arXiv subjects

Lev Shchur

Publications and source records attributed to Lev Shchur.

At least 19 recordsLinked to original sources

arXiv2026

Algorithmic overlaps in the Baxter-Wu model: cluster dynamics under Novotny-Evertz updates

We study the spatial overlap of successive spin configurations generated by Markov chain Monte Carlo simulations of the Baxter-Wu model. Using the Novotny-Evertz sublattice-freezing single-cluster update, we track the mean and variance of the algorithmic overlap across the critical region. We show that, even in this three-spin model, the overlap acts as an algorithmic observable that follows the thermodynamics of the transition: the single-cluster overlap mean behaves like an order parameter, dropping from a finite ordered-phase plateau toward zero across $T_c$. The overlap does not diverge at criticality, instead it remains finite and its finite-size value decays as a clean power law, $U_2(T_c)\sim L^{-\psi}$, over eleven sizes with an exponent $\psi{=}0.378(4)$ smaller than the value $\approx0.42$ found for the Ising and Potts models under standard Fortuin-Kasteleyn cluster dynamics, indicating that it reflects the Novotny-Evertz sublattice-freezing dynamics rather than any static property of the model.

cond-mat.stat-mech↗

arXiv2026

Algorithmic overlaps as thermodynamic variables: from local to cluster Monte Carlo dynamics in critical phenomena

We investigate the spatial overlap of successive spin configurations in Markov chain Monte Carlo simulations using the local Metropolis algorithm and the Swendsen-Wang and Wolff cluster algorithms. We examine the dynamics of these algorithms for models in different universality classes: Ising model, Potts model with three components, and four-state Potts model. The overlap of two successive Wolff clusters reflects critical behavior and can be used as an order parameter for the algorithm's dynamics. In the case of the Swendsen-Wang algorithm, similar behavior is demonstrated by the variance of the overlap of two consecutive lattice configurations, which behaves like an order parameter. Nothing similar is observed for the Metropolis algorithm, where the dynamics in the critical region are determined by the spin-flip frequency, which is equivalent to the acceptance rate. Thus, the critical behavior of the Wolff cluster overlap and of the variance of the configuration overlap in the Swendsen-Wang algorithm are naturally related to the critical behavior of geometric objects -- Fortuin-Kasteleyn clusters. Interestingly, in all cases the geometric quantity -- the configuration overlap or its variance -- reflects the thermodynamics of the phase transition.

cond-mat.stat-mech↗

arXiv2025

Determining the boundary of dynamical chaos in the generalized Chirikov map via machine learning

We investigate the boundary separating regular and chaotic dynamics in the generalized Chirikov map, an extension of the standard map with a control parameter $K$ and a phase-shifted (phase $\tau$) secondary sequence of kicks with a control parameter $K_\alpha$. Lyapunov maps were computed across the parameter space $(K, K_\alpha; \tau)$ and used to train a convolutional neural network (ResNet18) for binary classification of dynamical regimes. The trained model reproduces the known critical control parameter $K_c$ for the onset of global chaos in the standard map and identifies two-dimensional boundaries $(K, K_\alpha)$ in the generalized map for varying phase shifts $\tau$. The results reveal systematic deformation of the boundary as $\tau$ increases, highlighting the sensitivity of the system to phase modulation and demonstrating the ability of machine learning to extract interpretable features of complex Hamiltonian dynamics. This framework allows precise characterization of stability boundaries in nontrivial nonlinear systems.

nlin.CD↗

arXiv2024

Machine Learning Domain Adaptation in Spin Models with Continuous Phase Transitions

The main question raised in the article is whether a neural network trained on a spin lattice model in one universality class can be used to test a model in another universality class. The quantities of interest are the critical phase transition temperature and the correlation length exponent. In other words, the question of transfer learning is how ``universal'' the trained network is and under what conditions. For this purpose, we applied a supervised learning procedure to three two-dimensional models for which critical properties are precisely known: the Ising model, the four-state Potts model, and the Baxter-Wu model. We consider two datasets: one with spins configurations and one with binding energy configurations. We find that estimates of the critical temperature agree well with the known results for both datasets, but not with the results of cross-testing using the energy datasets of the two models: the four-state Potts model and the Ising model. Estimates of the critical length exponent are less regular, and appear to be more accurate for energy datasets. A good example is the cross-testing using the energy dataset between Ising model and Baxter-Wu model in both training and testing directions.12

cond-mat.stat-mech↗

arXiv2024

Phase probabilities in first-order transitions using machine learning

We set out to explore the possibility of investigating the critical behavior of systems with first-order phase transition using deep machine learning. We propose a machine learning protocol with ternary classification of instantaneous spin configurations using known values of disordered phase energy and ordered phase energy. The trained neural network is used to predict whether a given sample belong to one or another phase of matter. This allows us to estimate for the first time the probability that configurations with a certain energy belong to the ordered phase, coexistence phase, and disordered phase. Based on these probabilities, we obtained estimates of the values of the critical energies and latent heat for the Potts model with 10 and 20 components, which undergoes a strong discontinuous transition. We also found that the probabilities may reflect geometric transitions in the coexistence phase.

cond-mat.stat-mech↗

arXiv2024

Influence of anisotropy on the study of critical behavior of spin models by machine learning methods

In this paper, we applied a deep neural network to study the issue of knowledge transferability between statistical mechanics models. The following computer experiment was conducted. A convolutional neural network was trained to solve the problem of binary classification of snapshots of the Ising model's spin configuration on a two-dimensional lattice. During testing, snapshots of the Ising model spins on a lattice with diagonal ferromagnetic and antiferromagnetic connections were fed to the input of the neural network. Estimates of the probability of samples belonging to the paramagnetic phase were obtained from the outputs of the tested network. The analysis of these probabilities allowed us to estimate the critical temperature and the critical correlation length exponent. It turned out that at weak anisotropy the neural network satisfactorily predicts the transition point and the value of the correlation length exponent. Strong anisotropy leads to a noticeable deviation of the predicted values from the precisely known ones. Qualitatively, strong anisotropy is associated with the presence of oscillations of the correlation function above the Stephenson disorder temperature and further approach to the point of the fully frustrated case.

cond-mat.dis-nn↗

arXiv2024

Comparison of the microcanonical population annealing algorithm with the Wang-Landau algorithm

The development of new algorithms for simulations in physics is as important as the development of new analytical methods. In this paper, we present a comparison of the recently developed microcanonical population annealing (MCPA) algorithm with the rather mature Wang-Landau algorithm. The comparison is performed on two cases of the Potts model that exhibit a first-order phase transition. We compare the simulation results of both methods with exactly known results, including the finite-dimensional dependence of the maximum of the specific heat capacity. We evaluate the Binder cumulant minimum, the ratio of peaks in the energy distribution at the critical temperature, the energies of the ordered and disordered phases, and interface tension. Both methods exhibit similar accuracy at selected sets of modeling parameters.

cond-mat.stat-mech↗

arXiv2024

Blume-Capel model analysis with microcanonical population annealing method

We present a modification of the Rose-Machta algorithm (Phys. Rev. E 100 (2019) 063304) and estimate the density of states for a two-dimensional Blume-Capel model, simulating $10^5$ replicas in parallel for each set of parameters. We perform a finite-size analysis of the specific heat and Binder cumulant, determine the critical temperature along the critical line, and evaluate the critical exponents. The results obtained are in good agreement with those obtained previously using various methods -- Markov Chain Monte Carlo simulation, Wang-Landau simulation, transfer matrix, and series expansion. The simulation results clearly illustrate the typical behavior of specific heat along the critical lines and through the tricritical point.

cond-mat.stat-mech↗

arXiv2023

Finite-size analysis in neural network classification of critical phenomena

We analyze the problem of supervised learning of ferromagnetic phase transitions from the statistical physics perspective. We consider two systems in two universality classes, the two-dimensional Ising model and two-dimensional Baxter-Wu model, and perform careful finite-size analysis of the results of the supervised learning of the phases of each model. We find that the variance of the neural network (NN) output function (VOF) as a function of temperature has a peak in the critical region. Qualitatively, the VOF is related to the classification rate of the NN. We find that the width of the VOF peak displays the finite-size scaling governed by the correlation length exponent, $\nu$, of the universality class of the model. We check this conclusion using several NN architectures -- a fully connected NN, a convolutional NN and several members of the ResNet family -- and discuss the accuracy of the extracted critical exponents $\nu$.

cond-mat.stat-mech↗

arXiv2021

Spectrum of chain oscillation in Poiseuille flow

We simulate solid particles moving in the two-dimensional channel with the Poiseuille flow. We found that the collective chain excitation emerges with the increasing number of particles in the chain. We measured the spectrum of the chain oscillations varying the width of the channel and found the spectrum intensity became sharper for the more significant confinement ratio. The simulations were done using the Immersed boundary and Lattice Boltzmann methods. We compare our results with the experiments of the drop movement in the quasi-two-dimensional channel and with the simulations of other groups. The paper's main result is that the combination of the velocity gradient in Poiseuille flow and the proximity of the particle to the wall can induce the collective excitations in the chain of the particles moving in the two-dimensional channel.

physics.comp-ph↗

arXiv2021

Mean-field interactions in evolutionary spatial games

We introduce a mean-field term to an evolutionary spatial game model. Namely, we consider the game of Nowak and May, based on the Prisoner's dilemma, and augment the game rules by a self-consistent mean-field term. This way, an agent operates based on local information from its neighbors and nonlocal information via the mean-field coupling. We simulate the model and construct the steady-state phase diagram, which shows significant new features due to the mean-field term: while for the game of Nowak and May, steady states are characterized by a constant mean density of cooperators, the mean-field game contains steady states with a continuous dependence of the density on the payoff parameter. Moreover, the mean-field term changes the nature of transitions from discontinuous jumps in the steady-state density to jumps in the first derivative. The main effects are observed for stationary steady states, which are parametrically close to chaotic states: the mean-field coupling drives such stationary states into spatial chaos. Our approach can be readily generalized to a broad class of spatial evolutionary games with deterministic and stochastic decision rules.

cond-mat.stat-mech↗

arXiv2020

Algorithm for the replica redistribution in the implementation of parallel annealing method on the hybrid supercomputer architecture

The parallel annealing method is one of the promising approaches for large scale simulations as potentially scalable on any parallel architecture. We present an implementation of the algorithm on the hybrid program architecture combining CUDA and MPI. The problem is to keep all general-purpose graphics processing unit devices as busy as possible redistributing replicas and to do that efficiently. We provide details of the testing on Intel Skylake/Nvidia V100 based hardware running in parallel more than two million replicas of the Ising model sample. The results are quite optimistic because the acceleration grows toward the perfect line with the growing complexity of the simulated system.

physics.comp-ph↗

arXiv2019

Acceptance rate is a thermodynamic function in local Monte Carlo algorithms

We study properties of Markov chain Monte Carlo simulations of classical spin models with local updates. We derive analytic expressions for the mean value of the acceptance rate of single-spin-flip algorithms for the one-dimensional Ising model. We find that for the Metropolis algorithm the average acceptance rate is a linear function of energy. We further provide numerical results for the energy dependence of the average acceptance rate for the 3- and 4-state Potts model, and the XY model in one and two spatial dimensions. In all cases, the acceptance rate is an almost linear function of the energy in the critical region. The variance of the acceptance rate is studied as a function of the specific heat. While the specific heat develops a singularity in the vicinity of a phase transition, the variance of the acceptance rate stays finite.

cond-mat.stat-mech↗

arXiv2019

Matrix multiplication and universal scalability of the time on the Intel Scalable processors

Matrix multiplication is one of the core operations in many areas of scientific computing. We present the results of the experiments with the matrix multiplication of the big size comparable with the big size of the onboard memory, which is 1.5 terabyte in our case. We run experiments on the computing board with two sockets and with two Intel Xeon Platinum 8164 processors, each with 26 cores and with multi-threading. The most interesting result of our study is the observation of the perfect scalability law of the matrix multiplication, and of the universality of this law.

cond-mat.stat-mech↗

arXiv2018

On the geometric structures in evolutionary games on square and triangular lattices

We study a model of a spatial evolutionary game, based on the Prisoner's dilemma for two regular arrangements of players, on a square lattice and on a triangular lattice. We analyze steady state distributions of players which evolve from irregular, random initial configurations. We find significant differences between the square and triangular lattice, and we characterize the geometric structures which emerge on the triangular lattice.

physics.soc-ph↗

arXiv2017

Dynamic fractals in spatial evolutionary games

We investigate critical properties of a spatial evolutionary game based on the Prisoner's Dilemma. Simulations demonstrate a jump in the component densities accompanied by drastic changes in average sizes of the component clusters. We argue that the cluster boundary is a random fractal. Our simulations are consistent with the fractal dimension of the boundary being equal to 2, and the cluster boundaries are hence asymptotically space filling as the system size increases.

physics.soc-ph↗

arXiv2013

The two-dimensional 4-state Potts model in a magnetic field

We present a solution of the non-linear renormalization group equations leading to the dominant and subdominant singular behaviours of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non- vanishing magnetic field. The solutions i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and ii) prove recently proposed relations among logarithmic exponents.

hep-lat↗

arXiv2010

Duality of critical interfaces in Potts model: numerical check

We report on numerical investigation of fractal properties of critical interfaces in two-dimensional Potts models. Algorithms for finding percolating interfaces of Fortuin-Kasteleyn clusters, their external perimeters and interfaces of spin clusters are presented. Fractal dimensions are measured and compared to exact theoretical predictions.

cond-mat.stat-mech↗