Martin Gmitra

A self-adjusted Monte Carlo simulation as a model for financial markets with central regulation

Properties of the self-adjusted Monte Carlo algorithm applied to 2d Ising ferromagnet are studied numerically. The endogenous feedback form expressed in terms of the instant running averages is suggested in order to generate a biased random walk of the temperature that converges to criticality without an external tuning. The robustness of a stationary regime with respect to partial accessibility of the information is demonstrated. Several statistical and scaling aspects have been identified which allow to establish an alternative spin lattice model of the financial market. It turns out that our model alike model suggested by Bornholdt [Int. J. Mod. Phys. C 12 (2001) 667], may be described by Levy-type stationary distribution of feedback variations with unique exponent α1∼3.3. However, the differences reflected by Hurst exponents suggest that resemblances between the studied models seem to be non-trivial.

Neural network approach to magnetic dot arrays modeling

Two dimensional square lattice general model of the magnetic dot array is introduced. In this model the intradot self-energy is predicted via the neural network and interdot magnetostatic coupling is approximated by the collection of several dipolar terms. The model has been applied to disk-shaped cluster involving 193 ultrathin dots and 772 interaction centers. In this case among the intradot magnetic structures retrieved by neural networks the important role play single-vortex magnetization modes. Several aspects of the model have been understood numerically by means of the simulated annealing method.Abstract Two-dimensional square lattice model of the magnetic dot array is presented. The intradot self-energy is predicted using the neural network approachand interdot magnetostatic coupling is approximated by the several dipolar terms. The model is applied to disk-shaped cluster involving 193 ultrathin square dots with 772 interaction centers. Among the intradot magnetic structures retrieved by neural networks the important role play single-vortex intradot magnetostatic modes. Several aspects of the model have been understood by the numerical study using the simulated annealing method.

THE SELF-ORGANIZED MULTI-LATTICE MONTE CARLO SIMULATION

Self-organized Monte Carlo simulations of 2D Ising ferromagnet on the square lattice are performed. The essence of the suggested simulation method is an artificial dynamics consisting of the well-known single-spin-flip Metropolis algorithm supplemented by a random walk on the temperature axis. The walk is biased towards the critical region through a feedback based on instantaneous energy and magnetization cumulants, which are updated at every Monte Carlo step and filtered through a special recursion algorithm. The simulations revealed the invariance of the temperature probability distribution function, once some self-organized critical steady regime is reached, which is called here noncanonical equilibrium. The mean value of this distribution approximates the pseudocritical temperature of canonical equilibrium. In order to suppress finite-size effects, the self-organized approach is extended to multi-lattice systems, where the feedback basis on pairs of instantaneous estimates of the fourth-order magnetization cumulant on two systems of different size. These replica-based simulations resemble, in Monte Carlo lattice systems, some of the invariant statistical distributions of standard self-organized critical systems.

The neural network two-scale model of the magnetic dot array

Abstract We have constructed and investigated numerically a two-scale model of the truncated periodic planar array of the square dots on the square lattice, where large-interdot and small-intradot scales are described by particular models. The large-scale degrees of freedom are block spins interacting via dipolar interactions, whereas the small scales are described by the effective spin lattice Hamiltonian. The adaptive coupling of models belonging to different scales is mediated by interface model. Its main part is the Kohonen neural network. The low-energy states are investigated (for assemblies of 3×3 and 12×12 dots) using hybrid algorithm combining the energy minimization and neural network predictions. The effectiveness of simulation and variety of intradot configurations were enhanced by considering the point group symmetry aspect related to dot shape. The simulation for a given set of parameters recovers the state formed by “S” and “C” intradot configurations arranged into antiferromagnetically ordered array chains.

The magnetic reversal in dot arrays recognized by the self-organized adaptive neural network

Abstract The remagnetization dynamics of monolayer dot array superlattice XY 2-D spin model with dipole–dipole interactions is simulated. Within the proposed model of array, the square dots are described by the spatially modulated exchange-couplings. The dipole–dipole interactions are approximated by the hierarchical sums and spin dynamics is considered in regime of the Landau–Lifshitz equation. The simulation of reversal for 40 000 spins exhibits formation of nonuniform intra-dot configurations with nonlinear wave/anti-wave pairs developed at intra-dot and inter-dot scales. Several geometric and parametric dependences are calculated and compared with oversimplified four-spin model of reversal. The role of initial conditions and the occurrence of coherent rotation mode is also investigated. The emphasis is on the classification of intra-dot or inter-dot (interfacial) magnetic configurations done by adaptive neural network with varying number of neurons.

The dynamical response to the node defect in thermally activated remagnetization of magnetic dot array

Abstract The influence of nonmagnetic central node defect on dynamical properties of regular square-shaped segment of magnetic dot array under the thermal activation is investigated via computer simulations. Using stochastic Landau–Lifshitz–Gilbert equation we simulate hysteresis and relaxation processes. The remarkable quantitative and qualitative differences between magnetic dot arrays with nonmagnetic central node defect and magnetic dot arrays without defects have been found.

Properties Of Iterative Monte Carlo Single Histogram Reweighting

We present an iterative Monte Carlo algorithm for which the temperature variable is attracted by a critical point. The algorithm combines techniques of single histogram reweighting and linear filtering. The ferromagnetic 2D Ising model is studied numerically as an illustration. In that case, the iterations reach a stationary regime with an invariant probability distribution function of temperature which peaked near the pseudocritical temperature of the specific heat. The sequence of generated temperatures is analyzed in terms of stochastic autoregressive model. The error of histogram reweighting can be better understood within the suggested model. The presented model yields a simple relation, connecting the variance of pseudocritical temperature and the parameter of linear filtering.

The dot-vacancy contribution to two-fold anisotropy of magnetic dot array

A study of the impact of the magnetic dot-vacancy and biasing field on the basic magnetic properties of magnetic dot array is reported. Magnetic bias is a constant magnetic field applied simultaneously with the remagnetization field, which is oriented perpendicularly to the remagnetization field. Simulations, which are presented, deal with the magnetic dot arrays in rotating magnetic field and in-line remagnetization field with the dot-vacancies in various positions. The results led us to conclude, that amplitude of two-fold magnetization contribution increases with asymmetry of the position of dot-vacancy regarding to the center of array. The non-zero bias and the presence of magnetic defect are resulting in asymmetric behavior of magnetization response. For characterization of asymmetrical hysteresis loops was defined the specific coefficient of asymmetry. This quantitative characterization of asymmetry allows to determine the range of relevant biases leading to external two-fold contributions to magnetic anisotropy of array.

The Evolutionary Approach to the Optimization of Finite-Size Effects in Magnetic Dot Arrays

We considered simple 2D model of the magnetic dot array with uniaxial anisotropic dots. For this model we formulated the inverse magnetic problem, where desired static ground-state properties are attained by varying directions of the local anisotropy vectors. The problem is investigated numerically using a method based on evolutionary optimization.