Weiss Model Research Articles

Layer-specific BOLD effects in gradient and spin-echo acquisitions in somatosensory cortex.

Previous studies have shown varied BOLD signals with gradient echo (GE) across cortical depth. To interpret these variations, and understand the effects of vascular geometry and size, the magnitudes and layer distributions of GE and spin-echo (SE) BOLD functional MRI signals were compared in the somatosensory cortex of squirrel monkeys during tactile stimulation and in a resting state at high spatial resolution and high field. A block-design stimulation was used to identify tactile-evoked activation signals in somatosensory Areas 3b and 1. Layer-specific connectivities were calculated using resting-state data. Signal power spectra were compared by depth and pulse sequence. The measured ratios of transverse relaxation rate changes were compared with Anderson and Weiss's model. SE signals showed a 26% lower percentage signal change during tactile stimulation compared with GE, along with a slower time course. SE signals remained consistent but weaker in lower layers, whereas GE signals decreased with cortical depth. This pattern extended to resting-state power spectra. Resting-state functional connectivity indicated larger connectivity between the top layers of Area 3b and Area 1 for GE, with minimal changes for SE. Comparisons with theory suggest vessel diameters ranging from 19.4 to 9 microns are responsible for BOLD effects across cortical layers at 9.4 T. These results provide further evidence that at high field, SE BOLD signals are relatively free of contributions from sources other than microvascular changes in response to neural activity, whereas GE signals, even in the superficial layers, are not dominated by very large vessels.

Some remarks on the effect of the Random Batch Method on phase transition

In this article, we focus on two toy models : the Curie–Weiss model and the system of N particles in linear interactions in a double well confining potential. Both models, which have been extensively studied, describe a large system of particles with a mean-field limit that admits a phase transition. We are concerned with the numerical simulation of these particle systems. To deal with the quadratic complexity of the numerical scheme, corresponding to the computation of the O(N2) interactions per time step, the Random Batch Method (RBM) has been suggested. It consists in randomly (and uniformly) dividing the particles into batches of size p>1, and computing the interactions only within each batch, thus reducing the numerical complexity to O(Np) per time step. The convergence of this numerical method has been proved in other works.This work is motivated by the observation that the RBM, via the random constructions of batches, artificially adds noise to the particle system. The goal of this article is to study the effect of this added noise on the phase transition of the nonlinear limit, and more precisely we study the effective dynamics of the two models to show how a phase transition may still be observed with the RBM but at a lower critical temperature.

Phase Diagram and Specific Heat of a Nonequilibrium Curie–Weiss Model

Phase Diagram and Specific Heat of a Nonequilibrium Curie–Weiss Model

The Effect of Sociodemographic Factors on Female Educational Attainment: An Application of the Multipopulation Curie–Weiss Model

In recent times, there have been a lot of advocacies to encourage females of all ages to pursue education to the highest level. The Government of Ghana and its stakeholders have championed this to achieve the Sustainable Development Goal 4 in this regard. However, progress has been on a snail’s pace. In this study, we employed the multipopulation Curie–Weiss model to establish the influence of sociodemographic factors on the educational attainment of all females in Ghana. We utilized data from the Ghana Demographic and Health Survey (2008 and 2014). The parameters of the interacting and noninteracting multipopulation Curie–Weiss models were estimated using the partial least squares and ordinary least squares methods, respectively. Our findings suggest that marital status, place of residence, and wealth status of females in Ghana influence their educational attainment. Therefore, to get more females of all ages to attain higher levels of education, the Government of Ghana and its stakeholders should channel their efforts towards eradicating poverty at all levels, preventing early and teenage marriages, providing better infrastructure, and creating better living conditions for all females especially those in the poorest regions of the country.

Generative diffusion in very large dimensions

Generative models based on diffusion have become the state of the art in the last few years, notably for image generation. Here, we analyze them in the high-dimensional limit, where data are formed by a very large number of variables. We use methods from statistical physics and focus on two well-controlled high-dimensional cases: a Gaussian model and the Curie–Weiss model of ferromagnetism. In the latter case, we highlight the mechanism of symmetry breaking in the inverse diffusion, and point out that, in order to reconstruct the relative asymmetry of the two low-temperature states, and thus to obtain the correct probability weights, one needs a database with a number of points much larger than the dimension of each data point. We characterize the scaling laws in the number of data and in the number of dimensions for an efficient generation.

Improved Metropolis–Hastings algorithms via landscape modification with applications to simulated annealing and the Curie–Weiss model

AbstractIn this paper, we propose new Metropolis–Hastings and simulated annealing algorithms on a finite state space via modifying the energy landscape. The core idea of landscape modification rests on introducing a parameter c, such that the landscape is modified once the algorithm is above this threshold parameter to encourage exploration, while the original landscape is utilized when the algorithm is below the threshold for exploitation purposes. We illustrate the power and benefits of landscape modification by investigating its effect on the classical Curie–Weiss model with Glauber dynamics and external magnetic field in the subcritical regime. This leads to a landscape-modified mean-field equation, and with appropriate choice of c the free energy landscape can be transformed from a double-well into a single-well landscape, while the location of the global minimum is preserved on the modified landscape. Consequently, running algorithms on the modified landscape can improve the convergence to the ground state in the Curie–Weiss model. In the setting of simulated annealing, we demonstrate that landscape modification can yield improved or even subexponential mean tunnelling time between global minima in the low-temperature regime by appropriate choice of c, and we give a convergence guarantee using an improved logarithmic cooling schedule with reduced critical height. We also discuss connections between landscape modification and other acceleration techniques, such as Catoni’s energy transformation algorithm, preconditioning, importance sampling, and quantum annealing. The technique developed in this paper is not limited to simulated annealing, but is broadly applicable to any difference-based discrete optimization algorithm by a change of landscape.

On Magnetic Models in Wavefunction Ensembles.

In a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions. In this perspective we study how to construct Gibbs ensembles for magnetic quantum spin models. We show that with free boundary conditions and distinguishable "spins" there are no finite-temperature phase transitions because of high dimensionality of the phase space. Then we focus on the simplest case, namely the mean-field (Curie-Weiss) model, in order to discover whether phase transitions are even possible in this model class. This strategy at least diminishes the dimensionality of the problem. We found that, even assuming exchange symmetry in the wavefunctions, no finite-temperature phase transitions appear when the Hamiltonian is given by the usual energy expression of quantum mechanics (in this case the analytical argument is not totally satisfactory and we relied partly on a computer analysis). However, a variant model with additional "wavefunction energy" does have a phase transition to a magnetized state. (With respect to dynamics, which we do not consider here, wavefunction energy induces a non-linearity which nevertheless preserves norm and energy. This non-linearity becomes significant only at the macroscopic level.) The three results together suggest that magnetization in large wavefunction spin chains appears if and only if we consider indistinguishable particles and block macroscopic dispersion (i.e., macroscopic superpositions) by energy conservation. Our principle technique involves transforming the problem to one in probability theory, then applying results from large deviations, particularly the Gärtner-Ellis Theorem. Finally, we discuss Gibbs vs. Boltzmann/Einstein entropy in the choice of the quantum thermodynamic ensemble, as well as open problems.

Cramér-type moderate deviation of normal approximation for unbounded exchangeable pairs

In Stein’s method, the exchangeable pair approach is commonly used to estimate the approximation errors in normal approximation. In this paper, we establish a Cramér-type moderate deviation theorem of normal approximation for unbounded exchangeable pairs. As applications, Cramér-type moderate deviation theorems for the sums of local statistics and general Curie–Weiss model are obtained.

The equation of state for metal-doped ferroelectrics within the Weiss model

This paper presents a theoretical model for describing the thermodynamic properties of doped ferroelectric crystals based on a modified Weiss mean-field approach. Accounting for quadrupole and octupole terms in the expression for the effective field within the Weiss model makes it possible to move from the Langevin equation to the Landau–Ginzburg equation. Furthermore, the coefficients of the Landau–Ginzburg equation can be expressed in terms of the physical parameters of the crystal lattice. For these parameters, analytical expressions are proposed that describe their change when adding dopants in ceramic matrix composites. Perovskite barium titanate ceramics with a variety of inclusions is considered as an application example of the developed method. The obtained agreement between the analytical and experimental results for barium titanate ceramics with lanthanum/magnesium/zirconium dopants gives us hope of the applicability of the present theory to the calculation of other doped ferroelectrics as well.

Approximating the magnetization in the Curie–Weiss model

In this paper, we quantify approximations to the magnetization, a key quantity for the Curie–Weiss model, via the Stein method for the Markov chain whose stationary distribution coincides with the Curie–Weiss model.

Energy Landscape of the Two-Component Curie–Weiss–Potts Model with Three Spins

In this paper, we investigate the energy landscape of the two-component spin systems, known as the Curie–Weiss–Potts model, which is a generalization of the Curie–Weiss model consisting of $$q\ge 3$$ spins. In the energy landscape of a multi-component model, the most important element is the relative strength between the inter-component interaction strength and the component-wise interaction strength. If the inter-component interaction is stronger than the component-wise interaction, we can expect all the components to be synchronized in the course of metastable transition. However, if the inter-component interaction is relatively weaker, then the components will be desynchronized in the course of metastable transition. For the two-component Curie–Weiss model, the phase transition from synchronization to desynchronization has been precisely characterized in studies owing to its mean-field nature. The purpose of this paper is to extend this result to the Curie–Weiss–Potts model with three spins. We observe that the nature of the phase transition for the three-spin case is entirely different from the two-spin case of the Curie–Weiss model, and the proof as well as the resulting phase diagram is fundamentally different and exceedingly complicated.

Estimation in tensor Ising models

AbstractThe $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geqslant 2$. This is a natural generalization of the matrix Ising model that provides a convenient mathematical framework for capturing, not just pairwise, but higher-order dependencies in complex relational data. In this paper, we consider the problem of estimating the natural parameter of the $p$-tensor Ising model given a single sample from the distribution on $N$ nodes. Our estimate is based on the maximum pseudolikelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. We derive general conditions under which the MPL estimate is $\sqrt N$-consistent, that is, it converges to the true parameter at rate $1/\sqrt N$. Our conditions are robust enough to handle a variety of commonly used tensor Ising models, including spin glass models with random interactions and models where the rate of estimation undergoes a phase transition. In particular, this includes results on $\sqrt N$-consistency of the MPL estimate in the well-known $p$-spin Sherrington–Kirkpatrick model, spin systems on general $p$-uniform hypergraphs and Ising models on the hypergraph stochastic block model (HSBM). In fact, for the HSBM we pin down the exact location of the phase transition threshold, which is determined by the positivity of a certain mean-field variational problem, such that above this threshold the MPL estimate is $\sqrt N$-consistent, whereas below the threshold no estimator is consistent. Finally, we derive the precise fluctuations of the MPL estimate in the special case of the $p$-tensor Curie–Weiss model, which is the Ising model on the complete $p$-uniform hypergraph. An interesting consequence of our results is that the MPL estimate in the Curie–Weiss model saturates the Cramer–Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic statistical efficiency in the estimability regime, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.

A probabilistic approach to convex (ϕ)-entropy decay for Markov chains

We study the exponential dissipation of entropic functionals along the semigroup generated by a continuous time Markov chain and the associated convex Sobolev inequalities, including MLSI and Beckner inequalities. We propose a method that combines the Bakry–Émery approach and coupling arguments, which we use as a probabilistic alternative to the discrete Bochner identities. In particular, the validity of the method is not limited to the perturbative setting and we establish convex entropy decay for interacting random walks beyond the high temperature/weak interaction regime. In this framework, we show that the exponential contraction of the Wasserstein distance implies MLSI. We also revisit classical examples often obtaining new inequalities and sometimes improving on the best known constants. In particular, we analyse the zero range dynamics, hardcore and Bernoulli–Laplace models and the Glauber dynamics for the Curie–Weiss and Ising model.

Metastability of Synchronous and Asynchronous Dynamics.

Metastability is a ubiquitous phenomenon in nature, which interests several fields of natural sciences. Since metastability is a genuine non-equilibrium phenomenon, its description in the framework of thermodynamics and statistical mechanics has progressed slowly for a long time. Since the publication of the first seminal paper in which the metastable behavior of the mean field Curie–Weiss model was approached by means of stochastic techniques, this topic has been largely studied by the scientific community. Several papers and books have been published in which many different spin models were studied and different approaches were developed. In this review, we focus on the comparison between the metastable behavior of synchronous and asynchronous dynamics, namely, stochastic processes in discrete time in which, at each time, either all the spins or one single spin is updated. In particular, we discuss how two different stochastic implementations of the very same Hamiltonian give rise to different metastable behaviors.

Improved Room-Temperature Ferromagnetism in Nd-Doped SnO2 Nanostructures through Defect Engineering

Many efforts have been made to introduce room-temperature ferromagnetism (RTFM) in metal oxide semiconductors doped with either magnetic or nonmagnetic ions for their suitable application in spintronics in the past few years. However, it is not yet well understood whether the origin of RTFM in these systems is intrinsic or because of magnetic clusters. Hence, we report detailed experimental investigations on RTFM in Nd-doped SnO2 nanostructures synthesized using a sol–gel process, which has potential spintronics application. In the present work, we observe that defects/oxygen vacancies play a crucial role in improving the RTFM in SnO2 doped with Nd ions. The presence of oxygen vacancies in the prepared samples was confirmed by Raman scattering, X-ray photoelectron spectroscopy (XPS), and photoluminescence (PL) spectroscopy. The enhancement of the density of oxygen vacancies was further quantified by the deconvoluted core-level O 1s XPS spectra. All samples exhibited RTFM with a small contribution of paramagnetic (PM) ordering. These phases (FM + PM) were established by fitting all M–H curves using a two-phase theoretical model. Furthermore, the Curie–Weiss and spin wave (CWSW) model fitted the M–T curve well, which also indicated the existence of the two phases in our systems. The overlapping of bound magnetic polarons occurred because of the exchange interaction between Nd3+ ions and electrons trapped in oxygen vacancies; this overlap is responsible for introducing RTFM in Nd-doped SnO2. Hence, the origin of RTFM can be mostly explained by the bound magnetic polaron (BMP) model.

Large deviations for the skew-detailed-balance lifted-Markov processes to sample the equilibrium distribution of the Curie–Weiss model

Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of ‘lifting’ consists in duplicating the configuration space into two copies σ = ± and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance lifted-Markov-chain introduced by Turitsyn et al (2011 Physica D 240 410) is revisited for the Curie–Weiss mean-field ferromagnetic model, where the dynamics for the magnetization is closed. The large deviations at various levels for empirical time-averaged observables are analyzed and compared with their detailed-balance counterparts, both for the discrete extensive magnetization M and for the continuous intensive magnetization for large system-size N.

Fluctuations of the Magnetization in the p-Spin Curie–Weiss Model

In this paper we study the fluctuations of the magnetization in the p-spin Curie–Weiss model, for $$p \geqslant 3$$ p ⩾ 3 . We provide a complete description of the asymptotic distribution of the magnetization in the p-spin Curie–Weiss model, complementing the well-known results in the 2-spin case. Our results unearth various new phase transitions, such as the existence of a certain ‘critical’ curve in the parameter space, where the limiting distribution of the magnetization is a discrete mixture, with local Gaussian fluctuations around each of the atoms. The number of atoms (mixture components) is either two or three depending on the sign of one of the parameters and the parity of p. Another interesting revelation is the existence of certain ‘special’ points in the parameter space where the magnetization converges to a non-Gaussian limiting distribution at rate $$N^{\frac{1}{4}}$$ N 1 4 .

Magnetic properties and non-fermi liquid behaviour in mechanically alloyed FeCu

FexCu100−x alloys where x (wt%) = 25, 35, 50, 65, and 75 were prepared via mechanical alloying. Microstructural characterization revealed a single-phase face-centered cubic structure for a wide range of compositions. Atomic volumes were estimated from measured lattice parameters suggesting a high volume state consistent with Weiss model predictions of the ferromagnetic ground state. Saturation magnetic moments at 2 K monotonically decreased from 2.28 μB/Fe for starting Fe powder to 0.65 μB/Fe in Fe25Cu75 alloy. Samples with Fe content higher than 35% exhibited magnetic transition temperatures (Tc) higher than 350 K. On the other hand, the Fe25Cu75 sample exhibit a Tc around 250 K as determined from the gradient method. According to Banerjee's criterion, the magnetic transition is determined to be second order for the Fe25Cu75 alloy. Moreover, modified Arrott plots were used to reveal the critical behavior of the Fe25Cu75. Temperature dependencies of the resistivity were measured for all samples. At low temperatures, the Fe25Cu75 alloy exhibited T3/2 dependence of the resistivity. At higher temperatures (100–300 K) all mechanically alloyed FeCu exhibited unusual T linear dependence resistivity. Such behavior is often called “strange metallicity” or non-Fermi Liquid behavior. The possible origin of such behavior was also discussed.

Local Central Limit Theorem for Multi-group Curie–Weiss Models

AbstractWe study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is, for the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high-temperature regime.

A hierarchical mean field model of interacting spins

We consider a system of hierarchical interacting spins under dynamics of spin-flip type with a ferromagnetic mean field interaction, scaling with the hierarchical distance, coupled with a system of linearly interacting hierarchical diffusions of Ornstein–Uhlenbeck type. In particular, the diffusive variables enter in the spin-flip rates, effectively acting as dynamical magnetic fields. In absence of the diffusions, the spin-flip dynamics can be thought of as a modification of the Curie–Weiss model. We study the mean field and the two-level hierarchical model, in the latter case restricting to a subcritical regime, corresponding to high temperatures, obtaining macroscopic limits at different spatio-temporal scales and studying the phase transitions in the system. We also formulate a generalization of our results to the kth level hierarchical case, for any k finite, in the subcritical regime. We finally address the supercritical regime, in the zero-temperature limit, for the two-level hierarchical case, proceeding heuristically with the support of numerics.