Mean Field Research Articles

CDMFT+HFD: An extension of dynamical mean field theory for nonlocal interactions applied to the single band extended Hubbard model

We examine the phase diagram of the extended Hubbard model on a square lattice, for both attractive and repulsive nearest-neighbor interactions, using CDMFT+HFD, a combination of Cluster Dynamical Mean Field theory (CDMFT) and a Hartree-Fock mean-field decoupling of the inter-cluster extended interaction. For attractive non-local interactions, this model exhibits a region of phase separation near half-filling, in the vicinity of which we find islands of d-wave superconductivity, decaying rapidly as a function of doping, with disconnected regions of extended s-wave order at smaller (higher) electron densities. On the other hand, when the extended interaction is repulsive, a Mott insulating state at half-filling is destabilized by hole doping, in the strong-coupling limit, in favor of d-wave superconductivity. At the particle-hole invariant chemical potential, we find a first-order phase transition from antiferromagnetism (AF) to d-wave superconductivity as a function of the attractive nearest-neighbor interaction, along with a deviation of the density from the half-filled limit. A repulsive extended interaction instead favors charge-density wave (CDW) order at half-filling.

Comparison of Mechanical Property Simulations with Results of Limited Flexural Tests of Different Multi-Layer Carbon Fiber-Reinforced Polymer Composites.

This article is focused on the experimental study of flexural properties in different multi-layer carbon fiber-reinforced polymer (CFRP) composites and correlations with the results of finite element method (FEM) simulations of mechanical properties. The comparison of the results shows the possibility of reducing the number of experimental specimens for testing. The experimental study of flexural properties for four types of carbon fiber-reinforced polymer matrix composites with twill weaves (2 × 2) was carried out. As input materials, pre-impregnated carbon laminate GG 204 T and GG 630 T (prepreg) and two types of carbon fiber fabrics (GG 285 T and GG 300 T (fabric)) were used. Multi-layer samples were manufactured from two types of prepregs and two types of fabrics, which were hand-impregnated during sample preparation. The layers were stacked using same orientation. All specimens for flexural test were cut with the longer side in the weft direction. Pre-impregnated carbon laminates were further impregnated with resin DT 121H. Carbon fabrics were hand-impregnated with epoxy matrix LG 120 and hardener HG 700. To fulfill the aim of this research, finite element method (FEM)-based simulations of mechanical properties were performed. The FEM simulations and analysis were conducted in Hexagon's MSC Marc Mentat 2022.3 and Digimat 2022.4 software. This paper presents the results of actual experimental bending tests and the results of simulations of bending tests for different composite materials (mentioned previously). We created material models for simulations based on two methods-MF (Mean Field) and FE (Finite Element), and the comparative results show better agreement with the MF model. The composites (GG 285 T and GG 300 T) showed better flexural results than composites made from pre-impregnated carbon laminates (GG 204 T and GG 630 T). The difference in results for the hand-impregnated laminates was about 15% higher than for prepregs, but this is still within an acceptable tolerance as per the reported literature. The highest percentage difference of 14.25% between the simulation and the real experiment was found for the software tool Digimat FE 2022.4-GG 630 T composite. The lowest difference of 0.5% was found for the software tool Digimat MF 2022.4-GG 204 T composite. By comparing the results of the software tools with the results of the experimental measurements, it was found that the Digimat MF 2022.4 tool is closer to the results of the experimental measurements than the Digimat FE 2022.4 tool.

A mean field description of steady state isotropic structural superplastic deformation

A model developed previously to treat steady state isotropic superplastic deformation turned out to be of generic value and relevant to many classes of materials from metallic materials, ceramics, intermetallic compounds, composites, bulk metallic glasses, and geological materials. The present paper demonstrates the applicability of the concept of “oblate spheroidal ensembles of atoms containing quantified excess free volume present in a general grain boundary” to predict quantitatively the response of general, high-angle grain boundaries when subjected to external stimuli, e.g., an external stress, irrespective of the nature of the atomic bonds (metallic, ionic, or covalent) in preference to purely geometric descriptions of the grain boundary. The emergence of “oblate spheroids” as a response to an external stimulus is regarded in this analysis as self-organizing and a basic phenomenon that cuts across material classes and the nature of the chemical bonds.

Constrained synthetic wind fields from high-resolution 3D WindScanner measurements

The limited spatial and temporal resolution of wind measurements within the inflow of a wind turbine requires statistical modeling of synthetic wind fields, integrating actual measurements or derived statistics along with selected turbulence models. The short-range WindScanner technology enhances atmospheric wind measurements with high resolution in both spatial and temporal dimensions. This contribution utilizes 3D turbulent inflow measurements from three synchronized WindScanner to generate synthetic turbulent wind fields. Both, mean wind field parameters and turbulent time series are analyzed, and different input configurations for the constrained wind field generation are evaluated.Our findings indicate the presence of periods characterized by dominant horizontal shear and veer events, with wind speed and direction differences up to 1.53 ms−1 or 8.45° across the rotor span. Additionally, the study reveals that the optimal measurement configuration for constrained turbulence modeling varies depending on the specific evaluated location and velocity component being analyzed. Another observation is that excessive constraints, placed too closely, may lead to overfitting, thereby diminishing the representativeness of the synthetic field for the lateral velocity component.

Statistical relations between spectropolarimetric observables and the polar strength of the stellar dipolar magnetic field

Global magnetic fields of early-type stars are commonly characterised by the mean longitudinal magnetic field 〈Bz〉 and the mean field modulus 〈B〉, derived from the circular polarisation and intensity spectra, respectively. Observational studies often report a root mean square (rms) of 〈Bz〉 and an average value of 〈B〉. In this work, I used numerical simulations to establish statistical relationships between these cumulative magnetic observables and the polar strength, Bd, of a dipolar magnetic field. I show that in the limit of many measurements randomly distributed in rotational phase, 〈Bz〉rms = 0.179−0.043+0.031 Bd and 〈B〉avg = 0.691−0.023+0.020 Bd. The same values can be recovered with only three measurements, provided that the observations are distributed uniformly in the rotational phase. These conversion factors are suitable for ensemble analyses of large stellar samples, where each target is covered by a small number of magnetic measurements.

Convergence rate of LQG mean field games with common noise

Convergence rate of LQG mean field games with common noise

Parameters of dust particle chains levitated vertically in a gas discharge plasma

The structural parameters of the dust particle chains suspended vertically in the electric field of a gas discharge are studied here. The investigations are performed by a numerical multi-block model based on a mean field approximation. The model describes the movement of ions and dust particles under the action of an external electric field, the electric field (Coulomb) of each charged dust particle, and the field of bulk plasma charge (ions and electrons) that screens the charges of dust particles. The gravity and the ion drag forces acting on the dust particles are also taken into account. Self-consistent chain parameters are calculated and compared for different number (1, 3, 5, and 7) and two different diameters (3.4 and 8.94 mcm) of dust particles in the chains. It is shown that an “ion wake” is formed behind the dust structures, and it grows with the number of dust particles and their size. With an increase in the number of dust particles in the chain, the charge of the first (top) dust particle increases. In the case of small dust particles, with an addition of new particles into the chain, the centers of the chains almost remain at the position of a single particle in the vertical direction. The chains of big dust particles move as a whole down in the direction of gravity with an increase in the number of particles.

Markov processes on quasi-random graphs

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is 1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{\\sqrt{N}}$$\\end{document} plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

Broadscale spatial synchrony in a West Nile virus mosquito vector across multiple timescales

Insects often exhibit irruptive population dynamics determined by environmental conditions. We examine if populations of the Culex tarsalis mosquito, a West Nile virus (WNV) vector, fluctuate synchronously over broad spatial extents and multiple timescales and whether climate drives synchrony in Cx. tarsalis, especially at annual timescales, due to the synchronous influence of temperature, precipitation, and/or humidity. We leveraged mosquito collections across 9 National Ecological Observatory Network (NEON) sites distributed in the interior West and Great Plains region USA over a 45-month period, and associated gridMET climate data. We utilized wavelet phasor mean fields and wavelet linear models to quantify spatial synchrony for mosquitoes and climate and to calculate the importance of climate in explaining Cx. tarsalis synchrony. We also tested whether the strength of spatial synchrony may vary directionally across years. We found significant annual synchrony in Cx. tarsalis, and short-term synchrony during a single period in 2018. Mean minimum temperature was a significant predictor of annual Cx. tarsalis spatial synchrony, and we found a marginally significant decrease in annual Cx. tarsalis synchrony. Significant Cx. tarsalis synchrony during 2018 coincided with an anomalous increase in precipitation. This work provides a valuable step toward understanding broadscale synchrony in a WNV vector.

A mean field game model of staking system

A mean field game model of staking system

Investigation of the magnetocaloric effect and critical behavior of the perovskite La0.65Ca0.35−xBaxMnO3 (x=0, 0.1, 0.2, 0.3)

Polycrystalline samples of La[Formula: see text]Ca[Formula: see text]BaxMnO3 (x = 0, 0.1, 0.2, 0.3) were prepared using a conventional solid-state reaction method. The prepared samples demonstrated good single-phase quality, characterized by a space group of Pbnm. There was a significant increase in the relative cooling power (at 5[Formula: see text]T), with values as follows: RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text], RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text], RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text], and RCP[Formula: see text][Formula: see text]J[Formula: see text]⋅[Formula: see text]kg[Formula: see text]. It is noteworthy that the sample with a doping level of x = 0.2 underwent a transition from a first-order phase transition to a second-order phase transition. The critical behavior of the x = 0.2 and 0.3 samples was analyzed using modified Arrott plots and the Kouvel–Fisher method to determine critical exponents. The results obtained from both methods were in close agreement, indicating that the critical behavior of the x = 0.2 and 0.3 samples can be described by the mean field model.

Chiral condensate and the equation of state at nonzero baryon density from the hadron resonance gas model with a repulsive mean field

We study the QCD equation of state and the chiral condensate using the hadron resonance gas model with repulsive mean-field interactions. We find that the repulsive interactions improve the agreement with the lattice results on the derivatives of the pressure with respect to the baryon chemical potential up to eighth order. From the temperature dependence of the chiral condensate we estimate the crossover temperature as a function of baryon chemical potential, Tpc(μB). We find that the chiral crossover line starts to deviate significantly from the chemical freeze-out line already for μB>400MeV. Furthermore, we find that the chiral pseudocritical line can be parametrized as Tpc(μB)/Tpc(0)=1−κ2[μB/Tpc(0)]2−κ4[μB/Tpc(0)]4 with κ2=0.0150(2) and κ4=3.1(6)×10−5, which are in agreement with lattice QCD results for small values of μB. For the first time we find a tiny but nonzero value of κ4 in our study. Published by the American Physical Society 2024

The Dynamics of the Hubbard Model Through Stochastic Calculus and Girsanov Transformation

As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=UN$$\\end{document} fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the difference of expected number of particles at the two lattice sites is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.

Multiple-Population Discrete-Time Mean Field Games with Discounted and Total Payoffs: The Existence of Equilibria

In the paper we present a model of discrete-time mean-field game with several populations of players. Mean-field games with multiple populations of the players have only been studied in the literature in the continuous-time setting. The main results of this article are the first stationary and Markov mean-field equilibrium existence theorems for discrete-time mean-field games of this type. We consider two payoff criteria: β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-discounted payoff and total payoff. The results are provided under some rather general assumptions on one-step reward functions and individual transition kernels of the players. In addition, the results for total payoff case, when applied to a single population, extend the theory of mean-field games also by relaxing some strong assumptions used in the existing literature.

Multi-armed bandit approach for mean field game-based resource allocation in NOMA networks

Facing the exponential demand for massive connectivity and the scarcity of available resources, next-generation wireless networks have to meet very challenging performance targets. Particularly, the operators have to cope with the continuous prosperity of the Internet of things (IoT) along with the ever-increasing deployment of machine-type devices (MTDs). In this regard, due to its compelling benefits, non-orthogonal multiple access (NOMA) has sparked a significant interest as a sophisticated technology to address the above-mentioned challenges. In this paper, we consider a hybrid NOMA scenario, wherein the MTDs are divided into different groups, each of which is allocated an orthogonal resource block (RB) so that the members of each group share a given RB to simultaneously transmit their signals. Firstly, we model the densely deployed network using a mean field game (MFG) framework while taking into consideration the effect of the collective behavior of devices. Then, in order to reduce the complexity of the proposed technique, we apply the multi-armed bandit (MAB) framework to jointly address the resource allocation and the power control problem. Thereafter, we derive two distributed decision-making algorithms that enable the users to autonomously regulate their transmit power levels and self-organize into coalitions based on brief feedback received from the base station (BS). Simulation results are given to underline the equilibrium properties of the proposed resource allocation algorithms and to reveal the robustness of the proposed learning process.

Microscopic understanding of NMR signals by dynamic mean-field theory for spins

A recently developed dynamic mean-field theory for disordered spins (spinDMFT) is shown to capture the spin dynamics of nuclear spins very well. The key quantities are the spin autocorrelations. In order to compute the free induction decay (FID), pair correlations are needed in addition. They can be computed on spin clusters of moderate size which are coupled to the dynamic mean fields determined in a first step by spinDMFT. We dub this versatile approach non-local spinDMFT (nl-spinDMFT). It is a particular asset of nl-spinDMFT that one knows from where the contributions to the FID stem. We illustrate the strengths of nl-spinDMFT in comparison to experimental data for CaF2. Furthermore, spinDMFT provides the dynamic mean fields explaining the FID of the nuclear spins of 13C in adamantane up to some static noise. The spin Hahn echo in adamantane is free from effects of static noise and agrees excellently with the spinDMFT results without further fitting.

Aging properties of the voter model with long-range interactions

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, Si=±1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P(r)∝r−α . Employing both analytical and numerical methods, we compute the two-time correlation function G(r;t,s) ( t⩾s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A(t,s)=G(r=0;t,s) , decays algebraically for α > 1 as [L(t)/L(s)]−λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ=1/(α−1) for 1<α⩽2 . For α⩽1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G(r;t,s) obeys a scaling symmetry G(r;t,s)=g[r/L(s);L(t)/L(s)] for α > 2. Similarly, for 1<α⩽2 , one has G(r;t,s)=g[r/L(t);L(t)/L(s)] , where the length L regulating two-time correlations now differs from the coherence length as L∝Lδ , with δ=1+2(2−α) .

Bifurcation of solutions of Ornstein–Zernike equation in the gas–liquid coexistence region of simple fluids

One objective of this paper is to investigate bifurcation of solutions of Ornstein–Zernike equation in liquid state theory for the gas–liquid coexistence region of simple fluids. The analysis uses a discrete form of the Ornstein–Zernike equation in real space, together with a closure relation which provides nearly self consistent thermodynamic properties of fluids governed by interaction potentials like the Lennard-Jones potential. Accurate asymptotic forms of correlation functions are incorporated in the real space algorithm so as to mitigate the influence of cutoff length used in the discrete approximations. The global error reduction of this algorithm is similar to that of Simpson’s rule. It is found that there is no spinodal on the vapor side on a sub-critical isotherm. However, there is a density at which compressibility vanishes, but this lies on a nonphysical branch. There are fold bifurcation points on the vapor and liquid sides together with a ‘no-real-solution-region’ in between. This is followed by a region of negative compressibility extending up to the spinodal on the liquid side. Importantly, it is found that in regions where compressibility is non-positive, there are infinite number of solutions to OZE which violate an integral constraint that is needed for applications to thermodynamics. The emerging picture is somewhat similar to that in the case of the hypernetted-chain-closure. It is also found that complex solutions connect the physical solutions that exist outside the ‘no-real-solution-region’ as well as in region of negative compressibility. The second objective of the paper, in the light of bifurcation scenario, is to evolve an interpretation of the coupling-parameter expansion of correlation functions pertaining to the region of negative compressibility. It is shown that the expansion provides an accurate description of results of the restricted ensembles, implied in mean field theories and simulation methods. Bifurcation of singlet density to an in-homogeneous distribution at a spinodal is also established using a simple equation, which uses the direct correlation function. Thus the paper re-establishes the fact that restriction to a homogeneous singlet density is the root cause of all inconsistencies found in the gas–liquid coexistence region.

An optimal advertising model with carryover effect and mean field terms

We consider a class of optimal advertising problems under uncertainty for the introduction of a new product into the market, on the line of the seminal papers of Vidale and Wolfe (Oper Res 5:370–381, 1957) and Nerlove and Arrow (Economica 29:129–142, 1962). The main features of our model are that, on one side, we assume a carryover effect (i.e. the advertisement spending affects the goodwill with some delay); on the other side we introduce, in the state equation and in the objective, some mean field terms that take into account the presence of other agents. We take the point of view of a planner who optimizes the average profit of all agents, hence we fall into the family of the so-called “Mean Field Control” problems. The simultaneous presence of the carryover effect makes the problem infinite dimensional hence belonging to a family of problems which are very difficult in general and whose study started only very recently, see Cosso et al. [Ann Appl Probab 33(4):2863–2918, 2023]. Here we consider, as a first step, a simple version of the problem providing the solutions in a simple case through a suitable auxiliary problem.

Actor-Critic Reinforcement Learning Algorithms for Mean Field Games in Continuous Time, State and Action Spaces

Actor-Critic Reinforcement Learning Algorithms for Mean Field Games in Continuous Time, State and Action Spaces