Curie-Weiss Model Research Articles

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.

Symmetry Shapes Thermodynamics of Macroscopic Quantum Systems.

We derive a systematic approach to the thermodynamics of quantum systems based on the underlying symmetry groups. We first show that the entropy of a system can be described in terms of group-theoretical quantities that are largely independent of the details of its density matrix. We then apply our technique to generic N identical interacting d-level quantum systems. Using permutation invariance, we find that, for large N, the entropy displays a universal asymptotic behavior in terms of a function s(x) that is completely independent of the microscopic details of the model, but depends only on the size of the irreducible representations of the permutation group S_{N}. In turn, the equilibrium state of the system and macroscopic fluctuations around it are shown to satisfy a large deviation principle with a rate function f(x)=e(x)-β^{-1}s(x), where e(x) only depends on the ground state energy of particular subspaces determined by group representation theory, and β is the inverse temperature. We apply our theory to the transverse-field Curie-Weiss model, a minimal model of phase transition exhibiting an interplay of thermal and quantum fluctuations.

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

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

Distributional Approximation for General Curie–Weiss Models with Size-dependent Inverse Temperatures

Distributional Approximation for General Curie–Weiss Models with Size-dependent Inverse Temperatures

Opinion Models, Election Data, and Political Theory.

A unifying setup for opinion models originating in statistical physics and stochastic opinion dynamics are developed and used to analyze election data. The results are interpreted in the light of political theory. We investigate the connection between Potts (Curie-Weiss) models and stochastic opinion models in the view of the Boltzmann distribution and stochastic Glauber dynamics. We particularly find that the q-voter model can be considered as a natural extension of the Zealot model, which is adapted by Lagrangian parameters. We also discuss weak and strong effects (also called extensive and nonextensive) continuum limits for the models. The results are used to compare the Curie-Weiss model, two q-voter models (weak and strong effects), and a reinforcement model (weak effects) in explaining electoral outcomes in four western democracies (United States, Great Britain, France, and Germany). We find that particularly the weak effects models are able to fit the data (Kolmogorov-Smirnov test) where the weak effects reinforcement model performs best (AIC). Additionally, we show how the institutional structure shapes the process of opinion formation. By focusing on the dynamics of opinion formation preceding the act of voting, the models discussed in this paper give insights both into the empirical explanation of elections as such, as well as important aspects of the theory of democracy. Therefore, this paper shows the usefulness of an interdisciplinary approach in studying real world political outcomes by using mathematical models.

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.

Conformal Fisher information metric with torsion

We consider torsion in parameter manifolds that arises via conformal transformations of the Fisher information metric, and define it for information geometry of a wide class of physical systems. The torsion can be used to differentiate between probability distribution functions that otherwise have the same scalar curvature and hence define similar geometries. In the context of thermodynamic geometry, our construction gives rise to a new scalar—the torsion scalar defined on the manifold, while retaining known physical features related to other scalar quantities. We analyse this in the context of the Van der Waals and the Curie–Weiss models. In both cases, the torsion scalar has non trivial behaviour on the spinodal curve. We also briefly comment on the one dimensional classical Ising model and show that the torsion scalar diverges exponentially near criticality.

Infinite Volume Gibbs States and Metastates of the Random Field Mean-Field Spherical Model

For the discrete random field Curie–Weiss models, the infinite volume Gibbs states and metastates have been investigated and determined for specific instances of random external fields. In general, there are not many examples in the literature of non-trivial limiting metastates for discrete or continuous spin systems. We analyze the infinite volume Gibbs states of the mean-field spherical model, a model of continuous spins, in a general random external field with independent identically distributed components with finite moments of some order larger than four and non-vanishing variances of the second moments. Depending on the parameters of the model, we show that there exist three distinct phases: ordered ferromagnetic, ordered paramagnetic, and spin glass. In the ordered ferromagnetic and ordered paramagnetic phases, we show that there exists a unique infinite volume Gibbs state almost surely. In the spin glass phase, we show the existence of chaotic size dependence, provide a construction of the Aizenman–Wehr metastate, and consider both the convergence in distribution and almost sure convergence of the Newman–Stein metastates. The limiting metastates are non-trivial and their structure is universal due to the presence of Gaussian fluctuations and the spherical constraint.

Statistics of the two star ERGM

In this paper, we explore the two star Exponential Random Graph Model, which is a two parameter exponential family on the space of simple labeled graphs. We introduce auxiliary variables to express the two star model as a mixture of the β model on networks. Using this representation, we study asymptotic distribution of the number of edges, and the sampling variance of the degrees. In particular, the limiting distribution for the number of edges has similar phase transition behavior to that of the magnetization in the Curie-Weiss Ising model of Statistical Physics. Using this, we show existence of consistent estimates for both parameters. Finally, we prove that the centered partial sum of degrees converges as a process to a Brownian bridge in all parameter domains, irrespective of the phase transition.

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.

Motion of Lee–Yang Zeros

We consider the zeros of the partition function of the Ising model with ferromagnetic pair interactions and complex external field. Under the assumption that the graph with strictly positive interactions is connected, we vary the interaction (denoted by t) at a fixed edge. It is already known that each zero is monotonic (either increasing or decreasing) in t; we prove that its motion is local: the entire trajectories of any two distinct zeros are disjoint. If the underlying graph is a complete graph and all interactions take the same value $$t\ge 0$$ (i.e., the Curie-Weiss model), we prove that all the principal zeros (those in $$i[0,\pi /2)$$ ) decrease strictly in t.

Uniqueness of stationary distribution and exponential convergence for distribution dependent SDEs

The existence and uniqueness of stationary distributions and the exponential convergence in $ L^p $-Wasserstein distance are derived for SDEs driven by distribution dependent noise without uniformly dissipative drift. By using the exponential convergence and the Talagrand inequality of associated decoupled equations, global and local exponential convergences are established, and explicit convergence rate is obtained. Our results can be applied to the Curie-Weiss model and the granular media model in double-well landscape with quadratic interaction driven by distribution dependent noise.

When does the chaos in the Curie-Weiss model stop to propagate?

When does the chaos in the Curie-Weiss model stop to propagate?

Models for Quantum Measurement of Particles with Higher Spin.

The Curie-Weiss model for quantum measurement describes the dynamical measurement of a spin-12 by an apparatus consisting of an Ising magnet of many spins 12 coupled to a thermal phonon bath. To measure the z-component s=-l,-l+1,⋯,l of a spin l, a class of models is designed along the same lines, which involve 2l order parameters. As required for unbiased measurement, the Hamiltonian of the magnet, its entropy and the interaction Hamiltonian possess an invariance under the permutation s→s+1 mod 2l+1. The theory is worked out for the spin-1 case, where the thermodynamics is analyzed in detail, and, for spins 32,2,52, the thermodynamics and the invariance are presented.

Limit Theorems for Multi-group Curie–Weiss Models via the Method of Moments

We study a multi-group version of the mean-field or Curie-Weiss spin model. For this model, we show how, analogously to the classical (single-group) model, the three temperature regimes are defined. Then we use the method of moments to determine for each regime how the vector of the group magnetisations behaves asymptotically. Some possible applications to social or political sciences are discussed.

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.