Mean-field Techniques Research Articles

Fracture and mechanical behavior of vacuum-assisted microwave cured unidirectional carbon woven fabric reinforced epoxy composite

Woven fabric reinforced epoxy composite shown inherent favorable characteristics for aerospace industry applications. This paper comprehensively investigated the mechanical and fracture behavior of unidirectional carbon woven fabric reinforced epoxy composite using experimental and computational techniques. The composites were fabricated with two, four, and six ply laminates with cross-ply and inclined ply (45/–45) orientations. Laminates were fabricated using Vacuum-Assisted Resin Infusion Microwave Curing technique with a high fiber volume fraction of 50% for each laminate. Experimental analyses were performed to predict the behavior of composites under tensile, shear, and impact loading environment. Further, the mean-field homogenization technique coupled with the finite element method was employed to predict orthotropic material properties, fracture energy, and fracture parameters ( KICand GIC) of the composite. The results showed that fracture energy obtained by the computational technique was in good agreement with experimental results. The values of GICincreased with the number of plies both for cross and inclined plies orientation composites. KICvalues were higher for cross plies laminates than the inclined plies laminates.

Modelling of void shape effect on effective thermal conductivity of lotus-type porous materials

In this paper, a formula for estimating the effective thermal conductivity of lotus-type porous materials (LTPM) combining an analytical formula with a mean-field homogenisation technique is developed and validated. LTPMs are considered multi-phase materials in which each phase is defined by the pore shape. To estimate the effective thermal conductivities of multi-phase LTPMs, a two-step mean-field homogenisation method is implemented and validated. The validation is applied by comparing the results obtained with those of the representative volume element based finite element homogenisation method, which has been taken as a reference. The proposed formula is applied by replacing the first step of the two-step mean field homogenisation method using a formula for estimating the effective thermal conductivity of two-phase lotus-type materials. The good agreement between the results of the proposed formula and the reference results indicates that the proposed formula ensures an accurate estimation of the effective macroscopic thermal conductivities of multi-phase lotus-type porous materials.

Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory.

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44, 507-529 (doi:10.1016/j.ijsolstr.2006.04.038)) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy-Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.

Voter and Majority Dynamics with Biased and Stubborn Agents

We study binary opinion dynamics in a fully connected network of interacting agents. The agents are assumed to interact according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. We focus on the scenario where the agents are biased towards one of the opinions called the preferred opinion. Using suitably constructed branching processes, we show that under both rules the mean time to reach consensus is varTheta (log N), where N is the number of agents in the network. Furthermore, under the majority rule model, we show that consensus can be achieved on the preferred opinion with high probability even if it is initially the opinion of the minority. We also study the majority rule model when stubborn agents with fixed opinions are present. We find that the stationary distribution of opinions in the network in the large system limit using mean field techniques.

Slave-spin-1 formulation: A simple approach to time-dependent transport through an interacting two-level system

We introduce and develop a slave-spin mean-field technique for describing generic interacting two-level systems under time-dependent drivings, where an auxiliary $S=1$ spin is added to describe the localized character of the electrons. We show that the approach efficiently captures the main effects of the strong correlations as well as the dynamical nature of the driving, while remaining simple enough to allow for an analytical treatment. Our formalism provides a flexible solution method, which can be applied to different device configurations at an extremely small numerical cost. Furthermore, it leads to a very practical description of adiabatically driven systems in terms of frozen static solutions.

Unstable modes and bistability in delay-coupled swarms.

It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms-laying the groundwork for comparisons to robotics experiments.

Collaboratively Learning the Best Option on Graphs, Using Bounded Local Memory

We consider multi-armed bandit problems in social groups wherein each individual has bounded memory and shares the common goal of learning the best arm/option. We say an individual learns the best option if eventually (as t - ∞) it pulls only the arm with the highest expected reward. While this goal is provably impossible for an isolated individual due to bounded memory, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion (i.e., communication) as long as the communication networks/graphs satisfy some mild conditions. In this work, we model and analyze a type of learning dynamics which are wellobserved in social groups. Specifically, under the learning dynamics of interest, an individual sequentially decides on which arm to pull next based on not only its private reward feedback but also the suggestion provided by a randomly chosen neighbor. To deal with the interplay between the randomness in the rewards and in the social interaction, we employ the mean-field approximation method. Considering the possibility that the individuals in the networks may not be exchangeable when the communication networks are not cliques, we go beyond the classic mean-field techniques and apply a refined version of mean-field approximation. Notably, our results hold even if the communication graphs are highly sparse.

Minimal-size real-space d -wave pairing operator in CuO2 planes

A novel minimal-size pairing operator $\Delta^{\dagger}_{D0}$ with $d$-wave symmetry in CuO$_2$ planes is introduced. This pairing operator creates on-site Cooper pairs at the four oxygens that surround a copper atom. Via the time evolution of $\Delta^{\dagger}_{D0}$, an additional inter-orbital pairing operator $\Delta^{\dagger}_{Dpd}$ with $d$-wave symmetry is generated that pairs fermions located in a Cu and its four surrounding O's. The subsequent time evolution of $\Delta^{\dagger}_{Dpd}$ generates an intra-orbital $d$-wave pairing operator $\Delta^{\dagger}_{Dpp}$ involving the four O atoms that surround a Cu, as well as the $d$-wave operator $\Delta^{\dagger}_D$ traditionally used in single-band models for cuprates. Because we recover the larger size operators extensively used in the three-orbital Hubbard model, we suggest that long-range order using the canonical extended operators occurs together with long-range order in the new minimal operators. However, our minimal $d$-wave operators could be more practical to study $d$-wave superconductivity because in the finite-size relatively small systems accessible to computational techniques it is easier to observe long-range order using local operators. Moreover, an effective model with the usual tight-binding hopping of the CuO$_2$ planes supplemented by an attractive potential $V$ in the $d$-wave channel is introduced. Using mean-field techniques we show that a paired ground state is stabilized for any finite value of $V$.

High-temperature expansions and message passing algorithms

Improved mean-field techniques are a central theme of statistical physics methods applied to inference and learning. We revisit here some of these methods using high-temperature expansions for disordered systems initiated by Plefka, Georges and Yedidia. We derive the Gibbs free entropy and the subsequent self-consistent equations for a generic class of statistical models with correlated matrices and show in particular that many classical approximation schemes, such as adaptive TAP, expectation-consistency, or the approximations behind the vector approximate message passing algorithm all rely on the same assumptions, that are also at the heart of high-temperature expansions. We focus on the case of rotationally invariant random coupling matrices in the ‘high-dimensional’ limit in which the number of samples and the dimension are both large, but with a fixed ratio. This encapsulates many widely studied models, such as restricted Boltzmann machines or generalized linear models with correlated data matrices. In this general setting, we show that all the approximation schemes described before are equivalent, and we conjecture that they are exact in the thermodynamic limit in the replica symmetric phases. We achieve this conclusion by resummation of the infinite perturbation series, which generalises a seminal result of Parisi and Potters. A rigorous derivation of this conjecture is an interesting mathematical challenge. On the way to these conclusions, we uncover several diagrammatical results in connection with free probability and random matrix theory, that are interesting independently of the rest of our work.

Squeezed state metrology with Bragg interferometers operating in a cavity

Bragg interferometers, operating using pseudospin-1/2 systems composed of two momentum states, have become a mature technology for precision measurements. State-of-the-art Bragg interferometers are rapidly surpassing technical limitations and are soon expected to operate near the projection noise limit set by uncorrelated atoms. Despite the use of large numbers of atoms, their operation is governed by single-atom physics. Motivated by recent proposals and demonstrations of Raman gravimeters in cavities, we propose a scheme to squeeze directly on momentum states for surpassing the projection noise limit in Bragg interferometers. In our modeling, we consider the unique issues that arise when a spin squeezing protocol is applied to momentum pseudospins. Specifically, we study the effects of the momentum width of the atomic cloud and the coupling to momentum states outside the pseudospin manifold, as these atoms interact via a mode of the cavity. We show that appreciable levels of spin squeezing can be demonstrated in suitable parameter regimes in spite of these complications. Using this setting, we show how beyond mean-field techniques developed for spin systems can be adapted to study the dynamics of momentum states of interacting atoms. Our scheme promises to be feasible using current technology and is experimentally attractive because it requires no additional setup beyond what will be required to operate Bragg interferometers in cavities.

Linear-response description of superexchange-driven orbital ordering in K2CuF4

We study the nature of orbital and magnetic order in the layered perovskite ${\mathrm{K}}_{2}{\mathrm{CuF}}_{4}$, and compare to the case of the infinite-layer system ${\mathrm{KCuF}}_{3}$. To this end, we augment the local-density approximation $+$ dynamical mean-field theory technique with linear-response functions. We explain orbital and magnetic order, and their evolution with increasing pressure. We show that both the tetragonal (${\ensuremath{\varepsilon}}_{\mathrm{T}}$) and the Jahn-Teller (${\ensuremath{\varepsilon}}_{\mathrm{JT}}$) crystal-field splitting play a key role. We find that surprisingly, unlike in ${\mathrm{KCuF}}_{3},\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{\mathrm{T}}$ is comparable to, or even larger than, ${\ensuremath{\varepsilon}}_{\mathrm{JT}}$; in addition, ${\ensuremath{\varepsilon}}_{\mathrm{T}}$ is mostly determined by the layered structure itself and by the compression of the K cage, rather than by the deformations of the ${\mathrm{CuF}}_{6}$ octahedra. Next, we study the nature of orbital order. We calculate the superexchange transition temperature, finding ${T}_{\mathrm{KK}}\ensuremath{\sim}300\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, a value close to the one for ${\mathrm{KCuF}}_{3}$. Thus, in ${\mathrm{K}}_{2}{\mathrm{CuF}}_{4}$ as in ${\mathrm{KCuF}}_{3},\phantom{\rule{0.16em}{0ex}}{T}_{\mathrm{KK}}$ is too small to explain the existence of orbital order up to the melting temperature. We show, however, that in the case of the layered perovskite, an additional superexchange mechanism is at work. It is an orbital Zeeman term, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{h}}_{\mathrm{KK}}$, and it is active also above ${T}_{\mathrm{KK}}$. We show that due to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{h}}_{\mathrm{KK}}$, phases with different types of ordering can coexist at temperatures below ${T}_{\mathrm{KK}}$. Similar effects are likely to play a role in other layered correlated systems.

Micro-Kinetic Modelling of CO-TPD from Fe(100)—Incorporating Lateral Interactions

The experimentally determined temperature programmed desorption profile of CO from Fe(100) is characterized by four maxima, i.e., α1-CO, α2-CO, α3-CO, and β-CO (see e.g., Moon et al., Surf. Sci. 1985, 163, 215). The CO-TPD profile is modeled using mean-field techniques and kinetic Monte Carlo to show the importance of lateral interactions in the appearance of the CO-TPD-profile. The inclusion of lateral interactions results in the appearance of a new maximum in the simulated CO-TPD profile if modeled using the mean-field, quasi-chemical approach or kinetic Monte Carlo. It is argued that α2-CO may thus originate from lateral interactions rather than a differently bound CO on Fe(100). A detailed sensitivity analysis of the effect of the strength of the lateral interactions between the species involved (CO, C, and O), and the choice of the transition state, which affects the activation energy for CO dissociation, and the energy barrier for diffusion on the CO-TPD profile is presented.

Collaboratively Learning the Best Option on Graphs, Using Bounded Local Memory

We consider multi-armed bandit problems in social groups wherein each individual has bounded memory and shares the common goal of learning the best arm/option. We say an individual learns the best option if eventually (as $t\diverge$) it pulls only the arm with the highest expected reward. While this goal is provably impossible for an isolated individual due to bounded memory, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion (i.e., communication) as long as the communication networks/graphs satisfy some mild conditions. In this work, we model and analyze a type of learning dynamics which are well-observed in social groups. Specifically, under the learning dynamics of interest, an individual sequentially decides on which arm to pull next based on not only its private reward feedback but also the suggestion provided by a randomly chosen neighbor. To deal with the interplay between the randomness in the rewards and in the social interaction, we employ the \em mean-field approximation method. Considering the possibility that the individuals in the networks may not be exchangeable when the communication networks are not cliques, we go beyond the classic mean-field techniques and apply a refined version of mean-field approximation: Using coupling we show that, if the communication graph is connected and is either regular or has doubly-stochastic degree-weighted adjacency matrix, with probability → 1 as the social group size N → ∞, every individual in the social group learns the best option. If the minimum degree of the graph diverges as N → ∞, over an arbitrary but given finite time horizon, the sample paths describing the opinion evolutions of the individuals are asymptotically independent. In addition, the proportions of the population with different opinions converge to the unique solution of a system of ODEs. Interestingly, the obtained system of ODEs are invariant to the structures of the communication graphs. In the solution of the obtained ODEs, the proportion of the population holding the correct opinion converges to 1 exponentially fast in time. Notably, our results hold even if the communication graphs are highly sparse.

Elastic and inelastic local strain fields in composites with coated fibers or particles: theory and validation

This paper deals with mean field multiscale approaches for coated fiber- or particle-reinforced composites under nonlinear strain. The current work attempts to extend Dvorak’s well-known transformation field analysis for mean field approaches, in which the composite’s constitutive law is split into an elastic and an inelastic part. The classical Eshelby’s inhomogeneity problem considering eigenstrains is revisited in order to address the presence of a coating layer. For this scope, three different methodologies are employed, one for general ellipsoidal inhomogeneities, a modified composite cylinder method for long cylindrical fibers and a modified composite sphere method for spherical particles. After identifying proper interaction tensors for the inhomogeneity and its coating layer, the composite’s overall response is evaluated by extending classical mean field techniques, such as the Mori–Tanaka and the self-consistent methods. Numerical examples illustrate the differences in macroscopic and microscopic predictions between the general approach and the modified composite cylinder and sphere Assemblages.

The economics of netting in financial networks

Can the netting of on-balance-sheet interbank assets and liabilities be useful in thwarting financial contagion during a systemic crisis episode? In order to answer this question, in this paper we use mean-field approximation techniques and computer simulations to comparatively assess how contagion spreads out throughout an interbank network under different settlement modes. We find that a regulator forcing banks to net their credit/debt obligations instead of allowing them to regulate their mutual exposures on a gross basis succeeds in reducing the number of defaults and in preserving the aggregate amount of bank capital. Interbank netting takes its toll on retail depositors by increasing their potential losses, however. Hence, our analysis provides support for an optimal crisis-management policy mix that combines the enforcement of bilateral netting with a blanket deposit insurance scheme. The desirability of netting increases when the system is highly connected and susceptible to large shocks, especially when strains are first detected in banks located at the core of the network.

Asymptotically Optimal Load Balancing Topologies

We consider a system of N ~servers inter-connected by some underlying graph topology~G N . Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate lambda. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in G N . The above model arises in the context of load balancing in large-scale cloud networks and data centers, and has been extensively investigated in the case G N is a clique. Since the servers are exchangeable in that case, mean-field limits apply, and in particular it has been proved that for any lambda < 1, the fraction of servers with two or more tasks vanishes in the limit as N -> ınfty. For an arbitrary graph G N , mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph G N is said to be N -optimal or ∞N-optimal when the queue length process on G N is equivalent to that on a clique on an N -scale or ∞N-scale, respectively. We prove that if G N is an Erdos-Rényi random graph with average degree d(N), then with high probability it is N -optimal and ∞N-optimal if d(N) -> ınfty$ and d(N) / (∞N łog(N)) -> ınfty as N -> ınfty, respectively. This demonstrates that optimality can be maintained at N -scale and ∞N-scale while reducing the number of connections by nearly a factor N and ∞N/ łog(N) compared to a clique, provided the topology is suitably random. It is further shown that if G N contains Θ(N) bounded-degree nodes, then it cannot be N -optimal. In addition, we establish that an arbitrary graph G N is N -optimal when its minimum degree is N - o(N), and may not be N -optimal even when its minimum degree is c N + o(N) for any 0 < c < 1/2. Simulation experiments are conducted for various scenarios to corroborate the asymptotic results.

Asymptotically Optimal Load Balancing Topologies

We consider a system of N servers inter-connected by some underlying graph topology G N . Tasks with unit-mean exponential processing times arrive at the various servers as independent Poisson processes of rate λ. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in G N . The above model arises in the context of load balancing in large-scale cloud networks and data centers, and has been extensively investigated in the case G N is a clique. Since the servers are exchangeable in that case, mean-field limits apply, and in particular it has been proved that for any λ < 1, the fraction of servers with two or more tasks vanishes in the limit as N → ∞. For an arbitrary graph G N , mean-field techniques break down, complicating the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph G N is said to be N -optimal or √ N -optimal when the queue length process on G N is equivalent to that on a clique on an N -scale or √ N -scale, respectively. We prove that if G N is an Erdöo s-Rényi random graph with average degree d ( N ), then with high probability it is N -optimal and √ N -optimal if d ( N ) → ∞ and d ( N )/√ N log( N )) → ∞ as N → ∞, respectively. This demonstrates that optimality can be maintained at N -scale and √ N -scale while reducing the number of connections by nearly a factor N and √ N /log( N ) compared to a clique, provided the topology is suitably random. It is further shown that if G N contains Θ( N ) bounded-degree nodes, then it cannot be N -optimal. In addition, we establish that an arbitrary graph G N is N -optimal when its minimum degree is N - o ( N ), and may not be N -optimal even when its minimum degree is cN + o ( N ) for any 0 < c < 1/2. Simulation experiments are conducted for various scenarios to corroborate the asymptotic results.

Single-particle model of a strongly driven, dense, nanoscale quantum ensemble

We study the effects of interatomic interactions on the quantum dynamics of a dense, nanoscale, atomic ensemble driven by a strong electromagnetic field. We use a self-consistent, mean-field technique based on the pseudo-spectral time-domain method, and a full, three-directional basis to solve the coupled Maxwell-Liouville equations. We find that interatomic interactions generate a decoherence in the state of an ensemble on a much faster timescale than the excited state lifetime of individual atoms. We present a novel single-particle model of the driven, dense ensemble by incorporating interactions into a dephasing rate. This single-particle model reproduces the essential physics of the full simulation, and is an efficient way of rapidly estimating the collective dynamics of a dense ensemble.

Micromechanical modelling of porous viscoelastic composite materials

Modelling and predicting the effective behavior of non-ageing viscoelastic composites have attracted the attention of many researchers. Actually, predicting the effective behavior and the macroscopic overall response while taking into account the constituents properties and shape of inclusions as well as their volume fractions is a challenging topic. In this work, porous viscoelastic composites are considered. The porous effect is introduced as a solid voided inclusions embedded in a viscoelastic matrix. The effective behavior is modelled by delayed integropartial differential equations. The resolution of the resulting equations is done through a methodological approach based on the Volterra tensorial products and the dynamic Green‘s tensor. Thus, the localization equations relating the local and the global fields are derived. After that, the Mori-Tanaka mean field micromechanical model assumptions are applied to derive the Mori-Tanaka‘s localization tensor. Once this step is completed, the effective properties are obtained through mean field techniques. The effective properties are given through tensorial convolution products. A numerical algorithm is elaborated for the computation of direct and inverse tensorial convolution products. For the validation of the developed modelling a comparison with Laplace-Carson approach is done.

Study of methods for simulating multiphase construction materials

Computer-aided design and multiscale modeling and analysis can lead not only to a better understanding of the behavior of multiphase materials or increased performance, but also reduced energy consumption and raw materials, optimization and cost reduction. Generally, concept modeling involves multiscale analysis, which occurs at two or more levels, from the smallest scale (micro- or even nano-structure) to full system level (macroscopic scale). The purpose of this paper is to analyze the simulation feasibility of multiphase materials (traditional mortar), and also the comparison methods that can be applied in this regard.This paper introduces a multiscale modeling approach to simulate the effective (macroscopic) mechanical properties of the ordinary cement-based mortar. Simulations of multiphase materials have been done by mean-field homogenization techniques and the finite element method. Multiscale modeling helps to evaluate the stress–strain behavior and reveals important engineering properties, such as Young’s Modulus and Poisson’s ratio.