spreaders-2–removers (I2SR) rumor-spreading model with time delay and the nonlinear incidence is established in heterogeneous networks. Firstly, based on the mean-field equations corresponding to the model, the basic reproduction number is derived to ensure the existence of rumor-spreading equilibrium. Secondly, by applying Lyapunov stability theory and graph theory, the global stability of rumor-spreading equilibrium is analyzed in detail. In particular, aiming at the lowest control cost, the optimal control scheme is designed to optimize the intervention mechanism, and the optimal control conditions are derived using the Pontryagin’s minimum principle. Finally, some illustrative examples are provided to verify the effectiveness of the theoretical results. The results show that optimizing the intervention mechanism can effectively reduce the densities of spreaders-1 and spreaders-2 within the expected time, which provides guiding insights for public opinion managers to control rumors.

SEIOR Rumor Propagation Model Considering Hesitating Mechanism and Different Rumor-Refuting Ways in Complex Networks

Considering that the state transfer rules between nodes in existing rumor propagation models are mostly based on a single propagation mechanism, and most of the models have a single way of refuting rumors, in this paper, a new SEIOR rumor propagation model (ignorant (S), hesitators (E), spreaders (I), rumor debunkers (O), immunizers (R)) is proposed by introducing hesitators and rumor debunkers and combining different rumor-refuting ways of rumor debunkers. Firstly, the dynamics process of the propagation model is described by using the mean-field equations. Secondly, the equilibrium point and the basic regeneration number of the model are solved, and the existence and stability of the equilibrium point are discussed. Then, numerical simulations are used to analyze the effects of different factors on rumor propagation patterns. The results show that the rumor-spreading rate α2 has the greatest effect on rumor propagation. With the increase in α2, the degree of influence of the hesitator-converting rate α1 on the scale of rumor propagation first increases and then decreases. Different rumor-refuting ways have different inhibiting effects on the spread of rumors. Finally, based on the results of the theoretical proving and numerical analysis, some targeted measures to control the spread of rumors are proposed.

Numerical schemes for fully coupled mean-field forward backward stochastic differential equations

In this paper, based on the nonlinear Feynman-Kac formula and Itô-Taylor schemes for solving mean-field forward stochastic differential equations, we propose numerical schemes for solving fully coupled mean-field forward backward stochastic differential equations. Our numerical experiments show that the schemes are stable, effective, and can be of second-order accuracy when weak second-order schemes are used for solving mean-field forward stochastic differential equations.

Mean-field stochastic differential equations driven by <inline-formula><tex-math id="M1">$ G $</tex-math></inline-formula>-Brownian motion

The main purpose of this paper is to analyze the mean-field stochastic differential equations driven by $ G $-Brownian motion (mean-field $ G $-SDEs). We first investigate the existence and uniqueness of the solution for the mean-field $ G $-SDEs by utilizing the Banach fixed point theorem. Moreover, by the method of coupling by change of measures, the Harnack and log-Harnack inequalities for mean-field $ G $-SDEs with additive noise are established.

Linear-Quadratic-Gaussian mean-field controls of social optima

<p style='text-indent:20px;'>This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be <i>indefinite</i> in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some <i>fully-coupled</i> variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with <i>embedding representation</i>. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.</p>

On the mean-field Belavkin filtering equation

On the mean-field Belavkin filtering equation

Mean-field stochastic differential equations with a discontinuous diffusion coefficient

Mean-field stochastic differential equations with a discontinuous diffusion coefficient