Mean-field Interaction Research Articles

Langmuir adsorption model to assess the impact of silane coupling on nano-dispersion of silica in SBR

Surface active agents are often used to improve dispersion of nanoparticles. Quantitative correlation between these surface-active molecules and nanoscale dispersion is absent from the literature partly because a quantitative measure of nanoscale dispersion does not exist. Recently, we have developed the Virial-van der Waals method to quantify dispersion in nanocomposites using virial coefficients. In this paper, the Langmuir adsorption model is used to quantify the influence of surface-active agents on nano-scale dispersion in terms of the effective second virial coefficient B2∗. The impact of silane coupling agent on the nano-dispersion and silica aggregate structure in precipitated silica/SBR nanocomposites is demonstrated. It is shown that the higher viscosity SBR matrix led to a greater silica aggregate structural breakup, while lower viscosity matrix improved surface silanization. The isomeric content of the SBR, which impacts the dielectric behavior, impacted whether the system could be modeled through a mean-field or specific interactions. We earlier showed that larger aggregates improve dispersion, and this is reaffirmed in these results. After account is made for aggregate size, nano-scale dispersion improves with the addition of silane coupling agent. The behavior is well modeled using Langmuir monolayer adsorption.

On a Differential Generalized Nash Equilibrium Problem with Mean Field Interaction

On a Differential Generalized Nash Equilibrium Problem with Mean Field Interaction

Two-dimensional ideal magnetohydrodynamic waves on a rotating sphere under a non-Malkus field: I. Continuous spectrum and its ray-theoretical interpretation

Two-dimensional ideal incompressible magnetohydrodynamic (MHD) linear waves at the surface of a rotating sphere are studied as a model to imitate the outermost layer of the Earth's core or the solar tachocline. This thin conducting layer is permeated by a toroidal magnetic field the magnitude of which depends only on the latitude. The Malkus background field, which is proportional to the sine of the colatitude, provides two well-known groups of branches; on one branch, retrograde Alfvén waves gradually become fast magnetic Rossby (MR) waves as the field amplitude decreases, and on the other, prograde Alfvén waves undergo a gradual transition into slow MR waves. In the case of non-Malkus fields, we demonstrate that the associated eigenvalue problems can yield a continuous spectrum instead of Alfvén and slow MR discrete modes. The critical latitudes attributed to the Alfvén resonance eliminate these discrete eigenvalues and produce an infinite number of singular eigenmodes. The theory of slowly varying wave trains in an inhomogeneous magnetic field shows that a wave packet related to this continuous spectrum propagates toward a critical latitude corresponding to the wave and is eventually absorbed there. The expected behaviour whereby the retrograde propagating packets pertaining to the continuous spectrum approach the latitudes from the equatorial side and the prograde ones approach from the polar side is consistent with the profiles of their eigenfunctions derived using our numerical calculations. Further in-depth discussions of the Alfvén continuum would develop the theory of the “wave–mean field interaction” in the MHD system and the understanding of the dynamics in such thin layers.

Donnan equilibrium in charged slit-pores from a hybrid nonequilibrium molecular dynamics/Monte Carlo method with ions and solvent exchange.

Ion partitioning between different compartments (e.g., a porous material and a bulk solution reservoir), known as Donnan equilibrium, plays a fundamental role in various contexts such as energy, environment, or water treatment. The linearized Poisson-Boltzmann (PB) equation, capturing the thermal motion of the ions with mean-field electrostatic interactions, is practically useful to understand and predict ion partitioning, despite its limited applicability to conditions of low salt concentrations and surface charge densities. Here, we investigate the Donnan equilibrium of coarse-grained dilute electrolytes confined in charged slit-pores in equilibrium with a reservoir of ions and solvent. We introduce and use an extension to confined systems of a recently developed hybrid nonequilibrium molecular dynamics/grand canonical Monte Carlo simulation method ("H4D"), which enhances the efficiency of solvent and ion-pair exchange via a fourth spatial dimension. We show that the validity range of linearized PB theory to predict the Donnan equilibrium of dilute electrolytes can be extended to highly charged pores by simply considering renormalized surface charge densities. We compare with simulations of implicit solvent models of electrolytes and show that in the low salt concentrations and thin electric double layer limit considered here, an explicit solvent has a limited effect on the Donnan equilibrium and that the main limitations of the analytical predictions are not due to the breakdown of the mean-field description but rather to the charge renormalization approximation, because it only focuses on the behavior far from the surfaces.

Weakly nonlocal matter-wave droplets and soliton trains engineering in a Bose-Einstein condensate

We propose a comprehensive analysis of the modulational instability (MI) phenomenon in one-dimensional Bose-Einstein condensate (BEC) in the context of mutually symmetric components in the binary mixture. The proposed model considers weak nonlocal cubic mean-field and quadratic beyond mean-field interactions arising from quantum fluctuations. The instability regions are analyzed using linear stability analysis of a continuous wave, considering maximum perturbation wavenumber and growth rate. The regions for quantum droplets (QDs) and solitons are identified by varying the nonlocality parameter. Numerical simulations confirm analytical predictions, revealing that nonlocality influences the formation of various QD profiles and instability time. Furthermore, solitonic patterns exhibit nonlocality-dependent behavior. Our study opens new doors to a better characterization of QDs in binary mixtures in the presence of higher-order beyond mean-field effects.

Mean-field interacting multi-type birth–death processes with a view to applications in phylodynamics

Multi-type birth–death processes underlie approaches for inferring evolutionary dynamics from phylogenetic trees across biological scales, ranging from deep-time species macroevolution to rapid viral evolution and somatic cellular proliferation. A limitation of current phylogenetic birth–death models is that they require restrictive linearity assumptions that yield tractable message-passing likelihoods, but that also preclude interactions between individuals. Many fundamental evolutionary processes – such as environmental carrying capacity or frequency-dependent selection – entail interactions, and may strongly influence the dynamics in some systems. Here, we introduce a multi-type birth–death process in mean-field interaction with an ensemble of replicas of the focal process. We prove that, under quite general conditions, the ensemble’s stochastically evolving interaction field converges to a deterministic trajectory in the limit of an infinite ensemble. In this limit, the replicas effectively decouple, and self-consistent interactions appear as nonlinearities in the infinitesimal generator of the focal process. We investigate a special case that is rich enough to model both carrying capacity and frequency-dependent selection while yielding tractable message-passing likelihoods in the context of a phylogenetic birth–death model.

Exploring spatial segregation induced by competition avoidance as driving mechanism for emergent coexistence in microbial communities.

This study investigates the role of spatial segregation, prompted by competition avoidance, as a key mechanism for emergent coexistence within microbial communities. Recognizing these communities as complex adaptive systems, we challenge the sufficiency of mean-field pairwise interaction models, and we consider the impact of spatial dynamics. We developed an individual-based spatial simulation depicting bacterial movement through a pattern of random walks influenced by competition avoidance, leading to the formation of spatially segregated clusters. This model was integrated with a Lotka-Volterra metapopulation framework focused on competitive interactions. Our findings reveal that spatial segregation combined with low diffusion rates and high compositional heterogeneity among patches can lead to emergent coexistence in microbial communities. This reveals a novel mechanism underpinning the formation of stable, coexisting microbe clusters, which is nonetheless incapable of promoting coexistence in the case of isolated pairs of species. This study underscores the importance of considering spatial factors in understanding the dynamics of microbial ecosystems.

Photon superfluidity through dissipation

Superfluidity—frictionless flow—has been observed in various physical systems such as liquid helium, cold atoms, and exciton polaritons. Superfluidity is usually realized by cooling and suppressing all dissipation. Here we challenge this paradigm by demonstrating signatures of superfluidity, enabled by dissipation, in the flow of light within a room-temperature oil-filled cavity. Dissipation in the oil mediates effective photon-photon interactions which are noninstantaneous and nonlocal. Such interactions were expected to severely limit the emergence of superfluidity in conservative photonic systems. Surprisingly, when launching a photon fluid with sufficiently high density and low velocity against an obstacle in our driven-dissipative cavity, we observe a record suppression of backscattering. Our experiments also reveal the reorganization dynamics of photons into a nonscattering steady state and a qualitatively changing behavior of the optical phase as light propagates around the obstacle. The phase is locked between the laser and the obstacle but evolves with the intensity in the wake of the obstacle where the density of the photon fluid and its mean-field interaction energy decrease. Using a generalized Gross–Pitaevskii equation for photons coupled to a thermal field, we model our experiments and elucidate how the noninstantaneous and nonlocal character of interactions influences the suppression of scattering associated with superfluidity. Beyond providing the first signatures of cavity photon superfluidity, and of any superfluid both at room temperature and in steady state, our results pave the way for probing photon hydrodynamics in arbitrary potential landscapes using structured mirrors. Published by the American Physical Society 2024

Quantum liquid in lower dimensions: From the perspective of surface tension

We analyze the surface tension in ultra-cold atomic gases in a quasi one-dimensional and one-dimensional geometry. In recent years, experimental observations have confirmed the “clustering of atoms” to form droplets in ultra-cold atomic gases and the emergence of this new phase is attributed to the beyond mean-field interaction. However, two decades earlier, liquid formation was predicted due to the competition of two-body and three-body interactions. Here, we review both propositions and comment on the role of beyond mean-field and three-body interaction in liquid formation by calculating the surface tension.

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

Large-population asymptotics for the maximum of diffusive particles with mean-field interaction in the noises

We study the N→∞ limit of the normalized largest component in some systems of N diffusive particles with mean-field interaction. By applying a universal time change, the interaction in noises is transferred to the drift terms, and the asymptotic behavior of the maximum becomes well-understood due to existing results in the literature. We expect that the normalized maximum in the original setting has the same limiting distribution as that of i.i.d copies of a solution to the corresponding McKean–Vlasov SDE and we present some results and numerical simulations that support this conjecture.

Asymptotic analysis of particle cluster formation in the presence of anchoring sites

The aggregation or clustering of proteins and other macromolecules plays an important role in the formation of large-scale molecular assemblies within cell membranes. Examples of such assemblies include lipid rafts, and postsynaptic domains (PSDs) at excitatory and inhibitory synapses in neurons. PSDs are rich in scaffolding proteins that can transiently trap transmembrane neurotransmitter receptors, thus localizing them at specific spatial positions. Hence, PSDs play a key role in determining the strength of synaptic connections and their regulation during learning and memory. Recently, a two-dimensional (2D) diffusion-mediated aggregation model of PSD formation has been developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a stationary distribution consisting of multiple clusters, whose average size can be determined using an effective mean-field description of the particle concentration around each anchored cluster. In this paper, we derive corrections to the mean-field approximation by applying the theory of diffusion in singularly perturbed domains. The latter is a powerful analytical method for solving two-dimensional (2D) and three-dimensional (3D) diffusion problems in domains where small holes or perforations have been removed from the interior. Applications range from modeling intracellular diffusion, where interior holes could represent subcellular structures such as organelles or biological condensates, to tracking the spread of chemical pollutants or heat from localized sources. In this paper, we take the bounded domain to be the cell membrane and the holes to represent anchored clusters. The analysis proceeds by partitioning the membrane into a set of inner regions around each cluster, and an outer region where mean-field interactions occur. Asymptotically matching the inner and outer stationary solutions generates an asymptotic expansion of the particle concentration, which includes higher-order corrections to mean-field theory that depend on the positions of the clusters and the boundary of the domain. Motivated by a recent study of light-activated protein oligomerization in cells, we also develop the analogous theory for cluster formation in a three-dimensional (3D) domain. The details of the asymptotic analysis differ from the 2D case due to the contrasting singularity structure of 2D and 3D Green’s functions.Graphical abstract2D model of diffusion-based protein cluster formation in the presence of anchoring cites and particle recycling. a A set of N anchoring sites at positions xj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{x}}_j$$\\end{document}, j=1,…,N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j=1,\\ldots ,N$$\\end{document}, in a bounded domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. b Diffusing particles accumulate at the anchoring sites resulting in the formation of particle aggregates or clusters Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {U}}}_j$$\\end{document}. c The clusters are dynamically maintained by a combination of lateral diffusion outside the clusters and particle recycling

Regressing bubble cluster dynamics as a disordered many-body system

The coherent dynamics of bubble clusters are of fundamental and industrial importance, and are elusive due to the complex interactions of disordered bubble oscillations. Here we introduce and demonstrate a method for decomposition of the Lagrangian time series of bubble dynamics data by combining theory and principal component analysis. The decomposition extracts coherent features of bubble oscillations based on their energy, in a way similar to proper orthogonal decomposition of Eulerian flow field data. This method is applied to a dataset of spherical clusters under harmonic excitation at different amplitudes, with various nuclei density and polydispersity parameters. Results indicate that the underlying correlated mode of oscillations is isolated in a single dominant feature in cavitating regimes, independent of the nuclei's parameters. A systematic data analysis procedure further suggests that this feature is globally controlled by the dynamic cloud interaction parameter of Maeda and Colonius (J. Fluid Mech., vol. 862, 2019, pp. 1105–1134) that quantifies the mean-field interactions, regardless of initial polydispersity or nonlinearity. The method provides a simplified and comprehensive representation of complex bubble dynamics as well as a new path to reduced-order modelling of cavitation and nucleation.

A mean field game approach to optimal investment and risk control for competitive insurers

We consider an insurance market consisting of multiple competitive insurers with a mean field interaction via their terminal wealth under the exponential utility with relative performance. It is assumed that each insurer regulates her risk by controlling the number of policies. We respectively establish the constant Nash equilibrium (independent of time) on the investment and risk control strategy for the finite n-insurer game and the constant mean field equilibrium for the corresponding mean field game (MFG) problem (when the number of insurers tends to infinity). Furthermore, we examine the convergence relationship between the constant Nash equilibrium of finite n-insurer game and the mean field equilibrium of the corresponding MFG problem. Our numerical analysis reveals that, for a highly competitive insurance market consisting of many insurers, every insurer will invest more in risky assets and increase the total number of outstanding liabilities to maximize her exponential utility with relative performance.

Constant-pH Simulations of a Coarse-Grained Model of Polyfunctional Weak Charged Biopolymers

A coarse-grained model of linear polyfunctional weak charged biopolymers was implemented, formed of different proportions of acid-base groups resembling the composition of humic substances. These substances are mainly present in dissolved organic matter in natural water. The influence of electrostatic interactions computing methods, factors concerning the structure of the chain, different functional groups, and the ionic strength on polyelectrolytes were studied. Langevin dynamics with constant pH simulations were performed using the ESPResSO package and the Python-based Molecule Builder for ESPResSo (pyMBE) library. The coverage was fitted to a polyfunctional Frumkin isotherm, with a mean-field interaction between charged beads. The composition of the chain affects the charge while ionic strength affects both the charge and the radius of gyration. Additionally, the parameters intrinsic to the polyelectrolyte model were well reproduced by fitting the polyfunctional Frumkin isotherm. In contrast, the non-intrinsic parameters depended on the ionic strength. The method developed and applied to a polyfunctional polypeptide model, that resembles a humic acid, will be very useful for characterizing biopolymers with several acid-base functional groups, where their structure, the composition of the different functional groups, and the determination of the main intrinsic proton binding constants and their proportion are not exactly known.

Social Depolarization: Blume–Capel Model

This study belongs to an emerging area of research seeking ways to depolarize societies in the short run (around events such as elections) as well as in a sustainable fashion. We approach the depolarization process with a model of three homophilic groups (US Democrats, Republicans, and Independents interacting in the context of upcoming federal elections). We expand a previous polarization model, which assumed that each individual interacts with all other individuals in its group with mean-field interactions. We add a depolarization field, which is analogous to the Blume–Capel model’s crystal field. There are currently numerous depolarization efforts around the world, some of which act in ways similar to this depolarization field. We find that for low values of the depolarization field, the system continues to be polarized. When the depolarization field is increased, the polarization decreases.

Fluctuation-driven self-trapping in Bose-Bose mixtures

We investigate the collective dynamics of two-component Bose-Einstein condensates in a double well potential. By taking into account the Lee-Huang-Yang (LHY) correction from quantum fluctuation, we find that the LHY term, though much smaller than the mean field interactions, can significantly change the coherent oscillating behaviors of the bosonic Josephson junction in the regime of g12≃−g11g22. Besides the quantitative renormalization to the plasma frequencies of the Josephson oscillations, a series of unexpected nonzero fixed points beyond mean field emerges in the stationary state. More remarkably, unique macroscopic self-trapping states, which are usually absent in the mean field dynamics, can be sustained by the LHY nonlinearity. Our results reveal the nontrivial LHY effect on the Josephson dynamics, which can be tested in the current experiment. Published by the American Physical Society 2024

Conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching: Wellposedness, propagation of chaos, averaging principle

The conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching are investigated in this work. We use the L2-Wasserstein distance to measure the regularity of the coefficients in the probability measure argument. Also, we establish the propagation of chaos for the associated mean-field interaction particle system with common noise and provide an explicit bound on the convergence rate. Furthermore, an averaging principle is established for two time-scale conditional McKean-Vlasov equations, where much attention is paid to the convergence of the conditional distribution term.

Macroparticle Collisionality in PIC solver

This paper presents a new numerical technique for the calculation of the Coulomb interaction between nearby charged particles. The efficiency of this method is demonstrated by distributing charged particles in a three dimensional sphere for an amount of particles, Npar , ranging from 1 × 105 to 1.6 × 106. Using the presented method, the computation of the mean field and Coulomb near field interaction can be performed separately.

Evaluation of the excitation spectra with diffusion Monte Carlo on an auxiliary bosonic ground state.

We aim to improve upon the variational Monte Carlo (VMC) approach for excitations replacing the Jastrow factor by an auxiliary bosonic (AB) ground state and multiplying it by a fermionic component factor. The instantaneous change in imaginary time of an arbitrary excitation in the original interacting fermionic system is obtained by measuring observables via the ground-state distribution of walkers of an AB system that is subject to an auxiliary effective potential. The effective potential is used to (i) drive the AB system's ground-state configuration space toward the configuration space of the excitations of the original fermionic system and (ii) subtract from a diffusion Monte Carlo (DMC) calculation contributions that can be included in conventional approximations, such as mean-field and configuration interaction (CI) methods. In this novel approach, the AB ground state is treated statistically in DMC, whereas the fermionic component of the original system is expanded in a basis. The excitation energies of the fermionic eigenstates are obtained by sampling a fermion-boson coupling term on the AB ground state. We show that this approach can take advantage of and correct for approximate eigenstates obtained via mean-field calculations or truncated interactions. We demonstrate that the AB ground-state factor incorporates the correlations missed by standard Jastrow factors, further reducing basis truncation errors. Relevant parts of the theory have been tested in soluble model systems and exhibit excellent agreement with exact analytical data and CI and VMC approaches. In particular, for limited basis set expansions and sufficient statistics, AB approaches outperform CI and VMC in terms of basis size for the same systems. The implementation of this method in current codes, despite being demanding, will be facilitated by reusing procedures already developed for calculating ground-state properties with DMC and excitations with VMC.