Traditional Mean-field Theory Research Articles

Graph neural network modeling of grain-scale anisotropic elastic behavior using simulated and measured microscale data

Here we assess the applicability of graph neural networks (GNNs) for predicting the grain-scale elastic response of polycrystalline metallic alloys. Using GNN surrogate models, grain-averaged stresses during uniaxial elastic tension in low solvus high-refractory (LSHR) Ni Superalloy and Ti 7 wt%Al (Ti-7Al) are predicted as example face-centered cubic and hexagonal closed packed alloys, respectively. A transfer learning approach is taken in which GNN surrogate models are trained using crystal elasticity finite element method (CEFEM) simulations and then the trained surrogate models are used to predict the mechanical response of microstructures measured using high-energy X-ray diffraction microscopy (HEDM). The performance of using various microstructural and micromechanical descriptors for input nodal features to the GNNs is explored through comparisons to traditional mean-field theory predictions, reserved full-field CEFEM data, and measured far-field HEDM data. The effects of elastic anisotropy on GNN model performance and outlooks for the extension of the framework are discussed.

Mean-field theory of vector spin models on networks with arbitrary degree distributions

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

Charged nanorods at heterogeneously charged surfaces.

We study the spatial and orientational distribution of charged nanorods (rodlike counterions) as well as the effective interaction mediated by them between two plane-parallel surfaces that carry fixed (quenched) heterogeneous charge distributions. The nanorods are assumed to have an internal charge distribution, specified by a multivalent monopolar moment and a finite quadrupolar moment, and the quenched surface charge is assumed to be randomly distributed with equal mean and variance on the two surfaces. While equally charged surfaces are known to repel within the traditional mean-field theories, the presence of multivalent counterions has been shown to cause attractive interactions between uniformly charged surfaces due to the prevalence of strong electrostatic couplings that grow rapidly with the counterion valency. We show that the combined effects due to electrostatic correlations (caused by the coupling between the mean surface field and the multivalent, monopolar, charge valency of counterions) as well as the disorder-induced interactions (caused by the coupling between the surface disorder field and the quadrupolar moment of counterions) lead to much stronger attractive interactions between two randomly charged surfaces. The interaction profile turns out to be a nonmonotonic function of the intersurface separation, displaying an attractive minimum at relatively small separations, where the ensuing attraction can exceed the maximum strong-coupling attraction (produced by multivalent monopolar counterions between uniformly charged surfaces) by more than an order of magnitude.

Statistical mechanics of normal grain growth in one dimension: A partial integro-differential equation model

We develop a statistical-mechanical model of one-dimensional normal grain growth that does not require any drift-velocity parameterization for grain size, such as used in the continuity equation of traditional mean-field theories. The model tracks the population by considering grain sizes in neighbour pairs; the probability of a pair having neighbours of certain sizes is determined by the size-frequency distribution of all pairs. Accordingly, the evolution obeys a partial integro-differential equation (PIDE) over ‘grain size versus neighbour grain size’ space, so that the grain-size distribution is a projection of the PIDE's solution. This model, which is applicable before as well as after statistically self-similar grain growth has been reached, shows that the traditional continuity equation is invalid outside this state. During statistically self-similar growth, the PIDE correctly predicts the coarsening rate, invariant grain-size distribution and spatial grain size correlations observed in direct simulations. The PIDE is then reducible to the standard continuity equation, and we derive an explicit expression for the drift velocity. It should be possible to formulate similar parameterization-free models of normal grain growth in two and three dimensions.

Read-Green resonances in a topological superconductor coupled to a bath

We study a topological superconductor capable of exchanging particles with an environment. This additional interaction breaks particle-number symmetry and can be modelled by means of an integrable Hamiltonian, building on the class of Richardson-Gaudin pairing models. The isolated system supports zero-energy modes at a topological phase transition, which disappear when allowing for particle exchange with an environment. However, it is shown from the exact solution that these still play an important role in system-environment particle exchange, which can be observed through resonances in low-energy and -momentum level occupations. These fluctuations signal topologically protected Read-Green points and cannot be observed within traditional mean-field theory.

Temporal coherence, anomalous moments, and pairing correlations in the classical-field description of a degenerate Bose gas

The coherence properties of degenerate Bose gases have usually been expressed in terms of spatial correlation functions, neglecting the rich information encoded in their temporal behavior. In this paper we show, using a Hamiltonian classical-field formalism, that temporal correlations can be used to characterize familiar properties of a finite-temperature degenerate Bose gas. The temporal coherence of a Bose-Einstein condensate is limited only by the slow diffusion of its phase, and thus the presence of a condensate is indicated by a sharp feature in the temporal power spectrum of the field. We show that the condensate mode can be obtained by averaging the field for a short time in an appropriate phase-rotating frame, and that for a wide range of temperatures, the condensate obtained in this approach agrees well with that defined by the Penrose-Onsager criterion based on one-body (spatial) correlations. For time periods long compared to the phase diffusion time, the field will average to zero, as we would expect from the overall U(1) symmetry of the Hamiltonian. We identify the emergence of the first moment on short time scales with the concept of U(1) symmetry breaking that is central to traditional mean-field theories of Bose condensation. We demonstrate that the short-time averaging procedure constitutes a general analog of the 'anomalous' averaging operation of symmetry-broken theories by calculating the anomalous thermal density of the field, which we find to have form and temperature dependence consistent with the results of mean-field theories.

Quantum dynamics of cavity-assisted photoassociation of Bose-Einstein-condensed atoms

We explore the quantum dynamics of photoassociation of Bose-Einstein condensed atoms into molecules using an optical cavity field. Inside of an optical resonator, photoassociation of quantum degenerate atoms involves the interaction of three coupled quantum fields for the atoms, molecules, and the photons. The feedback created by a high-Q optical cavity causes the cavity field to become a dynamical quantity whose behavior is linked in a nonlinear manner to the atoms inside and where vacuum fluctuations have a more important role than in free space. We develop and compare several methods for calculating the dynamics of the atom-molecule conversion process with a coherently driven cavity field. We first introduce an alternate operator representation for the Hamiltonian from which we derive an improved form of mean field theory and an approximate solution of the Heisenberg-Langevin (HL) equations that properly accounts for quantum noise in the cavity field. It is shown that our improved mean field theory corrects several deficiencies in traditional mean field theory based on expectation values of annihilation/creation operators. Also, we show by direct comparison to numerical solutions of the density matrix equations that our approximate quantum solution of HL equations gives an accurate description of weakly or undriven cavities where mean field theories break down.

Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions

Traditional mean-field theory is a generic variational approach for analyzing symmetry breaking phases. However, this simple approach only applies to symmetry breaking states with short-range entanglement. In this paper, we describe a generic approach for studying two-dimensional (2D) quantum phases with long-range entanglement (such as topological phases). The method is based on (a) a general class of trial wave functions known as tensor-product states and (b) a 2D real-space renormalization group algorithm for efficiently calculating expectation values for these states. We demonstrate our method by studying several simple 2D quantum spin models exhibiting both symmetry breaking phase transitions and topological phase transitions. Our approach can be viewed as a unified mean-field theory for both symmetry breaking phases and topological phases.

Analytical results of the generalized mean-field theory for diluted magnetic semiconductors with RKKY interaction

Analytical results have been obtained in the framework of the generalized mean-field theory for diluted semiconductors with RKKY interaction. That theory accounts for the non-equivalency of different lattice sites by introducing the distribution function of local effective magnetic fields for non-regular (random) systems with magnetic interaction. The procedure is described that permits to deduce the analytical expression for that function. Corresponding improvement of the traditional mean-field theory could be observed by comparing results of such a generalized analytical model with exact results known for some simple cases, with numerical results of different authors considering the disorder of magnetic impurities’ arrangement, and with experimental data, as well.

The electrical double layer for a fully asymmetric electrolyte around a spherical colloid: An integral equation study

The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation for a totally asymmetric primitive model electrolyte around a spherical macroparticle is obtained and solved numerically in the case of size-asymmetric systems. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular-dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential versus charge relationship, radial distribution functions, mean electrostatic potential, and cumulative reduced charge for representative examples of 1:1 and 2:2 salts with a size-asymmetry ratio of 2. Our results are collated with those of the modified Gouy-Chapman (MGC) and unequal radius modified Gouy-Chapman (URMGC) theories and with those of HNC/MSA in the restricted primitive model (RPM) to assess the importance of size-asymmetry effects. One of the most striking characteristics found is that, contrary to the general belief, away from the point of zero charge the properties of an asymmetric electrical double layer (EDL) are not those corresponding to a symmetric electrolyte with the size and charge of the counterion, i.e., counterions do not always dominate. This behavior suggests the existence of a new phenomenology in the EDL that genuinely belongs to a more realistic size-asymmetric model where steric correlations are taken into account consistently. Such novel features cannot be described by traditional mean-field theories such as MGC, URMGC, or even by enhanced formalisms, such as HNC/MSA, if they are based on the RPM.

First-order mean spherical approximation for inhomogeneous fluids

The first-order mean-spherical approximation (FMSA) [Y. Tang, J. Chem. Phys., 118, 4140 (2003)] is extended to the studies of inhomogeneous fluids by combining with Rosenfeld's perturbative method [Y. Rosenfeld, J. Chem. Phys. 98, 8126 (1993)]. In the extension, the key input-direct correlation function of FMSA-is applied to constructing the free energy density functional. Preserving its high fidelity at the bulk limit, the FMSA shows satisfactory performance for Yukawa fluids near hard and attractive walls. The results are better than or comparable to several other theories reported before for the geometry. The FMSA is found, in particular, more satisfactory than the traditional mean-field theory for predicting density profiles around hard walls. The FMSA is also compared with the full MSA for inhomogeneous fluids, showing no appreciable differences. The inhomogeneous FMSA goes successfully through the self-consistency test for reproducing the radial distribution function of the bulk Yukawa fluid. As far as the computation is concerned, the FMSA can be executed much faster than any nonmean-field theories, and the speed is virtually identical to that of the mean-field theory.

Elementary Excitations of the Symmetric Spin-Orbital Model: TheXYLimit

The elementary excitations of the 1D, symmetric, spin-orbital model are studied in two anisotropic cases: the pure $\mathrm{XY}$ and the dimerized $\mathrm{XXZ}$ models. While they preserve the symmetry between spin and orbital degrees of freedom, these models allow for a picture of the low-lying excitations. In the pure $\mathrm{XY}$ case, a phase separation takes place between two phases with free-fermion-like gapless excitations. In the dimerized case the low-energy effective Hamiltonian reduces to the 1D Ising model with gapped excitations. All the elementary excitations involve simultaneous flips of the spin and orbital degrees of freedom: a clear indication of the breakdown of the traditional mean-field theory.

Optical effects of multipole interaction in aggregated structures: Rigorous scattering solution for a periodic chain of cylinders.

Recent experimental studies on the optical properties of gold colloidal aggregates have found interesting characteristics, such as double absorption peaks and enhanced depolarized scattering, that are not explainable in terms of traditional mean-field theories. Based on the observation that the aggregate clusters have a locally chainlike structure of touching spheres, a simple model for the calculation of their absorption and scattering characteristics is proposed that consists of a randomly oriented chain of cylindrical particles under the excitation of an electromagnetic field. The consideration of cylindrical geometry considerably simplifies the mathematics while retaining the essential physics of capacitive coupling between the particles. In the optical frequency regime, a rigorous scattering solution for the model shows that, besides the single-particle Mie resonance, there exists a low-frequency collective excitation arising from the anisotropic, high-multipole interactions between the metal particles. It results in a second absorption peak whose peak position is controlled by the capacitive coupling between the adjacent particles. Both the polarized and the depolarized scattering also show accompanying increases at the collective excitation frequency. Good agreement is found between the calculated results and the experimental observations.

Path-integral methods for treating quantal behavior in solids: Mean-field theory and the effects of fluctuations.

Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ``internal structure.'' The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.