Metropolis Dynamics Research Articles

Stochastic Gradient Descent-like relaxation is equivalent to Metropolis dynamics in discrete optimization and inference problems

Is Stochastic Gradient Descent (SGD) substantially different from Metropolis Monte Carlo dynamics? This is a fundamental question at the time of understanding the most used training algorithm in the field of Machine Learning, but it received no answer until now. Here we show that in discrete optimization and inference problems, the dynamics of an SGD-like algorithm resemble very closely that of Metropolis Monte Carlo with a properly chosen temperature, which depends on the mini-batch size. This quantitative matching holds both at equilibrium and in the out-of-equilibrium regime, despite the two algorithms having fundamental differences (e.g. SGD does not satisfy detailed balance). Such equivalence allows us to use results about performances and limits of Monte Carlo algorithms to optimize the mini-batch size in the SGD-like algorithm and make it efficient at recovering the signal in hard inference problems.

Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs.

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven by the heat-bath dynamics, their stationary characteristics, such as magnetization, are correctly reproduced by MFA as confirmed by Monte Carlo simulations. It turns out that MFA reproduces the same result as the generating functional analysis that is expected to provide the exact description of such models. We argue that on directed regular random graphs, the neighbors of a given vertex are typically uncorrelated, and that is why MFA for models with heat-bath dynamics provides their exact description. For models with Metropolis dynamics, certain additional correlations become relevant, and MFA, which neglects these correlations, is less accurate. Models with heat-bath dynamics undergo continuous phase transition, and at the critical point, the power-law time decay of the order parameter exhibits the behavior of the Ising mean-field universality class. Analogous phase transitions for models with Metropolis dynamics are discontinuous.

Transient Behavior of Damage Spreading in the Two-Dimensional Blume–Capel Ferromagnet

We study the transient behavior of damage propagation in the two-dimensional spin-$1$ Blume-Capel model using Monte Carlo simulations with Metropolis dynamics. We find that, for a particular region in the second-order transition regime of the crystal field--temperature phase diagram of the model, the average Hamming distance decreases exponentially with time in the weakly damaged system. Additionally, its rate of decay appears to depend linearly on a number of Hamiltonian parameters, namely the crystal field, temperature, applied magnetic field, but also on the amount of damage. Finally, a comparative study using Metropolis and Glauber dynamics indicates a slower decay rate of the average Hamming distance for the Glauber protocol.

Metastability for Glauber Dynamics on the Complete Graph with Coupling Disorder

Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values -1 or +1. Each spin i in [n]={1,2,dots , n} interacts with a magnetic field h in [0,infty ), while each pair of spins i,j in [n] interact with each other at coupling strength n^{-1} J(i)J(j), where J=(J(i))_{i in [n]} are i.i.d. non-negative random variables drawn from a probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature beta in (0,infty ). We show that there are critical thresholds beta _c and h_c(beta ) such that, in the limit as nrightarrow infty , the system exhibits metastable behaviour if and only if beta in (beta _c, infty ) and h in [0,h_c(beta )). Our main result is a sharp asymptotics, up to a multiplicative error 1+o_n(1), of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of J, while the correction terms do. The leading order of the correction term is sqrt{n} times a centred Gaussian random variable with a complicated variance depending on beta ,h, on the law of J and on the metastable state. The critical thresholds beta _c and h_c(beta ) depend on the law of J, and so does the number of metastable states. We derive an explicit formula for beta _c and identify some properties of beta mapsto h_c(beta ). Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant.

Oscillations in the nucleation preexponential.

Oscillations in the preexponential of the nucleation rate are due to the discrete nature of small nuclei. An accurate elementary expression to describe such oscillations is derived in the limit of a high nucleation barrier. The result is applied to the standard Becker-Döring equation in two and three dimensions, and to the lowest-energy nucleation path in a cold lattice gas with Glauber and Metropolis dynamics (equivalent to an Ising model on a square lattice) where oscillation effects can be more pronounced.

Metastability for the Ising model on the hexagonal lattice

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the mixing time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

Stability of skyrmion crystal phase in antiferromagnetic triangular lattice with DMI and single-ion anisotropy

We study a frustrated antiferromagnetic Heisenberg model on a triangular lattice with the Dzyaloshinskii–Moriya interaction (DMI) in the presence of an external magnetic field and focus on the effects of the single-ion anisotropy on the overall phase diagram topology and the skyrmion lattice (SkX) phase stability. Phase diagrams in the temperature-field plane for both easy-axis and easy-plane anisotropy of a varying strength are constructed in the regimes of a moderate and strong DMI by parallel tempering Monte Carlo simulations. For the considered range of parameters the phase diagrams featuring up to four ordered phases are identified. The SkX phase is found to be stabilized within some temperature and field window for sufficiently large DMI and not too strong anisotropy. We demonstrate that both types of the anisotropy can have beneficial effects on the SkX phase stabilization. Namely, in the systems with moderate DMI a small easy-plane anisotropy extends the field window of the SkX phase appearance by shifting its lower bound to smaller values and in the systems with larger DMI a small easy-axis anisotropy shifts the entire field range of the SkX phase stabilization towards smaller values. The Metropolis dynamics is employed to probe the persistence of the SkX phase upon quenching the field to zero or small finite values. It is confirmed that low temperatures, small DMI and small easy-plane anisotropy values support the skyrmions persistence, i.e., increase their lifetime.

Self-Adaptive Acceptance Rate-Driven Markov Chain Monte Carlo Method Applied to the Study of Magnetic Nanoparticles

A standard canonical Markov Chain Monte Carlo method implemented with a single-macrospin movement Metropolis dynamics was conducted to study the hysteretic properties of an ensemble of independent and non-interacting magnetic nanoparticles with uniaxial magneto-crystalline anisotropy randomly distributed. In our model, the acceptance-rate algorithm allows accepting new updates at a constant rate by means of a self-adaptive mechanism of the amplitude of Néel rotation of magnetic moments. The influence of this proposal upon the magnetic properties of our system is explored by analyzing the behavior of the magnetization versus field isotherms for a wide range of acceptance rates. Our results allows reproduction of the occurrence of both blocked and superparamagnetic states for high and low acceptance-rate values respectively, from which a link with the measurement time is inferred. Finally, the interplay between acceptance rate with temperature in hysteresis curves and the time evolution of the saturation processes is also presented and discussed.

Classical and Quantum Gases on a Semiregular Mesh

The main objective of a statistical mechanical calculation is drawing the phase diagram of a many-body system. In this respect, discrete systems offer the clear advantage over continuum systems of an easier enumeration of microstates, though at the cost of added abstraction. With this in mind, we examine a system of particles living on the vertices of the (biscribed) pentakis dodecahedron, using different couplings for first and second neighbor particles to induce a competition between icosahedral and dodecahedral orders. After working out the phases of the model at zero temperature, we carry out Metropolis Monte Carlo simulations at finite temperature, highlighting the existence of smooth transitions between distinct “phases”. The sharpest of these crossovers are characterized by hysteretic behavior near zero temperature, which reveals a bottleneck issue for Metropolis dynamics in state space. Next, we introduce the quantum (Bose-Hubbard) counterpart of the previous model and calculate its phase diagram at zero and finite temperatures using the decoupling approximation. We thus uncover, in addition to Mott insulating “solids”, also the existence of supersolid “phases” which progressively shrink as the system is heated up. We argue that a quantum system of the kind described here can be realized with programmable holographic optical tweezers.

Metastability for the dilute Curie–Weiss model with Glauber dynamics

We analyse the metastable behaviour of the dilute Curie-Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are Bernoulli random variables with mean $p\in (0,1)$. This model can be also viewed as an Ising model on the Erd\H{o}s-R\'enyi random graph with edge probability $p$. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature $\beta$. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where $N\to\infty$, $\beta>\beta_c=1$ and $h$ is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie-Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities.

Classical critical dynamics in quadratically driven Kerr resonators

Driven-dissipative kerr lattices with two-photon driving are experimentally relevant systems known to exhibit a symmetry-breaking phase transition, which belongs to the universality class of the thermal Ising model for the parameter regime studied here. In this work, we perform finite-size scaling of this system as it is quenched to the transition and the dynamical critical exponent is found to be compatible with $z\approx2.18$ corresponding with metropolis dynamics in classical simulations. Furthermore, we show that the Liouvillian gap scales with the same exponent, similar to scaling of the Hamiltonian gap at quantum phase transitions in closed systems.

From early nucleation past the percolation threshold: Status of the Kolmogorov-Avrami theory on a cold Ising lattice.

A high-precision test of the Kolmogorov-Avrami (KA) model of "geometric" interactions between growing nuclei is performed for an Ising ferromagnet driven by Metropolis dynamics. As long as the interface boundaries of interacting nuclei are rough, the KA model is adequate for arbitrary large densities of the new phase, well beyond the percolation transition, as in the case of homogeneous nucleation. For a prenucleated system the KA description breaks down at sufficiently low temperature, when the length of a kinetically smooth interface exceeds the distance between seeds.

Convergence Time to Equilibrium of the Metropolis Dynamics for the GREM

We study the convergence time to equilibrium of the Metropolis dynamics for the Generalized Random Energy Model with an arbitrary number of hierarchical levels, a finite and reversible continuous-time Markov process, in terms of the spectral gap of its transition probability matrix. This is done by deducing bounds to the inverse of the gap using a Poincar\'e inequality and a path technique. We also apply convex analysis tools to give the bounds in the most general case of the model.

Tunneling of the hard‐core model on finite triangular lattices

We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.

Aging in Metropolis dynamics of the REM: a proof

We consider Metropolis dynamics of the Random Energy Model. We prove that the classical two-time correlation function that allows one to establish aging converges almost surely to the arcsine law distribution function, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. To do this we link the two-time correlation function to a certain continuous-time clock process which, after proper rescaling, is proven to converge to a stable subordinator almost surely in the random environment and in the fine $$J_1$$ -topology of Skorohod. This fine topology then enables us to deduce from the arcsine law for stable subordinators the asymptotic behavior of the two-time correlation function that characterizes aging.

Q-Neighbor Ising model on random networks

A modified kinetic Ising model with Metropolis dynamics, so-called [Formula: see text]-neighbor Ising model, is investigated on random graphs. In this model, each spin interacts only with [Formula: see text] spins randomly chosen from its neighborhood. Investigations are performed by means of Monte Carlo (MC) simulations and the analytic pair approximation (PA). The range of parameters such as the size of the [Formula: see text]-neighborhood and the mean degree of nodes of the random graph is determined for which the model exhibits continuous or discontinuous ferromagnetic (FM) phase transition with decreasing temperature. It is also shown that, in the case of discontinuous transition for large enough and fixed mean degree of nodes, the width of the hysteresis loop oscillates with the parameter [Formula: see text], expanding for even and shrinking for odd values of [Formula: see text]. Predictions of the PA show satisfactory quantitative agreement with results of MC simulations.

Low-Temperature Behavior of the Multicomponent Widom–Rowlison Model on Finite Square Lattices

We consider the multicomponent Widom-Rowlison with Metropolis dynamics, which describes the evolution of a particle system where $M$ different types of particles interact subject to certain hard-core constraints. Focusing on the scenario where the spatial structure is modeled by finite square lattices, we study the asymptotic behavior of this interacting particle system in the low-temperature regime, analyzing the tunneling times between its $M$ maximum-occupancy configurations, and the mixing time of the corresponding Markov chain. In particular, we develop a novel combinatorial method that, exploiting geometrical properties of the Widom-Rowlinson configurations on finite square lattices, leads to the identification of the timescale at which transitions between maximum-occupancy configurations occur and shows how this depends on the chosen boundary conditions and the square lattice dimensions.

Activated aging dynamics and effective trap model description in the random energy model

We study the out-of-equilibrium aging dynamics of the random energy model (REM) ruled by a single spin-flip Metropolis dynamics. We focus on the dynamical evolution taking place on time-scales diverging with the system size. Our aim is to show to what extent the activated dynamics displayed by the REM can be described in terms of an effective trap model. We identify two time regimes: the first one corresponds to the process of escaping from a basin in the energy landscape and to the subsequent exploration of high energy configurations, whereas the second one corresponds to the evolution from a deep basin to the other. By combining numerical simulations with analytical arguments we show why the trap model description does not hold in the former but becomes exact in the second.

Tricriticality in the q-neighbor Ising model on a partially duplex clique.

We analyze a modified kinetic Ising model, a so-called q-neighbor Ising model, with Metropolis dynamics [Phys. Rev. E 92, 052105 (2015)PLEEE81539-375510.1103/PhysRevE.92.052105] on a duplex clique and a partially duplex clique. In the q-neighbor Ising model each spin interacts only with q spins randomly chosen from its whole neighborhood. In the case of a duplex clique the change of a spin is allowed only if both levels simultaneously induce this change. Due to the mean-field-like nature of the model we are able to derive the analytic form of transition probabilities and solve the corresponding master equation. The existence of the second level changes dramatically the character of the phase transition. In the case of the monoplex clique, the q-neighbor Ising model exhibits a continuous phase transition for q=3, discontinuous phase transition for q≥4, and for q=1 and q=2 the phase transition is not observed. On the other hand, in the case of the duplex clique continuous phase transitions are observed for all values of q, even for q=1 and q=2. Subsequently we introduce a partially duplex clique, parametrized by r∈[0,1], which allows us to tune the network from monoplex (r=0) to duplex (r=1). Such a generalized topology, in which a fraction r of all nodes appear on both levels, allows us to obtain the critical value of r=r^{*}(q) at which a tricriticality (switch from continuous to discontinuous phase transition) appears.

Theoretical and experimental comparative study about magnetic behavior of Fe-Al alloys in the high Fe concentration region

In this research, annealing samples of Fe100-qAlq (with q ≤ 600/32; Δq = 100/32) were obtained by arc-melting. These were heat treated at 1000 °C during a week followed by quenching in ice water. The ferromagnetic iron matrix was modified with the non-magnetic Al atom. These samples were analyzed by X-ray diffraction and Mössbauer spectroscopy at room temperature. All alloys present disordered body centered cubic (BCC) structure or A2 with mean crystallite sizes of about 34 nm. It was observed that when q = 3.125% the iron's hardness falls over 80%. Furthermore, Mössbauer spectroscopy shows that all alloys are disordered and ferromagnetic. The spectra were fitted using a hyperfine magnetic field distribution with ΔH = 1T. Statistical and hyperfine parameters of the distributions were obtained as a function of q. After fixing the Mössbauer spectra, it was found that the mean hyperfine magnetic field (MHMF) presents a linear decrease with the Al concentration. This behavior is strongly related with the diluted character of the impurity. Through Mössbauer spectroscopy and/or X-ray diffraction, an anomalous behavior was observed around q = 12.500. Moreover, by using the Monte Carlo method and the local environment model, the long-range order parameter of the samples was possible to obtain. Additionally, only 2 samples reached the perfect disorder, which indicated that the method of preparation does not guarantee the state of perfect disorder. The local environment model allowed finding the variations in the hyperfine field of the Fe when one atom of Fe is replaced by one Al atom in the first or second coordination shell of the 57Fe. Finally, the Monte Carlo method with Metropolis dynamics and the ½ Ising model were used to simulate the magnetic behavior of the alloys, used in this study. The results were compared with experimental phase diagrams of these alloys, allowing us to find the functionality of the exchange energy in the system as a function of Al concentration. For this system, the phase diagram was fitted using a cubic function of the JFeFe exchange energy with the aluminum concentration.