Mean-field Method Research Articles

Pairing phase favored by magnetic frustration.

The interplay between magnetic frustration and pairing is investigated by adopting a BCS-like pairing mechanism on the frustrated $J_1-J_2$ Ising model on the square lattice. The ground-state and thermal phase transitions of the model are analyzed using a fermionic formulation within a cluster mean-field method. In this approach, the lattice system is divided into identical clusters, where the intracluster dynamic is exactly solved, and the intercluster interactions are replaced by self-consistent mean fields. We introduce a framework with two pairing couplings: an intracluster local coupling, $g$, which controls the electron pairing and its mobility within the clusters, and an intercluster coupling, $g'$, which adjusts the pairing mechanism between clusters. Tuning $g'/g$ allows evaluating how the pairing phase evolves from a weak pairing coupling between clusters (clustered system) to a strong one ($g' \rightarrow g$, homogeneous system). In the range $0 \le g'/g \le 1$, we find that a gradual increase in $g'/g$ favors the pairing phase and induces a change in criticality. In particular, our results reveal the presence of tricriticality for a certain range of $g'/g$. In addition, an increase in competing magnetic interactions weakens the magnetic orders, causing the pairing phase to occur at lower strengths of pairing interactions, especially when $g' = g$. Therefore, our findings support that magnetic frustration favors the pairing phase, contributing to the onset of a superconducting state.

Micromechanics Modeling on Mechanical Properties in Mg Alloys with Bimodal Grain Size Distribution

Bimodal grain structure (BGS) Mg alloys containing a high fraction of fine grains (FGs) and a low fraction of coarse grains (CGs) show a good combination of strength and plasticity. Here, taking the ZK60 alloy as an example, the influences of CG size, volume fraction, and texture intensity on mechanical properties and the hetero-deformation-induced (HDI) effect were examined using the Mori–Tanaka mean-field method combined with strain gradient theory of plasticity. The results indicate that the overall mechanical properties decrease with an increase in CG size because the limited HDI effect cannot compensate for the strength and plasticity decrease derived from larger CGs. A higher aspect ratio of CG along the loading direction can weaken the HDI effect and subsequently reduce the overall mechanical properties. Optimal comprehensive mechanical properties can be achieved when the CG volume fraction is approximately 30%. Furthermore, an increasing basal texture intensity in CG results in higher yield strength and lower ultimate tensile strength, while the uniform elongation reaches a maximum value when ~60% of CGs possess hard orientations with Euler angles of (0~30°, 0°, 0°).

Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks

The modeling and analysis of epidemic processes in networks have attracted much attention over the past few decades. A major underlying assumption is that the recovery process and infection process are homogeneous, allowing the associated theoretical studies to be conducted in a convenient manner. However, the recovery and infection processes usually exhibit heterogeneous rates in the real world, which makes it difficult to characterize the general relations between the dynamics and the underlying network structure. In this work, we focus on the susceptible–infected–susceptible (SIS) epidemic process with heterogeneous recovery rates in a finite-size complete graph. Specifically, we study the metastable-state statistical properties of SIS epidemic dynamics with two different nodal recovery rates in complete graphs. We propose approximate solutions to the metastable-state expectation and the variance in the number of infected nodes within the framework of the mean-field approximation method. We also derive several upper and lower bounds of the steady-state probability that a node is in the infected state. We verify the proposed approximate solutions of the mean and variance via simulations. This work provides insights into the fluctuations in the statistical properties of epidemic processes with complex dynamical behaviors in networks.

Triaxial deformation, shell structure, shell corrections and the connection to alpha-clustering in nuclei

Abstract We explore the connection between the appearance of quasi-stable structures in mean-field type calculations, which arise as a result of the evolution of the underlying shell structure as a function of deformation, and $\alpha$-clustering in light even-even nuclei. The Nilsson-Strutinsky mean-field approach employs a macroscopic liquid-drop whose energy is modified by a shell correction term derived using the Strutinsky method. This method reflects the variations in the energies of the single-particle states with deformation. As such, there is no obvious connection to clustering. 

Here we use the changing level scheme of the deformed harmonic oscillator as a function of triaxial deformation to fully explore the variation in stability of $\alpha$-cluster structures in light even-even nuclei. The energies of the harmonic oscillator levels are used to deduce the energy required to disrupt the $\alpha$-cluster as a function of the triaxial deformation. We find that there is good agreement between variations in the shell correction energy in the mean-field method and the energy required to disrupt the $\alpha$-cluster. This provides a necessary link between understanding of the appearance of quasi-stable $\alpha$-cluster structures and quasi-stable shapes appearing in mean-field calculations.

Restricted Open-Shell Cluster Mean-Field theory for Strongly Correlated Systems.

The cluster-based Mean Field method (cMF) and it is second order perturbative correction was introduced by Jiménez-Hoyos and Scuseria to reduce the cost of modeling strongly correlated systems by dividing an active space up into small clusters, which are individually solved in the mean-field presence of each other. In that work, clusters with unpaired electrons are treated by allowing the α and β orbitals to spin polarize. While that provided significant energetic stabilization, the resulting cMF wave function was spin-contaminated, making it difficult to use as a reference state for spin-pure post-cMF methods. In this work, we propose the Restricted Open-shell cMF (RO-cMF) method, extending the cMF approach to systems with open-shell clusters, while not permitting spin-polarization. While the resulting RO-cMF energies are necessarily higher in energy than the unrestricted orbital cMF, the new RO-cMF provides a simple reference state for post-cMF methods that recover the missing intercluster correlations. We provide a detailed explanation of the method, and report demonstrative calculations of exchange coupling constants for three systems: a di-iron complex, a dichromium complex, and a dimerized organic radical. We also report the first perturbatively corrected RO-cMF-PT2 results as well.

Heavy Metal Rejection Performance and Mechanical Performance of Cellulose-Nanofibril-Reinforced Cellulose Acetate Membranes.

In this research, cellulose acetate (CA) and CA nanocomposite membranes, reinforced with mass fractions of cellulose nanofibrils (CNF), are prepared using the phase separation technique. The membranes are extensively characterized using several techniques: Fourier Transform Infrared (FTIR) spectroscopy confirms the chemical structures, while Scanning Electron Microscopy (SEM) reveals their surface morphology. Mechanical characterization is conducted to explore the mechanical behavior of the membranes under wet and dry conditions through tensile testing. The mechanical properties of CA and CA-CNF membranes are also estimated using the Mori-Tanaka mean-field homogenization method and compared to experimental findings. The flux performance for pure and dam water, assessed at 3 bar, demonstrates that CNF reinforcement notably enhances the CA membrane's performance, particularly in flux rate and fouling resistance. The CA membrane shows high efficiency in removing Fe2+, Ba2+, and Al3+ from dam water, while CA-CNF membranes exhibit a varied range of removal efficiencies for the same ions, with the 0.5 wt % CNF variant showing superior resistance to surface fouling. Additionally, while CNF increases tensile strength and stiffness, it leads to earlier failure under smaller deformations, especially at higher concentrations. This research provides a detailed assessment of CA and CA-CNF membranes, examining their chemical, structural, and mechanical properties alongside their effectiveness in water treatment applications.

Variational Bayes latent class analysis for EHR-based phenotyping with large real-world data

IntroductionBayesian approaches to patient phenotyping in clinical observational studies have been limited by the computational challenges associated with applying the Markov Chain Monte Carlo (MCMC) approach to real-world data. Approximate Bayesian inference via optimization of the variational evidence lower bound, variational Bayes (VB), has been successfully demonstrated for other applications.MethodsWe investigate the performance and characteristics of currently available VB and MCMC software to explore the practicability of available approaches and provide guidance for clinical practitioners. Two case studies are used to fully explore the methods covering a variety of real-world data. First, we use the publicly available Pima Indian diabetes data to comprehensively compare VB implementations of logistic regression. Second, a large real-world data set, Optum™ EHR with approximately one million diabetes patients extended the analysis to large, highly unbalanced data containing discrete and continuous variables. A Bayesian patient phenotyping composite model incorporating latent class analysis (LCA) and regression was implemented with the second case study.ResultsWe find that several data characteristics common in clinical data, such as sparsity, significantly affect the posterior accuracy of automatic VB methods compared with conditionally conjugate mean-field methods. We find that for both models, automatic VB approaches require more effort and technical knowledge to set up for accurate posterior estimation and are very sensitive to stopping time compared with closed-form VB methods.DiscussionOur results indicate that the patient phenotyping composite Bayes model is more easily usable for real-world studies if Monte Carlo is replaced with VB. It can potentially become a uniquely useful tool for decision support, especially for rare diseases where gold-standard biomarker data are sparse but prior knowledge can be used to assist model diagnosis and may suggest when biomarker tests are warranted.

Self-induced Bose glass phase in quantum quasicrystals

We study the emergence of Bose glass phases in self sustained bosonic quasicrystals induced by a pair interaction between particles of Lifshitz–Petrich type. By using a mean-field variational method designed in momentum space as well as Gross–Pitaevskii simulations we determine the phase diagram of the model. The study of the local and global superfluid fraction allows the identification of supersolid, super quasicrystal, Bose glass and insulating phases. The Bose glass phase emerges as a quasicrystal phase in which the global superfluidity is essentially zero, while the local superfluidity remains finite in certain ring structures of the quasicrystalline pattern. Furthermore, we perform continuous space Path Integral Monte Carlo simulations for a case in which the interaction between particles stabilizes a quasicrystal phase. Our results show that as the strength of the interaction between particles is increased the system undergoes a sequence of states consistent with the super quasicrystal, Bose glass, and quasicrystal insulator thermodynamic phases.

A simplicial SSIS epidemic model with the outgoing pressure

A new simplicial epidemic model that considers the pressure of out-going is proposed to describe the characteristics of clustering on disease transmission more accurately. In addition, the probability evolution equations of nodes in each state are obtained by the quenched mean-field method. Furthermore, we analyze the conditions of the existence and the stability of the equilibrium points. Subsequently, the sensitivity analysis of the parameters is investigated, and it can be concluded that the degree about pairwise transmission rate has great impact on the propagation threshold. Our simulation results indicate that the system produces forward bifurcation or backward bifurcation via the one-parameter bifurcation diagram, and the bistable state of the system appears under certain conditions. Meanwhile, we obtain the transition conditions of the system from the disease-free equilibrium state to the bistable state through the divisional diagram. It is also noticed that the pressure of out-going plays a crucial role in the spreading process of diseases. On the one hand, the increasing of the pressure of out-going leads to the decreasing of the disease transmission threshold and a faster outbreak of disease. On the other hand, an increase in the individuals without the pressure of out-going causes the increasing of transmission threshold and a slower outbreak of disease.

Chimera states and eigen microstates of nonidentical power-law coupled oscillators with heterogeneous phase lag

Chimera state is a special spatiotemporal patterns where coherent and incoherent states coexist, not only related to neuronal evolution but also to diseases such as parkinson, epilepsy, and schizophrenia. In many studies of biological networks, the interaction patterns between individuals often exhibit a scale-free heavy-tailed distribution. Hence, researches into chimera state within scale-free structures may help to understand the close relationship between function and structure in biological systems. Especially in neural systems, interactions between neurons depend on both electrical and chemical signals, naturally involving heterogeneous delay effects. In this study, we introduce power-law coupling, nonidentical natural frequency, and heterogeneous phase delay into the phase oscillator model, exploring the relationship between oscillator coupled strength, power-law exponent, and the emergence of chimera state. Other important thing is that we extend the Eigen Microstate (EM) method to investigate the criticality, attempting to reveal the mechanism of chimera state emergence through the mesoscopic structure of eigen microstate. We find that the critical analysis results of the EM method, relying solely on phase data, are entirely consistent with those obtained by the mean-field method, revealing the inevitable relationship between the emergence of chimera state and the structure–power law coupling. In addition, the spatiotemporal patterns of eigen microstate provide insights into the mesoscopic structure, can be used to distinguish spatial distribution of synchronization, chimera state, and disorder state, as well as their characteristic dynamics of oscillators. This expansion research of the EM method will advance the exploration of criticality in real oscillatory systems.

Classical spin models and basic magnetic interactions on 1/1-approximant crystals

We study classical spin models on the 1/1 Tsai-type approximant lattice using Monte Carlo and mean-field methods. Our aim is to understand whether the phase diagram differences between Gd- and Tb-based approximants can be attributed to anisotropy induced by the crystal-electric field. To address this question, we treat Gd ions as Heisenberg spins and Tb ions as Ising spins. Additionally, we consider the presence of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction to replicate the experimentally observed correlation between magnetic properties and electron concentration. Surprisingly, our findings show that the transition between ferromagnetic and antiferromagnetic order remains unaltered by the anisotropy, even when accounting for the dipole interaction. We conclude that a more comprehensive model, extending beyond the free-electron gas RKKY interaction, is likely required to fully understand the distinctions between Gd- and Tb-based approximants. Our work represents a systematic exploration of the impact of anisotropy on the ground-state properties of classical spin models in quasicrystal approximants. Published by the American Physical Society 2024

Model-agnostic neural mean field with a data-driven transfer function

As one of the most complex systems known to science, modeling brain behavior and function is both fascinating and extremely difficult. Empirical data is increasingly available from ex vivo human brain organoids and surgical samples, as well as in vivo animal models, so the problem of modeling the behavior of large-scale neuronal systems is more relevant than ever. The statistical physics concept of a mean-field model offers a tractable way to bridge the gap between single-neuron and population-level descriptions of neuronal activity, by modeling the behavior of a single representative neuron and extending this to the population. However, existing neural mean-field methods typically either take the limit of small interaction sizes, or are applicable only to the specific neuron models for which they were derived. This paper derives a mean-field model by fitting a transfer function called Refractory SoftPlus, which is simple yet applicable to a broad variety of neuron types. The transfer function is fitted numerically to simulated spike time data, and is entirely agnostic to the underlying neuronal dynamics. The resulting mean-field model predicts the response of a network of randomly connected neurons to a time-varying external stimulus with a high degree of accuracy. Furthermore, it enables an accurate approximate bifurcation analysis as a function of the level of recurrent input. This model does not assume large presynaptic rates or small postsynaptic potential size, allowing mean-field models to be developed even for populations with large interaction terms.

Scalar cosmological perturbations from quantum gravitational entanglement

Abstract A major challenge at the interface of quantum gravity and cosmology is to explain the emergence of the large-scale structure of the Universe from Planck scale physics. In this letter, we extract the dynamics of scalar isotropic cosmological perturbations from full quantum gravity, as described by the causally complete Barrett-Crane group field theory model. From the perspective of the underlying quantum gravity theory, cosmological perturbations are represented as nearest-neighbor two-body entanglement of group field theory quanta. Their effective dynamics is obtained via mean-field methods and described relationally with respect to a causally coupled physical Lorentz frame. We quantitatively study these effective dynamical equations and show that at low energies they are perfectly consistent with those of General Relativity, while for trans-Planckian scales quantum effects become important. These results therefore not only provide crucial insights into the potentially purely quantum gravitational nature of cosmological perturbations, but also offer rich phenomenological implications for the physics of the early Universe.

Nonadiabatic Exciton Dynamics and Energy Gradients in the Framework of FMO-LC-TDDFTB.

We introduce a novel methodology for simulating the excited-state dynamics of extensive molecular aggregates in the framework of the long-range corrected time-dependent density-functional tight-binding fragment molecular orbital method (FMO-LC-TDDFTB) combined with the mean-field Ehrenfest method. The electronic structure of the system is described in a quasi-diabatic basis composed of locally excited and charge-transfer states of all fragments. In order to carry out nonadiabatic molecular dynamics simulations, we derive and implement the excited-state gradients of the locally excited and charge-transfer states. Subsequently, the accuracy of the analytical excited-state gradients is evaluated. The applicability to the simulation of exciton transport in organic semiconductors is illustrated on a large cluster of anthracene molecules. Additionally, nonadiabatic molecular dynamics simulations of a model system of benzothieno-benzothiophene molecules highlight the method's utility in studying charge-transfer dynamics in organic materials. Our new methodology will facilitate the investigation of excitonic transfer in extensive biological systems, nanomaterials, and other complex molecular systems consisting of thousands of atoms.

Effect of individual activity level heterogeneity on disease spreading in higher-order networks.

The active state of individuals has a significant impact on disease spread dynamics. In addition, pairwise interactions and higher-order interactions coexist in complex systems, and the pairwise networks proved insufficient for capturing the essence of complex systems. Here, we propose a higher-order network model to study the effect of individual activity level heterogeneity on disease-spreading dynamics. Activity level heterogeneity radically alters the dynamics of disease spread in higher-order networks. First, the evolution equations for infected individuals are derived using the mean field method. Second, numerical simulations of artificial networks reveal that higher-order interactions give rise to a discontinuous phase transition zone where the coexistence of health and disease occurs. Furthermore, the system becomes more unstable as individual activity levels rise, leading to a higher likelihood of disease outbreaks. Finally, we simulate the proposed model on two real higher-order networks, and the results are consistent with the artificial networks and validate the inferences from theoretical analysis. Our results explain the underlying reasons why groups with higher activity levels are more likely to initiate social changes. Simultaneously, the reduction in group activity, characterized by measures such as "isolation," emerges as a potent strategy for disease control.

The dynamical analysis of simplicial SAIS epidemic model with awareness programs by media

In this paper, we propose an SAIS epidemic model with awareness programs by media on higher-order networks to study the impact of media and higher-order networks on disease transmission, in which the simplicial complex is utilized to construct a social network that describes connections between nodes, where a single link can connect more than two individuals. We use the cumulative density of awareness programs to quantify the impact of media and use the Microscopic Markov Chains method to get the probability evolution equation of each node. In order to solve the computational difficulties caused by the excessively high dimension of the equation, we use the mean field method to reduce the dimension of the equation to get the equilibrium of the system. Through dynamical analysis, we obtain the threshold of disease outbreak and conditions for the existence and stability of the disease-free equilibrium and the endemic equilibrium. The numerical simulations of SAIS model prove that the mean field method has good predictive performance. We also obtain the influence of parameters on threshold and system dynamical behavior through sensitivity analysis. The results show that: on one hand, the introduce of 2-simplex increases the probability of disease transmission in healthy populations, and the system appears discontinuous transition and bistable coexistence of health state and disease epidemic state, on the other hand, awareness programs by media can increase the threshold of disease outbreak, decrease the final density of infected individuals and inhibit the explosive growth of disease, thus effectively control the spread of diseases.

Mode analysis of spin field of thermal atomic ensembles

The spin dynamics in a thermal atomic vapor cell have been investigated thoroughly over the past decades and have proven to be successful in quantum metrology and memory owing to their long coherent time and manipulation convenience. The existing mean field analysis of spin dynamics among the whole cell is sometimes inaccurate due to the non-uniformity of the ensemble and spatial coupling of multi-physical fields interacting with the ensembles. Here we perform mode analysis onto the quasi-continuous spin field including atomic thermal motion to derive Bloch mode equations and obtain corresponding analytical solutions in diffusion regime. We demonstrate that the widely used mean field dynamics of thermal gas is a particular case in our solution, corresponding to the uniform spatial mode. This mode analysis approach offers a precise method for analyzing the dynamics of the spin ensemble in greater detail from a field perspective, enabling the effective determination of spatially non-uniform multi-physical fields coupling with the spin ensembles, which cannot be accurately analyzed by the mean field method. Furthermore, this work paves the way to address quantum noises and relaxation mechanisms associated with non-uniform fields and inter-atomic interactions, which limit further improvement of ultra-sensitive spin-based sensors.

Markov game for CV joint adaptive routing in stochastic traffic networks: A scalable learning approach

This study proposes a learning-based approach to tackle the challenge of joint adaptive routing in stochastic traffic networks with Connected Vehicles (CVs). We introduce a Markov Routing Game (MRG) to model the adaptive routing behavior of all vehicles in such networks, thereby incorporating both competitive route choices and real-time decision-making. We establish the existence of the Nash policy (i.e., optimal joint adaptive routing policy) within the MRG that enables vehicles to adapt optimally to real-time traffic conditions online through efficient communication. To enhance scalability, we innovate with a homogeneity-based mean-field approximation method and, based on that, further develop the Homogeneity-based Mean-Field Deep Reinforcement Learning (HMF-DRL) algorithm to learn the Nash policy within the MRG. Through numerical experiments on the Nguyen–Dupuis network, we demonstrate our algorithm’s ability to efficiently converge and learn the joint adaptive routing policy that significantly enhances traffic network efficiency. Furthermore, our study provides insights into the effects of travel demand, penetration of CVs, and levels of uncertainty on the performance of the joint adaptive routing policy. This paper presents a significant step towards improving network efficiency and reducing the travel time for a majority of vehicles amid uncertain traffic conditions.

Effects of Nanowire Doping on Plasmon-Enhanced N2 Dissociation.

Doping a transition metal element into plasmonic systems can tune the optical properties of the system, which will potentially facilitate the plasmon-enhanced catalytic process. In this study, we applied the linear-response time-dependent density functional theory (LR-TDDFT) method with real-time electron dynamics and mean-field Ehrenfest dynamics methods to computationally investigate the effects of doping silver nanowires on plasmon-enhanced N2 dissociation. We calculated the absorption spectra for different doped systems, applied an external electric field to the system, and performed mean-field Ehrenfest dynamics to examine how plasmonic excitation will affect the N2 activation or dissociation. In addition, we also studied how the transition metal dopant affects the system's electronic structure and potential energy surface.

Adaptive mean field multi-agent reinforcement learning

Large-scale Multi-Agent Reinforcement Learning (MARL) is fundamentally a challenge due to the curse of dimensionality. In a homogeneous multi-agent setting, mean field theory gives an effective way of scalable MARL by abstracting other agents to a virtual mean agent, assuming that the influence between agents is equal and infinitesimal. However, in some real scenarios, only several neighboring agents, rather than all agents, affect the decision-making of an agent, and different neighboring agents may have varying degrees of influence on the agent's decision-making. In this paper, not restricted to a homogeneous setting, we propose adaptive mean field MARL, which is based on the attention mechanism and can be used to deal with many-agent scenarios where there may be different influence relationships among agents. Specifically, we first derive the mean field approximation with adaptive weight and give the error bound of the approximation. Then, we propose adaptive mean field Q-Learning and describe how to obtain the adaptive weight. In addition, we discuss the differences between the proposed approach and existing mean-field MARL methods. Finally, we conduct experiments on simulation platforms, and the results show that the performance of the proposed approach outperforms that of the state-of-the-art method.