Standard Mean-field Approximation Research Articles

Violent relaxation in one-dimensional self-gravitating system: Deviation from the Vlasov limit due to finite-N effects.

We investigate the effect of a finite particle number N on the violent relaxation leading to the quasistationary state (QSS) in a one-dimensional self-gravitating system. From the theoretical point of view, we demonstrate that the local Poissonian fluctuations embedded in the initial state give rise to an additional term proportional to 1/N in the Vlasov equation. This term designates the strength of the local mean-field variations by fluctuations. Because it is of the mean-field origin, we interpret it differently from the known collision term in the way that it effects the violent relaxation stage. Its role is to deviate the distribution function from the Vlasov limit, in the collisionless manner, at a rate proportional to 1/N while the violent relaxation is progressing. This hypothesis is tested by inspecting the QSSs in simulations of various N. We observe that the core phase-space density can exceed the limiting density deduced from the Vlasov equationand its deviation degree is in accordance with the 1/N estimate. This indicates the deviation from the standard mean-field approximation of the violent relaxation process by that 1/N term. In conclusion, the finite-N effect has a significant contribution to the QSS apart from that it plays a role in the collisional stage that takes place long after. The conventional collisionless Vlasov equationmight not be able to describe the violent relaxation of a system of particles properly without the correction term of the local finite-N fluctuations.

The primitive model in classical density functional theory: beyond the standard mean-field approximation

The primitive model describes ions by point charges with an additional hard-core interaction. In classical density-functional theory (DFT) the mean-field electrostatic contribution can be obtained from the first order of a functional perturbation of the pair potential for an uncharged reference system of hard spheres. This mean-field electrostatic term particularly contributes at particle separations that are forbidden due to hard-core overlap. In this work we modify the mean-field contribution such that the pair potential is constant for distances smaller than the contact distance of the ions. We motivate our modification by the underlying splitting of the potential, which is similar to the splitting of the Weeks–Chandler–Andersen potential and leads to higher-order terms in the respective expansion of the functional around the reference system. The resulting formalism involves weighted densities similar to the ones found in fundamental measure theory. To test our modifications, we analyze and compare density profiles, direct and total correlation functions, and the thermodynamic consistency of the functional via a widely established sum rule and the virial pressure formula for our modified functional, for established functionals, and for data from computer simulations. We found that our modifications clearly show improvements compared to the standard mean-field functional, especially when predicting layering effects and direct correlation functions in high concentration scenarios; for the latter we also find improved consistency when calculated via different thermodynamic routes. In conclusion, we demonstrate how modifications toward higher order corrections beyond mean-field functionals can be made and how they perform, by this providing a basis for systematic future improvements in classical DFT for the description of electrostatic interactions.

Thermodynamic properties of the spin S=3/2 quantum ferromagnetic Blume-Capel model in a transverse crystal field.

The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.

Machine-learning free-energy functionals using density profiles from simulations

The formally exact framework of equilibrium Density Functional Theory (DFT) is capable of simultaneously and consistently describing thermodynamic and structural properties of interacting many-body systems in arbitrary external potentials. In practice, however, DFT hinges on approximate (free-)energy functionals from which density profiles (and hence the thermodynamic potential) follow via an Euler–Lagrange equation. Here, we explore a relatively simple Machine-Learning (ML) approach to improve the standard mean-field approximation of the excess Helmholtz free-energy functional of a 3D Lennard-Jones system at a supercritical temperature. The learning set consists of density profiles from grand-canonical Monte Carlo simulations of this system at varying chemical potentials and external potentials in a planar geometry only. Using the DFT formalism, we nevertheless can extract not only very accurate 3D bulk equations of state but also radial distribution functions using the Percus test-particle method. Unfortunately, our ML approach did not provide very reliable Ornstein–Zernike direct correlation functions for small distances.

Toward an Atomistic Understanding of Deep Eutectics Solvents Electrochemical Interfacial Structure

Green, stable, and wide electrochemical window deep eutectics solvents (DESs) are ideal candidates for electrochemical systems, such as batteries, supercapacitors, catalysis, electrodeposition, and many more. However, since these mixtures are highly dense and composed of only large asymmetric ionic and molecular components, short-range ion-ion, ion-molecule interactions play a dominant role both in the bulk as well as at the interface properties. This results in breakdown of the standard mean-field approximation, which is the basis of dilute solution theory. In addition, due to the hygroscopic nature of DESs, the presence of latent water is unavoidable. Therefore, understanding the DESs-electrode interface (electrical double layer structure, EDLS) together with the role of water at a molecular scale is of great importance for the widespread use of these solvents in electrochemical systems [1-3]. In this presentation, we will first review the benchmark study on conventional dilute solution electrical double layer structure and secondly present our recent findings and theoretical developments on the electrochemical interfacial structure of deep eutectic solvents using a combined experimental and a novel atomistic molecular dynamics approach. In the presentation, we focused on the computational approach. The use of such an atomistic-molecular approach allows investigation of all the possible bulk and interfacial interactions and, consequently, helps to shed light on the nanoscale electrochemical interfacial structure of DESs. Unlike the interfacial structures observed and proposed for other electrolyte-electrode interfaces (compact- diffuse double layer for dilute solution [4]; overscreening or crowding for concentrated solutions [5]), the simulations of DESs-electrode interfacial structure show an unexpected and previously unrecognized phenomenon: the electrochemical interface is composed of a mixed layer structure followed by a mixed charged clustered layer regardless of the surface polarization [1,2]. Some of the findings will be discussed in this presentation.

Variational EM method for blur estimation using the spike-and-slab image prior

Most state-of-the-art blind image deconvolution methods rely on the Bayesian paradigm to model the deblurring problem and estimate both the blur kernel and latent image. It is customary to model the image in the filter space, where it is supposed to be sparse, and utilize convenient priors to account for this sparsity. In this paper, we propose the use of the spike-and-slab prior together with an efficient variational Expectation Maximization (EM) inference scheme to estimate the blur in the image. The spike-and-slab prior, which constitutes the gold standard in sparse machine learning, selectively shrinks irrelevant variables while mildly regularizing the relevant ones. The proposed variational Expectation Maximization algorithm is more efficient than usual Markov Chain Monte Carlo (MCMC) inference and, also, proves to be more accurate than the standard mean-field variational approximation. Additionally, all the prior model parameters are estimated by the proposed scheme. After blur estimation, a non-blind restoration method is used to obtain the actual estimation of the sharp image. We investigate the behavior of the prior in the experimental section together with a series of experiments with synthetically generated and real blurred images that validate the method's performance in comparison with state-of-the-art blind deconvolution techniques.

Phenomenological three-orbital spin-fermion model for cuprates

A spin-fermion model that captures the charge-transfer properties of Cu-based high critical temperature superconductors is introduced and studied via Monte Carlo simulations. The strong Coulomb repulsion among $d$-electrons in the Cu orbitals is phenomenologically replaced by an exchange coupling between the spins of the itinerant electrons and localized spins at the Cu sites, formally similar to double-exchange models for manganites. This interaction induces a charge-transfer insulator gap in the undoped case (five electrons per unit cell). Adding a small antiferromagnetic Heisenberg coupling between localized spins reinforces the global tendency towards antiferromagnetic order. To perform numerical calculations the localized spins are considered classical, as in previous related efforts. In this first study, undoped and doped $8\times 8$ clusters are analyzed in a wide range of temperatures. The numerical results reproduce experimental features in the one-particle spectral function and the density-of-states such as $(i)$ the formation of a Zhang-Rice-like band with a dispersion of order $\sim 0.5$ eV and with rotational symmetry about wavevector $(\pi/2,\pi/2)$ at the top of the band, and $(ii)$ the opening of a pseudogap at the chemical potential upon doping. We also observed incipient tendencies towards spin incommensurability. This simple model offers a formalism intermediate between standard mean-field approximations, that fail at finite temperatures in regimes with short-range order, and sophisticated many-body techniques such as Quantum Monte Carlo, that suffer sign problems.

Discrete and Continuum Approximations for Collective Cell Migration in a Scratch Assay with Cell Size Dynamics.

Scratch assays are routinely used to study the collective spreading of cell populations. In general, the rate at which a population of cells spreads is driven by the combined effects of cell migration and proliferation. To examine the effects of cell migration separately from the effects of cell proliferation, scratch assays are often performed after treating the cells with a drug that inhibits proliferation. Mitomycin-C is a drug that is commonly used to suppress cell proliferation in this context. However, in addition to suppressing cell proliferation, mitomycin-C also causes cells to change size during the experiment, as each cell in the population approximately doubles in size as a result of treatment. Therefore, to describe a scratch assay that incorporates the effects of cell-to-cell crowding, cell-to-cell adhesion, and dynamic changes in cell size, we present a new stochastic model that incorporates these mechanisms. Our agent-based stochastic model takes the form of a system of Langevin equations that is the system of stochastic differential equations governing the evolution of the population of agents. We incorporate a time-dependent interaction force that is used to mimic the dynamic increase in size of the agents. To provide a mathematical description of the average behaviour of the stochastic model we present continuum limit descriptions using both a standard mean-field approximation and a more sophisticated moment dynamics approximation that accounts for the density of agents and density of pairs of agents in the stochastic model. Comparing the accuracy of the two continuum descriptions for a typical scratch assay geometry shows that the incorporation of agent growth in the system is associated with a decrease in accuracy of the standard mean-field description. In contrast, the moment dynamics description provides a more accurate prediction of the evolution of the scratch assay when the increase in size of individual agents is included in the model.

Pressure effect of magnetic and electronic properties of Mn2PtGa Heusler alloy

First-principles calculations are performed to investigate pressure effects on structure, magnetism, martensitic phase transition and Curie temperatures of Mn2PtGa Heusler alloy in framework of the density functional theory. It is shown that Mn2PtGa prefer to crystallize in the inverse Heusler type structure. Besides, we predict an extraordinary occurrence of pressure induced metallic ferrimagnetism to half-metallic ferromagnetism transition in cubic phase of Mn2PtGa alloy under hydrostatic pressure up to 43 GPa and the half-metallic ferromagnetism is found to be robust even the lattice further compression to 90 GPa. However, with the pressure up to 100 GPa, the spin-down gap starts to close and the half metallicity begin to disappear, while with the pressure increasing from 100 GPa to 300 GPa, the alloy returns to metallic characteristic. In addition, the energy difference between the austenitic and martensitic phases is found to increase with increasing pressure followed by a decrease when pressure reaches to 43 GPa, which implies a variation trend of martensitic phase transition temperature. Furthermore, Curie temperatures in both austenitic and martensitic phases are estimated under pressure by using the standard mean-field approximation which agrees well with the theoretical results in literature. The robustness of the half metallicity, magnetic transition and the high Curie temperature under pressure make Mn2PtGa alloy a promising candidate for applications in spintronic devices.

Dynamics of exciton-polaritons in discrete lattices under incoherent localized pumping

The paper deals with the spontaneous coherence building up between exciton-polaritons trapped in an array of deep potential wells in the presence of an incoherent pump. A theoretical approach based on a standard tight-binding mean-field approximation is used to reduce the continuous periodic problem to a discrete model. The typical dynamics of the nonlinear exciton-polariton system for the cases of spatially uniform and for localized pumps are discussed. Special attention is paid to the ``staggered'' coherent steady states with $\ensuremath{\pi}$ jumps in the phases between neighboring sites and to ``uniform'' states with a smooth phase distribution. It is shown that, apart from the states with a single frequency, mixed states with spectra with several harmonics can form in the system. The selection mechanism that controls the type of steady state growing from a weak noise is studied. It is found that in the case of localized pumps the decaying tails of the solutions play a crucial role in the dynamics of the polaritons. The applicability of the obtained theoretical results for a qualitative explanation of the complex phenomena observed in recent experiments is discussed.

Statistical systems with nonintegrable interaction potentials.

Statistical systems composed of atoms interacting with each other trough nonintegrable interaction potentials are considered. Examples of these potentials are hard-core potentials and long-range potentials, for instance, the Lennard-Jones and dipolar potentials. The treatment of such potentials is known to confront several problems, e.g., the impossibility of using the standard mean-field approximations, such as Hartree and Hartree-Fock approximations, the impossibility of directly introducing coherent states, the difficulty in breaking the global gauge symmetry, which is required for describing Bose-Einstein condensed and superfluid systems, the absence of a correctly defined Fourier transform, which hampers the description of uniform matter as well as the use of local-density approximation for nonuniform systems. A novel iterative procedure for describing such systems is developed, starting from a correlated mean-field approximation, allowing for a systematic derivation of higher orders, and meeting no problems listed above. The procedure is applicable to arbitrary systems, whether equilibrium or nonequilibrium. The specification for equilibrium systems is presented. The method of extrapolating the expressions for observable quantities from weak coupling to strong coupling is described.

Inhomogeneous phases in the quark-meson model with vacuum fluctuations

Inhomogeneous chiral-symmetry breaking phases at non-vanishing chemical potential and temperature are studied within a two-flavor quark-meson model in the chiral limit. The analysis is performed beyond the standard mean-field approximation by taking into account the Dirac-sea contributions of the quarks. Compared with the case where the Dirac sea is neglected, we find that the inhomogeneous phase shrinks, but in general does not disappear. It is shown within a Ginzburg-Landau analysis that the Lifshitz point of the inhomogeneous phase coincides with the tricritical point if the ratio between sigma-meson and constituent quark mass in vacuum is chosen to be $m_\sigma/M = 2$, corresponding to the fixed mass ratio in the Nambu--Jona-Lasinio model. In the present model, however, this ratio can be varied, offering the possibility to separate the two points. This is confirmed by our numerical calculations, which demonstrate a strong sensitivity of the size of the inhomogeneous phase on $m_\sigma$. Finally, we uncover a general instability of the model with respect to large wave numbers of the chiral modulations, which calls for further improvements beyond the present approximation.

The spin-imbalanced attractive Hubbard model in [formula omitted]: Phase diagrams and BCS–BEC crossover at low filling

We study the superconducting properties of the spin-imbalanced attractive Hubbard model (AHM) in the presence of a Zeeman magnetic field, using the broken symmetry Hartree approximation, mostly in the low density case. The evolution from the weak coupling (BCS-like limit) to the strong coupling limit of tightly bound local pairs (LP) with increasing attraction, at T=0 is investigated for d=3, both for the case of a fixed chemical potential and for a fixed number of particles. Also the influence of the spin-dependent Hartree term on the phase diagrams at T≠0 at weak coupling is studied and reentrant transition is found, which is demonstrated in the temperature characteristics of the order parameter and magnetization. The BCS–BEC crossover diagrams in the presence of a Zeeman magnetic field in three dimensions for simple cubic (sc) lattice are investigated and the results are compared with those obtained in the other lattice geometries. For strong attraction and in the dilute limit, the homogeneous magnetized superconducting phase (SCM) and the tricritical point are found in the (h−μ) and (h−n) diagrams. Next, we construct the finite temperature phase diagrams at weak and strong couplings in d=3 and for symmetric hopping parameters (t↑=t↓), going beyond the standard mean field approximation. The critical temperatures of the superconducting transition are determined within the self-consistent T-matrix approach. We perform a comparison of the results obtained from the (GG0)G0 and (GG)G0 T-matrix schemes, both for the AHM at fixed low electron concentration on an sc lattice and for the 3D continuum model of a dilute gas of fermions with contact attraction. A generalization of the Tc equations for the AHM in non-zero Zeeman magnetic field case is discussed and detailed numerical solutions to these generalized equations are shown. An interesting result is that a spin polarized superfluid state with a gapless region for the majority spin species can be stable in the strong coupling regime. The influence of spin dependent hopping integrals (mass imbalance) (t↑≠t↓) on the stability of the SCM phase is also studied. We also analyze the influence of the Hartree term on the BCS–BEC crossover diagrams in magnetic field and show that the presence of such a term restricts the range of the occurrence of the SCM phase.

The Timing and Targeting of Treatment in Influenza Pandemics Influences the Emergence of Resistance in Structured Populations

Antiviral resistance in influenza is rampant and has the possibility of causing major morbidity and mortality. Previous models have identified treatment regimes to minimize total infections and keep resistance low. However, the bulk of these studies have ignored stochasticity and heterogeneous contact structures. Here we develop a network model of influenza transmission with treatment and resistance, and present both standard mean-field approximations as well as simulated dynamics. We find differences in the final epidemic sizes for identical transmission parameters (bistability) leading to different optimal treatment timing depending on the number initially infected. We also find, contrary to previous results, that treatment targeted by number of contacts per individual (node degree) gives rise to more resistance at lower levels of treatment than non-targeted treatment. Finally we highlight important differences between the two methods of analysis (mean-field versus stochastic simulations), and show where traditional mean-field approximations fail. Our results have important implications not only for the timing and distribution of influenza chemotherapy, but also for mathematical epidemiological modeling in general. Antiviral resistance in influenza may carry large consequences for pandemic mitigation efforts, and models ignoring contact heterogeneity and stochasticity may provide misleading policy recommendations.

Game of Life on the Equal Degree Random Lattice

An effective matrix method is performed to build the equal degree random (EDR) lattice, and then a cellular automaton game of life on the EDR lattice is studied by Monte Carlo (MC) simulation. The standard mean field approximation (MFA) is applied, and then the density of live cells is given ρ=0.37017 by MFA, which is consistent with the result ρ=0.37±0.003 by MC simulation.

FISSION BARRIERS AT FINITE TEMPERATURE: A THEORETICAL DESCRIPTION WITH THE GOGNY FORCE

The evolution with temperature of nuclear properties relevant to fission are analyzed in the case of the 240 Pu nucleus by using the standard finite temperature mean field approximation (including pairing correlations) and the Gogny D1S force. To be more specific, potential energy curves, pairing correlation energies, level density parameter a, collective quadrupole mass, etc are considered. The impact of their evolution with temperature in the quantum decay rate, meaningful only in the low temperature regime, is also considered.

Derivation of effective field theories

A general self-consistency approach allows for a thorough treatment of the corrections to the mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a point of view on the effective field theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a coarse-graining procedure. We outline and justify our method by some preliminary calculations. Results are given for the critical temperature and the Landau parameters of the phi(4) theory--the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions--the problem for the calculation of the true critical temperature--is considered within the framework of the present approach. A comprehensive description of the ground-state properties of many-body systems is also demonstrated.

Critical dopant concentration in polyacetylene and phase diagram from a continuous four-Fermi model

The Optimized Perturbation Theory (OPT) method, at finite temperature and finite chemical potential, is applied to the field theory model for polyacetylene. The critical dopant concentration in trans-polyacetylene is evaluated and compared with the available experimental data and with previous calculations. The results obtained within the OPT go beyond the standard mean field (or large-N) approximation (MFA) by explicitly including finite N effects. A critical analysis of the possible theoretical prescriptions to implement and interpret these corrections to the mean field results, given the available data, is given. For typical temperatures probed in the laboratory, our results show that the critical dopant concentration is only weakly affected by thermal effects.

A neurologically plausible implementation of statistical inference applied to random dot motion

We propose a model of basic motion perception consisting of a hierarchical non-linear state space model (NSSM) developed within a variational Bayesian (VB) framework. Each level of the hierarchy is a `cause' that generates a prior distribution on the level below via a generative function. The temporal dynamics and generative functions between layers of the hierarchy are implemented as neural networks with non-linear activation functions. Optimization of model parameters and causes proceed concurrently as a combination of fixed-point rules and gradient decent schemes. To make the optimization problem tractable, the standard mean-field and Laplace approximations are employed. The precise factoring used in the mean-field approximation is designed to meet a balance between tractability, neurological plausibility and modeling power. In this approach inference and learning proceed concurrently, in an online and unsupervised fashion. Other work has produced similar implementations of NSSMs that have been successful in predicting low-dimensional temporal signals, but with highly restrictive assumptions on the form of the posterior distributions and with learning done in batches instead of online [1]. Competing work has relaxed many of these implementation assumptions and achieved prediction of high dimensional input but at the cost of discarding VB techniques in favor of inefficient discrete state-space methods [2]. Moreover, such techniques have been demonstrated only with pre-learned weights and simplistic statistical models. Our model is primarily tested on realistic high-dimensional input generated by randomly moving dots over a detection grid. The results from a spiking neuron implementation of the model based on the Neural Engineering Framework (NEF) are compared directly to single cell recordings in random dot motion perception and decision-making tasks.

Two-level interacting boson models beyond the mean field

The phase diagram of two-level boson Hamiltonians, including the interacting boson model (IBM), is studied beyond the standard mean field approximation using the Holstein-Primakoff mapping. The limitations of the usual intrinsic state (mean field) formalism concerning finite-size effects are pointed out. The analytic results are compared to numerics obtained from exact diagonalizations. Excitation energies and occupation numbers are studied in different model space regions (Casten triangle for IBM) and especially at the critical points.