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1
Content available remote Strong stationary duality for Möbius monotone Markov chains : examples
EN
We construct strong stationary dual chains for nonsymmetric random walks on square lattice, for random walks on hypercube and for some Ising models on the circle. The strong stationary dual chains are all sharp and have the same state space as original chains.We use Möbius monotonicity of these chains with respect to some natural orderings of the corresponding state spaces. This method provides an alternative way to study mixing times for studied models.
2
Content available remote Monte Carlo Simulations of the Ising Model on GPU
EN
Monte Carlo simulations of two- and three-dimensional Ising model on graphic cards (GPU) are described. The standard Metropolis algorithm has been employed. In the framework of the implementation developed by us, simulations were up to 100 times faster than their sequential CPU analogons. It is possible to perform simulations for systems containing up to 109 spins on Tesla C2050 GPU. As a physical application, higher cumulants for the 3d Ising model have been calculated.
PL
W artykule przedstawiono sposób wykorzystania modelu Isinga w kontekście zastosowania go do usuwania szumów z dokumentów mających ubytki spowodowane szkodami fizycznymi. Proponowana metoda opiera się na maksymalizowaniu prawdopodobieństwa rozkładu łącznego za pomocą algorytmu iteracyjnej zmiany mody warunkowej. Zastosowane rozszerzenie algorytmu ICM pozwala na zwiększenie efektywności przetwarzania dokumentów. Wyniki badań empirycznych potwierdzają przydatność algorytmu w zastosowaniach.
EN
This article discusses how to use the Ising model in the context of noise reduction being applied to the documents of losses caused physical harm. The proposed method is based on energy function minimizing using optimized Iterated Conditional Modes. Applied extension ICM algorithm allows to increase the efficiency of document processing. Empirical studies confirm the usefulness of the algorithm in practice.
EN
The Slit Island Method (SIM) is a technique for the estimation of the fractal dimension of an object by determining the area– perimeter relations for successive slits. The SIM could be applied for image analysis of irregular grayscale objects and their classification using the fractal dimension. It is known that this technique is not functional in some cases. It is emphasized in this paper that for specific objects a negative or an infinite fractal dimension could be obtained. The transformation of the input image data from unipolar to bipolar gives a possibility of reformulated image analysis using the Ising model context. The polynomial approximation of the obtained area-perimeter curve allows object classification. The proposed technique is applied to the images of cervical cell nuclei (Papanicolaou smears) for the preclassification of the correct and atypical cells.
EN
The paper shows Monte Carlo simulations of the Ising model on a square lattice with no external magnetic field. In particular, the uncertainty of the spin coupling interactions in the Ising model has been considered. The influence on the phase transition of the Gaussian noise in the spin coupling values has been demonstrated.
PL
W artykule przedstawiono symulacje Monte Carlo modelu Isinga na sieci kwadratowej przy braku zewnętrznego pola magnetycznego. W szczególności rozważono niepewność wartości energii sprzężenia oddziałujących spinów w modelu Isinga. Zademonstrowano wpływ na przejście fazowe obecności szumu gaussowskiego w wartościach stałej sprzężenia oddziałujących spinów.
6
Content available remote On Spins and Genes
EN
Many processes in natural and social sciences can be modeled by systems of interacting objects. It is usually very difficult to obtain analytic expressions describing time evolution and equilibrium behavior of such systems. Very often we rely only on computer simulations. Fortunately, in many cases one can construct useful approximation schemes and derive exact results which capture some specific features of a given process. A frequent approach is to replace interactions between objects by a mean interaction. Here we illustrate a self-consistent mean-field approximation in two examples: the Ising model of interacting spins and a simple model of a self-regulating gene.
PL
Naszym celem jest zrozumienie i przewidywanie zachowania się układów wielu oddziałujących obiektów, takich jak cząstki i spiny w fizyce statystycznej czy geny i białka w biologii molekularnej. Jako matematycy pragniemy udowadniać twierdzenia i wyprowadzać analityczne wzory. Bardzo szybko okazuje się, ze w istotnych zastosowaniach jest to niemożliwe. Co robić? Część z nas ucieka w wyrafinowane symulacje komputerowe. Czy nie ma innej drogi? Czy jesteśmy ograniczeni do wyboru pomiędzy Matematyka i Mathematica? Na pomoc przychodzi metoda samouzgodnionego pola średniego. Ferromagnetyczny model Isinga i samoregulujący się gen zilustrują nam te niezwykle uniwersalna metodę otrzymywania przybliżonych rozwiązań analitycznych
7
Content available remote Statistical Description of Magnetic Domains in the Two Dimensional Ising Model
EN
The Zipf-Mandelbrot power law and its connection with the inhomogeneity of the system has been used. We describe the statistical distributions of the domain masses in the Ising model near the phase transition induced by the temperature. For the large domain masses we observe the characteristic irregularities. The statistical distribution near the critical point appears to be of the Pareto type.
8
Content available remote Limit theorems for stochastic dynamical system arising in ising model analysis
EN
A simple stochastic dynamical system defined on the space of doubly-infinite sequences of real numbers is considered. Limit theorems for this system are proved. The results are applied to the physical model of wetting of the flat heterogeneous wall.
EN
Since few years the Belief Propagation [13, 14, 15] algorithm is reported as a very efficient tool to perform the optimization of systems which can be topologically transformed to the one of acceptable equivalent forms [9, 7]. The Ising system is often mentioned in these papers as a good example to present some basic foundations of BP. It is however rarely used as a tool to solve the Ising system itself. In this article we are going to present the analysis of critical properties, connected to the phase transition of magnetic system described by the Ising hamiltonian and the comparison of results to those obtained using evolutionary algorithm.
10
Content available remote Bulk critical exponents in a presence of the capillary condensation
EN
The properties of a simple fluid or magnet with strong one-axis anisotropy can be studied by means of the Ising model. Such a model, but in an confined geometry with identical boundary fields, is studied along various thermodynamic paths using the density-matrix renormalization group technique. It has been found that also in the presence of the capillary condensation the critical exponent β of the bulk system can be found.
11
Content available remote The wetting layers for long-range wall-particle potentials
EN
The complete wetting in 2D Ising strips subject to identical surface fields decaying as h1z-p is studied by means of the density-matrix renormalization-group technique. Using different criteria the thickness of a layer is determined along some isotherms above the wetting temperature. It is found that magnetization profiles are characterized by wide interfacial regime.
EN
The properties of a simple fluid, or Ising magnet, confined in an Lx ∞ geometry, are studied by means of numerical density-matrix renormalization-group techniques. We have proposed and have verified a few criterions to determine the wetting transition phase boundary with different ranges of surface forces.
13
Content available remote Curie temperatures for the Ising model on Archimedean lattices
EN
Critical temperatures for the ferro-paramagnetic transition in the Ising model are evaluated for five Archimedean lattices, basing on Monte Carlo simulations. The obtained Curie temperatures are 1.25, 1.40, 1.45, 2.15 and 2.80 [J/kB] for (3,12 do2), (4,6,12), (4,8 do 2), (3,4,6,4) and (3 do 4,6) lattices, respectively.
14
Content available remote Finite-size analysis via the critical energy-subspace method in the Ising models
EN
We briefly review some applications of the critical minimum energy-subspace method (CrMES) using the Wang?Landau sampling for the estimation of the density of states (DOS). These applications concern the two- and three-dimensional Ising models and their important conserved order parameter versions (COP), known also as the Ising models with fixed magnetization (IMFM). The recently developed CrMES scheme greatly facilitates methods for sampling the DOS of classical statistical models in large systems. In effect, the CrMES technique enables the estimation of critical behaviour using only a small part of the energy space. Specific heat curves are obtained, their peaks are located and their scaling behaviour is studied and compared with the results known from literature whenever such results exist.
15
Content available remote Corrections to the Kelvin equation for long-range wall-particle potentials
EN
The properties of a simple fluid, or Ising magnet, confined in an L× ∞ geometry, are studied by means of numerical density-matrix renormalization-group techniques. Whereas the particle-particle potential is short ranged, the wall-particle potential is long ranged decaying as h1/lp for various values of p-integer, where l = 1,2,…,L labels the columns across the strip and h1 is the reduced amplitude of the boundary field. For the shortrange wall-particle potential, according to the Kelvin equation, the bulk coexistence field scales as 1/L for large L; thermodynamics and scaling arguments predict higher-order corrections of the 1/L2 and 1/L5/3 types at partial and complete wetting, respectively. However, at complete wetting for a large range of surface fields and temperatures a correction to scaling of type 1/L4/3 has been found recently. We discuss the influence of long-range wall-fluid potentials on the scaling. Results are obtained for several values of h1 for strips of widths up to L = 690.
EN
Recent developments in the statistical theory of simple fluids in a film geometry near bulk criticality is reviewed.We summarize results obtained by exact or approximate, but very accurate methods within Ising model. Particular attention is paid to the properties of the measurable solvation force and its relation to the structure of the confined system, in the Ising model given by magnetization profiles. Relevance of the reviewed results for various physical systems is briefly discussed.
EN
The results of grand canonical Monte Carlo study of two-dimensional lattice gas model of network-forming particles on a triangular lattice are reported. The model takes into account the effects of molecular association, resulting from the orientation-dependent interactions as well as the effects of cooperative interactions, which lead to the weakening of the bond energies. A phase transition between the dilute and the condensed phase is considered. Phase diagrams for different systems are presented and it is shown that the systems studied belong to the universality class of two-dimensional Ising model.
18
Content available remote Study of continuous phase transition with Toom cellular automata
EN
The heuristic proof, based on computer simulation investigations, is presented that though stationary Toom cellular automata exhibit many features which are characteristic for an equilibrium system (e.g. rapid change in the order parameter, when noise is fine tuned, or slow decay of the two point correlation function), the stationary state is not Gibbsian. It means that it is impossible to define energy on the microscopic level in such a way that the dynamic system becomes representative of some equilibrium lattice model. Moreover, properties on the coarse-grained level: fluctuations, seem to be distinct from the corresponding ones of the Ising model.
19
Content available remote Computer simulations of one-dimensional coupled map lattices
EN
We perform computer simulations of some one-dimensional models of coupled map lattices (CML) with symmetry and diffusive nearest neighbour coupling, to study Ising-type transitions. Such transitions appear to be related to the presence of a dip (minimum) in the plot of the Lyapunov dimension versus coupling parameter.
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