International Journal of Modern Physics C

"Extended Ensemble Monte Carlo" is a generic term that indicates a set of algorithms, which are now popular in a variety of fields in physics and statistical information processing. Exchange Monte Carlo (Metropolis-Coupled Chain, Parallel Tempering), Simulated Tempering (Expanded Ensemble Monte Carlo) and Multicanonical Monte Carlo (Adaptive Umbrella Sampling) are typical members of this family. Here, we give a cross-disciplinary survey of these algorithms with special emphasis on the great flexibility of the underlying idea. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Secs. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. In Sec. 6, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 7. We stress that an extension is not necessary restricted to the space of energy or temperature. Even unphysical (unrealizable) configurations can be included in the ensemble, if the resultant fast mixing of the Markov chain offsets the increasing cost of the sampling procedure. Multivariate (multicomponent) extensions are also useful in many examples. In Sec. 8, we give a survey on extended ensembles with a state space whose dimensionality is dynamically varying. In the appendix, we discuss advantages and disadvantages of three types of extended ensemble algorithms.

New explicit hybrid Numerov type methods are presented in this paper. These efficient methods are constructed using a new approach, where we do not need the use of the intermediate high accuracy interpolatory nodes, since only the Taylor expansion of the internal points is needed. The methods share sixth algebraic order at a cost of five stages per step while their phase-lag order is 14 and partly satisfy the dissipation order conditions. It has be seen that the property of phase-lag is more important than the nonempty interval in constructing numerical methods for the solution of Schrödinger equation and related problems.^1–3 Numerical results over some well known problems in physics and mechanics indicate the superiority of the new methods.

A simple spin model is studied, motivated by the dynamics of traders in a market, where expectation bubbles and crashes occur. The dynamics is governed by interactions, which are frustrated across different scales: while ferromagnetic couplings connect each spin to its local neighborhood, an additional coupling relates each spin to the global magnetization. This new coupling is allowed to be anti-ferromagnetic. The resulting frustration causes a metastable dynamics with intermittency and phases of chaotic dynamics. The model reproduces main observations of real economic markets as power-law distributed returns and clustered volatility.

Recently, Lenski et al.^1–3 have carried out several experiments on bacterial evolution. Their findings support the theory of punctuated equilibrium in biological evolution. They have further quantified the relative contributions of adaptation, chance and history to bacterial evolution. In this paper, we show that a modified M-trait Bak–Sneppen model can explain many of the experimental results in a qualitative manner.

The structural stability of carbon nanocages, fullerens and toroids, have been investigated by performing molecular-dynamics computer simulations. The systems considered are C₁₂₀ and C₂₄₀ in ball and toroidal structures. Calculations have been realized by using an empirical many-body potential energy function for carbon. It has been found that C₁₂₀ ball is very unstable, and the other structures are relatively more strong against heat treatment.

We discuss the steps to construct Dirac operators, which have arbitrary fermion offsets, gauge paths, a general structure in Dirac space and satisfy the basic symmetries (gauge symmetry, hermiticity condition, charge conjugation, hypercubic rotations and reflections) on the lattice. We give an extensive set of examples and offer help to add further structures.

The dynamics of uniaxially (along the z direction) anisotropic Heisenberg ferromagnets (in three dimensions) in the presence of a magnetic field varying sinusoidally in time (along the x-direction only) is studied by Monte Carlo method using Metropolis rate. The time averaged (over a complete cycle of the oscillating field) value of z component of the magnetization continuously vanishes at a particular transition temperature associated with a dynamic symmetry breaking of m_z-h_x loop. The temperature variation of dynamic order parameter and the dynamic specific heat for different values of the anisotropy show that the transition temperature increases as the strength of anisotropy increases.

It is argued that telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in biological, economic and social systems. Telegraph reaction diffusion (TRD) is studied in one and two spatial dimensions. Some exact and approximate results are obtained. A coupled map lattice (CML) corresponding to the spatial prisoner's dilemma game is constructed and studied in the weak diffusion limit. A formula is derived for Lyapunov exponents and it is shown that periodic solutions are dense in the weak coupling regime and that this system is structurally stable.

We present molecular dynamics simulations of the influence of point defects on the shear elastic properties of copper. We find that vacancies do not influence these properties at all, while the introduction of interstitials causes a large reduction of the elastic coefficients. The simulations establish a phase diagram of the melting temperature as a function of the density of interstitials. A crystal having no free surface undergoes bulk mechanical melting as a result of the vanishing of C′ ≡ (C₁₁ - C₁₂)/2 once the specific volume reaches a critical value, equal to the experimental volume of liquid phase. This critical volume is history independent, in the sense that it does not depend on whether is it reached by heating the crystal or by adding defects at a constant temperature. These results generalize the Born model of melting for the case where point defects are present.

We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses Stürmer's extrapolation formula and assumes that a step of integration can be taken as a step of light ladder, which limits its use to shallow energies. The second method is an improvement of pre-existing iterative schemes, designed for stronger convergence and can be used at high-energies.

We consider pattern recognition schemes that are based upon Hamiltonian dynamical system. Different oscillatory modes are used for storing and encoding patterns, and the effect of resonance is used for determining the most excited mode. We also propose a new technique for pattern orthogonalization resorting to hidden dimensions. Numerical experiments confirm high storage capacity and absence of false memories for the proposed system. Hamiltonian systems may be important as classical analogs of quantum computing systems or quantum neural networks.

A causal, stochastic model of networked computers, based on information theory and nonequilibrium dynamical systems is presented. This provides a simple explanation for recent experimental results revealing the structure of information in network transactions. The model is based on non-Poissonian stochastic variables and pseudo-periodic functions. It explains the measured patterns seen in resource variables on computers in network communities. Weakly non-Poissonian behavior can be eliminated by a conformal scaling transformation and leads to a mapping onto statistical field theory. From this, it is possible to calculate the exact profile of the spectrum of fluctuations. This work has applications to anomaly detection and time-series analysis of computer transactions.