Mean-field Approximation Research Articles

Re-routing game: The inadequacy of mean-field approach in modeling the herd behavior in path switching

Coordination of vehicle routes is a feasible way to ease traffic congestion issues amid a fixed road infrastructure. Nevertheless, even when optimal route configurations are provided to individual drivers, it is hard to achieve due to the fact that greedy drivers may switch to other routes to lower individual costs. Recent research uses a mean-field cavity approach from spin glass studies to analyze the impact of path switching in optimized transportation networks. However, this method only provides a mean-field approximation, which does not account for the collective herd behavior in path switching due to uncoordinated individual decisions. In this study, we propose an exhaustive cavity approach to investigate the impact of uncoordinated path switching in a re-routing game and reveal that greedy drivers’ decisions can be highly correlated, leading to the failure of mean-field approaches. Our theoretical results fit well with simulations, and our developed framework can be generalized to analyze other games with multiple players and rounds. Our results shed light on the impact of herd behavior among uncoordinated human drivers in suppressing congestion through path coordination.

Extended Nambu–Jona-Lasinio model for quark and nuclear matters

In this work, we extend the two-flavor Nambu–Jona-Lasinio model to one capable of exploring quark and nuclear matter consistently. With an extra term standing for quark-nucleon interactions, nucleons could automatically emerge as color-singlet three-quark entities by following a process similar to mesons. Besides the quark part in mean field approximation, both mesons and nucleons could contribute to the thermodynamic potential thus possibly give rise to quarkyonic matter beyond mean field. In the study, two kinds of “confining” couplings are adopted for the new interaction term and two different quark masses are considered for comparison. It turns out that only confined nuclear matter or deconfined quark matter is possible for all the cases at zero temperature, thus quarkyonic matter is not favored at all. Even more strictly, only the case with stronger confinement effect and a smaller quark mass admits a physical first-order phase transition from nuclear matter to quark matter around twice saturation density.

Elementary Steps of Catalytic Reactions Occurring on Metallic Alloy Nanoparticles.

Catalytic reactions running in an adsorbed overlayer on metallic alloy nanoparticles are of high interest in the context of applications in the chemical industry. The understanding of the corresponding kinetics is, however, still limited. One of the reasons of this state of the art is the interplay between adsorption and adsorbate-influenced segregation of metal atoms inside alloy nanoparticles. I scrutinize this interplay by using a generic field model of segregation and the mean-field approximation in order to describe adsorption, desorption, and elementary catalytic reactions. Under steady-state conditions, the segregation is demonstrated to be manifested in the change of the dependence of the activation energies of desorption or elementary reactions on coverage, and the sign of this change is positive. The effect of this change on the apparent reaction orders is briefly discussed as well.

First-principles studies on the magnetic properties, exchange interactions and spin Hall conductivity of the half-Heusler alloy MgMnGe

This study presents first-principles estimates of the magnetic characteristics, exchange interactions and the spin Hall conductivity (SHC) of the half -Heusler alloy MgMnGe. Various configurations of spin-ordering are taken into consideration. It is discovered that MgMnGe has an antiferromagnetic ground state, with a degenerate in plane Néel vector. The Korringa-Kohn-Rostoker method has been used to calculate the Néel temperature and exchange interactions. The Néel temperature of this material obtained theoretically using mean field approximation is 447.2 K while the experimental reported value is largely higher than 300 K. The exchange interaction has been shown to be mostly mediated by the Mn1-Mn2 exchange. The spin Hall conductivity of MgMnGe is obtained by mapping the electronic structures to the Wannier basis via linear response theory. The spin Hall conductivity as a function of the Fermi level has been determined.

The interplay between liquid–liquid and ferroelectric phase transitions in supercooled water

The distinctive characteristics of water, evident in its thermodynamic anomalies, have implications across disciplines from biology to geophysics. Considered a valid hypothesis to rationalize its unique properties, a liquid-liquid phase transition in water below the freezing point, in the so-called supercooled regime, has nowadays been observed in several molecular dynamics simulations and is being actively researched experimentally. The hypothesis of ferroelectric phase transition in supercooled water can be traced back to 1977, due to Stillinger. In this work, we highlight intriguing and far-reaching implications of water: The ferroelectric and liquid-liquid phase transitions can be designed as two facets of the same underlying phenomenon. Our results are based on the analysis of extensive molecular dynamics simulations and are explained in the context of a classical density functional theory in mean-field approximation valid for a polar liquid, where dipolar interaction is treated perturbatively. The theory underpins the potential role of ferroelectricity in promoting the liquid-liquid phase transition, being the density-polarization coupling inherent in the dipolar interaction potential. The existence of ferroelectric order in supercooled low-density liquid water is confirmed by the observation in molecular dynamics simulations of collective modes in space-time polarization correlation functions, traceable to spontaneous breaking of continuous rotational symmetry. Our work sheds light on water's supercooled behavior and opens the door to experimental investigations of the static and dynamic behavior of water's polarization.

Role of harmonic and anharmonic interactions in controlling the cooperativity of spin-crossover molecular crystals

We present a theoretical model of spin transitions in homogeneous spin-crossover (SCO) materials, based on the description of the interactions between SCO molecules by the Lennard-Jones potential in which the equilibrium distances depend on the spin states of the linked molecules. By expanding the potential to order 3 considering a homogeneous lattice spacing, we demonstrate that the present model is isomorphic with an Ising-like model combining infinite long-range “ferromagnetic-like” interactions, competing with short-range “antiferromagnetic-like” between nearest neighbors in the case of harmonic interactions. When an anharmonic contribution is included, the short-range interaction becomes dependent on the average high-spin fraction, and changes its sign along the spin transition. The thermodynamic properties of the model were first investigated analytically in mean field approximation for both harmonic and anharmonic cases, for which we clarify the conditions for obtaining gradual and first-order transitions. Moreover, we demonstrated that within this approximation, the strength of the antiferromagnetic-like interactions are insufficient to produce multistep transitions. In contrast, the resolution of the previous emerging Ising-like model by Monte Carlo simulations demonstrated the possibility of obtaining gradual, first-order and two-step transitions as a result of the antagonism between long- and short-range interactions. Overall, the obtained results demonstrate the suitability of the present theoretical model to describe cooperative SCO materials. The key message of our paper is that our model can be used to predict complex behavior of spin-crossover crystals. Published by the American Physical Society 2024

Relativistic BEC extracted from a complex FRG flow equation

Abstract Based on the functional renormalization group (FRG) under the local potential approximation, we analyze the Bose-Einstein condensation (BEC) in the relativistic complex scalar theory. This framework leads to a complex flow equation of the effective potential, even with the well-known Litim regulator. In order to evaluate the condensate from such a complex effective potential, we impose a condition between chemical potential and mass, analogously to those in the free theory or the mean field theory. We elucidate that for the strongly (weakly) coupled theory, the phase diagrams computed from the FRG are more (less) deviated from that under the mean field approximation. This result implies that quantum fluctuations strongly affect the nonperturbative formation of the BEC.

Dual symmetries of dense isotopically and chirally asymmetric QCD

In the present paper, the dual symmetries of dense quark matter phase diagram found in some massless three- and two-color NJL models in the mean field approximation have been shown to exist at a more fundamental level as dual transformations of fields and chemical potentials leaving the Lagrangian invariant. As a result, the corresponding dual symmetries of the full phase diagram can be shown without any approximation. And it has been shown not only in the NJL models, but also in framework of two- and three-color massless QCD itself. This is quite interesting, since one might say that it is not very common to show something completely non-perturbatively in QCD.

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.

Investigations on full-Heusler alloys Mn2TaAl and Mn2WAl for spintronic and thermoelectric applications

Investigations on full-Heusler alloys Mn2TaAl and Mn2WAl for spintronic and thermoelectric applications

Ab-initio variational wave functions for the time-dependent many-electron Schrödinger equation

Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. We employ time-dependent Jastrow factors and backflow transformations, enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we employ the time-dependent variational Monte Carlo technique and introduce a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately describe the time evolution, providing insight into quantum dynamical effects in interacting electronic systems, beyond the capabilities of mean-field.

Charged aggregation.

We introduce an aggregation process that begins with equal concentrations of positively and negatively "charged" monomers. Oppositely charged monomers merge to form neutral dimers. These dimers are the seeds for subsequent aggregation events in which neutral clusters of necessarily even mass join irreversibly to form neutral aggregates of ever-increasing size. In the mean-field approximation with mass independent reaction rates, we solve for the reaction kinetics and show that the concentration of clusters of mass k,c_{k}(t), asymptotically scales as A_{k}/t, with A_{k} having a nontrivial dependence on k. We also investigate the phenomenon of gelation in charged aggregation when the reaction rate equals the product of the two incident cluster masses. Finally, we generalize our model to the case of three and more types of monomers.

Impact of correlations on nuclear binding energies

A strong effort will be dedicated in the coming years to extend the reach of ab initio nuclear-structure calculations to heavy doubly open-shell nuclei. In order to do so, the most efficient strategies to incorporate dominant many-body correlations at play in such nuclei must be identified. With this motivation in mind, the present work analyses the step-by-step inclusion of many-body correlations and their impact on binding energies of Calcium and Chromium isotopes. Employing an empirically-optimal Hamiltonian built from chiral effective field theory, binding energies along both isotopic chains are studied via a hierarchy of approximations based on polynomially-scaling expansion many-body methods. More specifically, calculations are performed based on (i) the spherical Hartree–Fock–Bogoliubov mean-field approximation plus correlations from second-order Bogoliubov many-body perturbation theory or Bogoliubov coupled cluster with singles and doubles on top of it, along with (ii) the axially-deformed Hartree–Fock–Bogoliubov mean-field approximation plus correlations from second-order Bogoliubov many-body perturbation theory built on it. The corresponding results are compared to experimental data and to those obtained via valence-space in-medium similarity renormalization group calculations at the normal-ordered two-body level that act as a reference in the present study. The spherical mean-field approximation is shown to display specific shortcomings in Ca isotopes that can be understood analytically and that are efficiently corrected via the consistent addition of low-order dynamical correlations on top of it. While the same setting cannot appropriately reproduce binding energies in doubly open-shell Cr isotopes, allowing the unperturbed mean-field state to break rotational symmetry permits to efficiently capture the static correlations responsible for the phenomenological differences observed between the two isotopic chains. Eventually, the present work demonstrates that polynomially-scaling expansion methods based on unperturbed states that possibly break (and restore) symmetries constitute an optimal route to extend ab initio calculations to heavy closed- and open-shell nuclei.

Hybrid star properties with the NJL and mean field approximation of QCD models: A Bayesian approach

Hybrid star properties with the NJL and mean field approximation of QCD models: A Bayesian approach

Finding patient zero in susceptible-infectious-susceptible epidemic processes.

Finding the source of an epidemic is important, because correct source identification can help to stop a budding epidemic or prevent new ones. We investigate the backward equationsof the N-intertwined mean-field approximation susceptible-infectious-susceptible (SIS) process. The backward equationsallow us to trace the epidemic back to its source on networks of sizes up to at least N=1500. Additionally, we show that the source of the "more realistic" Markovian SIS model cannot feasibly be found, even in a "best-case scenario," where the infinitesimal generator Q, which completely describes the epidemic process and the underlying contact network, is known. The Markovian initial condition s(0), which reveals the epidemic source, can be found analytically when the viral state vector s(t) is known at some time t as s(0)=s(t)e^{-Qt}. However, s(0) can hardly be computed, except for small times t. The numerical errors are largely due to the matrix exponential e^{-Qt}, which is severely ill-behaved.

An Efficient RI-MP2 Algorithm for Distributed Many-GPU Architectures.

Second-order Møller-Plesset perturbation theory (MP2) using the Resolution of the Identity approximation (RI-MP2) is a widely used method for computing molecular energies beyond the Hartree-Fock mean-field approximation. However, its high computational cost and lack of efficient algorithms for modern supercomputing architectures limit its applicability to large molecules. In this paper, we present the first distributed-memory many-GPU RI-MP2 algorithm explicitly designed to utilize hundreds of GPU accelerators for every step of the computation. Our novel algorithm achieves near-peak performance on GPU-based supercomputers through the development of a distributed memory algorithm for forming RI-MP2 intermediate tensors with zero internode communication, except for a single asynchronous broadcast, and a distributed memory algorithm for the energy reduction step, capable of sustaining near-peak performance on clusters with several hundred GPUs. Comparative analysis shows our implementation outperforms state-of-the-art quantum chemistry software by over 3.5 times in speed while achieving an 8-fold reduction in computational power consumption. Benchmarking on the Perlmutter supercomputer, our algorithm achieves 11.8 PFLOP/s (83% of peak performance) performing and the RI-MP2 energy calculation on a 314-water cluster with 7850 primary and 30,144 auxiliary basis functions in 4 min on 180 nodes and 720 A100 GPUs. This performance represents a substantial improvement over traditional CPU-based methods, demonstrating significant time-to-solution and power consumption benefits of leveraging modern GPU-accelerated computing environments for quantum chemistry calculations.

An Integrated Approach to Improving Itinerary Completion in Coordinated Care Networks

Problem definition: Coordinated care network (CCN) is a burgeoning paradigm where patients’ diagnosis and treatment plans are developed based on collaboration between multiple, colocated medical specialties to holistically address patients’ health needs. A primary performance metric for CCNs is how quickly patients can complete their itinerary of appointments at multiple medical services in the network. Rapid completion is critical to care delivery but also presents a major operational challenge. Because information about a patient’s condition and treatment options evolves over the course of the itinerary, care paths are not known a priori. Thus, appointments (except for the first one) cannot be reserved in advance, which may result in significant delays if capacity is not allocated properly. Methodology/results: We study capacity allocation for the patient’s first (root) appointment as the primary operational lever to achieve rapid itinerary completion in CCNs. We develop a novel queueing-based analytical framework to optimize this root appointment allocation, maximizing the proportion of patients completing care by prespecified deadlines. Our framework accounts for the complex interactions among all patients in the network through the blocking process, which contrasts with conventional siloed planning. We provide an exact characterization of the itinerary time and develop a mean-field approximation with convergence guarantees that permits tractable solutions for large-scale network problems. In a simulation case study of Mayo Clinic, our solution improves on-time completion from 60% under the current plan to more than 93%. Managerial implications: We demonstrate that root appointment allocation is a multifaceted problem and that ignoring any of those facets can lead to poor performance. Simultaneously accounting for all of these complexities makes manual template design or traditional optimization methods inadequate, highlighting the significance of our integrated approach. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0649 .

Thermalized and mixed meanfield ADP potentials for magnesium hydrides

We develop meanfield approximation and numerical quadrature schemes for the evaluation of Angular-Dependent interatomic Potentials (ADPs) for magnesium and magnesium hydrides at finite temperature (thermalization) and arbitrary atomic molar fractions (mixing) within a non-equilibrium statistical mechanical framework and derive local equilibrium relations. We numerically verify and experimentally validate the accuracy and fidelity of the resulting thermalized/mixed ADPs (TADPs) by means of selected numerical tests including free entropy, heat capacity, thermal expansion, molar volumes, equation of state and elastic constants. We show that the local equilibrium properties predicted by TADPs agree closely with those computed directly from ADP by means of Molecular Dynamics (MD).

Managing Resources for Shared Micromobility: Approximate Optimality in Large-Scale Systems

We consider the problem of managing resources in shared micromobility systems (bike sharing and scooter sharing). An important task in managing such systems is periodic repositioning/recharging/sourcing of units to avoid stockouts or excess inventory at nodes with unbalanced flows. We consider a discrete-time model; each period begins with an initial inventory at each node in the network, and then, customers (demand) materialize at the nodes. Each customer picks up a unit at the origin node and drops it off at a randomly sampled destination node with an origin-specific probability distribution. We model the above network inventory management problem as an infinite horizon discrete-time discounted Markov decision process (MDP) and prove the asymptotic optimality of a novel mean-field approximation to the original MDP as the number of stations becomes large. To compute an approximately optimal policy for the mean-field dynamics, we provide an algorithm with a running time that is logarithmic in the desired optimality gap. Lastly, we compare the performance of our mean field-based policy with state-of-the-art heuristics via numerical experiments, including experiments using Austin scooter-sharing data. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02023 .

Multimodal and Multifactor Branching Time Active Inference.

Active inference is a state-of-the-art framework for modeling the brain that explains a wide range of mechanisms. Recently, two versions of branching time active inference (BTAI) have been developed to handle the exponential (space and time) complexity class that occurs when computing the prior over all possible policies up to the time horizon. However, those two versions of BTAI still suffer from an exponential complexity class with regard to the number of observed and latent variables being modeled. We resolve this limitation by allowing each observation to have its own likelihood mapping and each latent variable to have its own transition mapping. The implicit mean field approximation was tested in terms of its efficiency and computational cost using a dSprites environment in which the metadata of the dSprites data set was used as input to the model. In this setting, earlier implementations of branching time active inference (namely, BTAIVMP and BTAIBF) underperformed in relation to the mean field approximation (BTAI3MF) in terms of performance and computational efficiency. Specifically, BTAIVMP was able to solve 96.9% of the task in 5.1 seconds, and BTAIBF was able to solve 98.6% of the task in 17.5 seconds. Our new approach outperformed both of its predecessors by solving the task completely (100%) in only 2.559 seconds.