Narrow Results

A new and efficient algorithm is presented for the calculation of the partition function in the S=±1 Ising model. As an example, we use the algorithm to obtain the thermal-dependence of the magnetic spin susceptibility of an Ising antiferromagnet for a 8×8 square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and the experimental data, and they are better than the mean field approach results. For the 8×8 lattice, the algorithm reduces the computation time by nine orders of magnitude.

Asymmetric exclusion processes (ASEP) on lattices with a junction, in which two or more parallel lattice branches combine into a single one, is important as a model for complex transport phenomena. This paper investigates the effect of unequal injection rates in ASEP with a junction. It is a generalization of the work of Pronina and Kolomeisky [J. Stat. Mech. P07010 (2005)], in which only equal injection rates are considered. It is shown that the unequal rates give rise to new phases and the phase diagram structure is qualitatively changed. The phase diagram and the density profiles are investigated by using Monte Carlo simulations, mean field approximation and domain wall approach. The analytical results are in good agreement with Monte Carlo simulations.

We investigate the inactive–active phase transition in an array of additive (exclusive-or) cellular automata (CA) under noise. The model is closely related with the Domany-Kinzel (DK) probabilistic cellular automaton (PCA), for which there are rigorous as well as numerical estimates on the transition probabilities. Here, we characterize the critical behavior of the noisy additive cellular automaton by mean field analysis and finite-size scaling and show that its phase transition belongs to the directed percolation universality class of critical behavior. As a by-product of our analysis, we argue that the critical behavior of the noisy elementary CA 90 and 102 (in Wolfram’s enumeration scheme) must be the same. We also perform an empirical investigation of the mean field equations to assess their quality and find that away from the critical point (but not necessarily very far away) the mean field approximations provide a reasonably good description of the dynamics of the PCA.

We consider the grand canonical ensemble of the static and extremal black holes, when the equivalence of the electric charge and mass of individual black hole is postulated. Assuming uniform distribution of black holes in space, we are finding the effective mass of test particle and mean time dilation at the admissible points of space, taking into account the gravitational action of surrounding black holes. Having specified the statistics that governs extremal black holes, we study its effect on those quantities. Here, the role of statistics is to assign a statistical weight to the configurations of certain fixed number of black holes. We borrow these weights from Bose–Einstein, Fermi–Dirac, classical and infinite statistics. Using mean field approximation, the aforementioned characteristics are calculated and visualized, which permits us to draw the conclusions on visible effect of each statistics.

Inspired from the connection between Lie symmetries and two-dimensional materials, we propose a new statistical lattice model based on a double hexagonal structure appearing in the G₂ symmetry. We first construct an Ising-1/2 model, with spin values σ = ±1, exhibiting such a symmetry. The corresponding ground state shows the ferromagnetic, the antiferromagnetic, the partial ferrimagnetic and the topological ferrimagnetic phases depending on the exchange couplings. Then, we examine the phase diagrams and the magnetization using the mean field approximation (MFA). Among others, it has been suggested that the present model could be localized between systems involving the triangular and the single hexagonal lattice geometries.

In this paper, the Monte Carlo simulation (MCS) and the mean field approximation (MFA) methods have been used to study the magnetic properties of double perovskite Ba₂CrMoO₆ in the framework of the Ising model, which is important for a better understanding of the magnetic behavior of this material. This compound is constituted of two magnetic cubic sublattices: one occupied by Mo${}^{5+}$ and other occupied Cr${}^{3+}$, the critical exponents and phase diagrams for this compound are determined. The critical behavior, compensation temperature and susceptibility are also determined.

Recently, the study of complex networks led to the analysis of the so-called scale-free models in statistical mechanics. This study has increased its importance, thanks to the wide range of applications in numerous physical contexts; for example, one important question is to understand the behavior of various models on such networks. We start first by investigating the Ising model in the mean field approximation and on scale-free networks, studying especially the Ising model with annealed dilution and Clock model, with particular attention devoted to focusing on similarities between the mean field approximations with or without scale-free statistics. A particular emphasis is given to the possible practical applications of these results in other disciplines such as medicine and social science.

In this paper, we investigate particle traffic on an m-input n-output (MINO) junction using totally asymmetric exclusion processes (TASEPs) under random sequential update. The model is suitable for description of biological transport. A general theoretical solution for traffic dynamics of TASEPs is developed based on a mean-field approximation. It is found that the low-density and high-density regions can be calculated qualitatively and quantitatively once the number of m and/or n is determined. The phase diagram, system current and density profiles are obtained through theoretical analysis and supported by Monte Carlo simulations. Comparison between a m-input n-output TASEP junction in random and parallel sequential updates has also been reported.

Over the past few decades, the study of epidemic propagation has caught widespread attention from many areas. The field of graphs contains a wide body of research, yet only a few studies explore epidemic propagation’s dynamics in “signed” networks. Motivated by this problem, in this paper we propose a new epidemic propagation model for signed networks, denoted as S-SIS. To explain our analysis, we utilized the mean field theory to demonstrate the theoretical results. When we compare epidemic propagation through negative links to those only having positive links, we find that a higher proportion of infected nodes actually spreads at a relatively small infection rate. It is also found that when the infection rate is higher than a certain value, the overall spreading in a signed network begins showing signs of suppression. Finally, in order to verify our findings, we apply the S-SIS model on Erdös–Rényi random network and scale-free network, and the simulation results is well consist with the theoretical analysis.

We prove that, for a smooth two-body potential, the quantum mean-field approximation to the nonlinear Schrödinger equation of the Hartree type is stable at the classical limit h → 0, yielding the classical Vlasov equation.