A MEAN FIELD APPROACH FOR ISING MODELS ON SCALE-FREE NETWORKS

2011 ◽  
Vol 25 (07) ◽  
pp. 453-464 ◽  
Author(s):  
G. IANNONE ◽  
ORLANDO LUONGO
Keyword(s):  
Ising Model ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Practical Applications ◽  
Scale Free ◽  
Wide Range ◽  
Scale Free Networks ◽  
The Mean ◽  
Mean Field Approximations

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.

scholarly journals The Combined Effect of Rotation and Magnetic Field on Finite-Amplitude Thermal Convection

1973 ◽  
Vol 26 (5) ◽  
pp. 617 ◽  
Author(s):  
R Van der Borght ◽  
JO Murphy
Keyword(s):  
Magnetic Field ◽  
Mean Field ◽  
Field Approximation ◽  
Finite Amplitude ◽  
Combined Effect ◽  
Mean Field Approximation ◽  
Free Boundaries ◽  
Wide Range ◽  
The Mean ◽  
Basic Equations

The combined effect of an imposed rotation and magnetic field on convective transfer in a horizontal Boussinesq layer of fluid heated from below is studied in the mean field approximation. The basic equations are derived by a variational technique and their solutions are then found over a wide range of conditions, in the case of free boundaries, by numerical and analytic techniques, in particular by asymptotic and perturbation methods. The results obtained by the different techniques are shown to be in excellent agreement. As for the linear theory, the calculations predict that the simultaneous presence' of a magnetic field and rotation may produce conflicting tendencies.

Processus de branchement en champ moyen et réaction PCR

2001 ◽  
Vol 33 (2) ◽  
pp. 391-403 ◽  
Author(s):  
Didier Piau
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Process Mean ◽  
Dna Mutations ◽  
Wide Range ◽  
The Mean ◽  
Efficiency Rate

Sun and Waterman model DNA mutations during the PCR reaction by a non-canonical branching process. Mean-field approximated values fit the simulated values surprisingly well. We prove this as a theoretical result, for a wide range of the parameters. Thus, we bound explicitly the biases, in law and in the mean, that the mean-field approximation induces in the random number of mutations of a DNA molecule, as a function of the initial number of molecules, of the number of PCR cycles, of the efficiency rate and of the mutation rate. The range where we prove that the approximation is good contains the observed mutation rates in many actual PCR reactions.

Phase Transitions and the Ising Model

2019 ◽  
pp. 423-446
Author(s):  
Robert H. Swendsen
Keyword(s):  
Phase Transitions ◽  
Ising Model ◽  
Landau Theory ◽  
Magnetic Moments ◽  
Mean Field ◽  
Higher Dimensions ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Ising Chain ◽  
The Mean

Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution for the Ising chain (one-dimensional model), including the introduction of the transfer matrix method. Higher dimensions are treated in the Mean Field Approximation (MFA), which is also extended to Landau theory. The Ising model is deceptively simple. It can be defined in a few words, but it displays astonishingly rich behavior. It originated as a model of ferromagnetism in which the magnetic moments were localized on lattice sites and had only two allowed values.

The Glauber and Metropolis Transition Rates on the Stationary States of the Ising Model

1997 ◽  
Vol 11 (13) ◽  
pp. 565-570
Author(s):  
G. L. S. Paula ◽  
W. Figueiredo
Keyword(s):  
Ising Model ◽  
Nearest Neighbor ◽  
Mean Field ◽  
Field Approximation ◽  
Two Dimensions ◽  
Mean Field Approximation ◽  
Stationary States ◽  
Transition Rates ◽  
The Mean ◽  
The One

We have applied the Glauber and Metropolis prescriptions to investigate the stationary states of the Ising model in one and two dimensions. We have employed the formalism of the master equation to follow the evolution of the system towards the stationary states. Although the Glauber and Metropolis transition rates lead the system to the same equilibrium states for the Ising model in the Monte Carlo simulations, we show that they can predict different results if we disregard the correlations between spins. The critical temperature of the one-dimensional Ising model cannot even be found by using the Metropolis algorithm and the mean field approximation. However, taking into account only correlations between nearest neighbor spins, the resulting stationary states become identical for both Glauber and Metropolis transition rates.

scholarly journals On Magnetic Inhibition of Thermal Convection

1972 ◽  
Vol 25 (6) ◽  
pp. 703 ◽  
Author(s):  
R Van der Borght ◽  
JO Murphy ◽  
EA Spiegel
Keyword(s):  
Magnetic Field ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Free Boundaries ◽  
Quantitative Estimates ◽  
Wide Range ◽  
The Face ◽  
The Mean ◽  
The Magnetic Field

The effect of an imposed vertical magnetic field on convective transfer in a horizontal Boussinesq layer of fluid heated from below is studied in the mean field approximation. Solutions are found over a wide range of conditions, for free boundaries, by a combination of numerical and analytic techniques. Quantitative estimates are made of the significant modifications to the heat transfer which are brought about by the presence of the magnetic field. It is found that the general properties of nonlinear steady cellular convection seem to persist in the face of magnetic inhibition.

Self-consistent approximations for field theories at finite temperature

1993 ◽  
Vol 71 (5-6) ◽  
pp. 285-294
Author(s):  
M. H. Thoma
Keyword(s):  
Finite Temperature ◽  
Mean Field ◽  
Field Approximation ◽  
Two Dimensions ◽  
Mean Field Approximation ◽  
Coupling Constant ◽  
Starting Point ◽  
Consistent Approximations ◽  
The Mean ◽  
Mean Field Approximations

Various mean field approximations at finite temperature are used for calculating ground state energies and propagators of the [Formula: see text] theory in two dimensions and quantum chromodynamics (QCD). In the case of the [Formula: see text] theory a symmetry restoration is observed above a critical coupling constant if a temperature independent renormalization is used. In the case of QCD the mean field approximation is insufficient but can be regarded as a starting point for more complicated approximations, which are discussed qualitatively.

The Curie Temperature of the Ferroelectric Superlattice Films with Surface Modification

2009 ◽  
Vol 64 (11) ◽  
pp. 723-728
Author(s):  
Bao-Bing Zheng ◽  
Xiao-Yu Kuang ◽  
Shao-Mei Chang ◽  
Ya-Ru Zhao ◽  
Wen-Qiang Li
Keyword(s):  
Surface Modification ◽  
Ising Model ◽  
Mean Field ◽  
Transverse Field ◽  
Exchange Interactions ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Model Parameters ◽  
Transverse Ising Model ◽  
The Mean

We examine the critical behaviour of a finite alternating ferroelectric superlattice based on the transverse Ising model within the framework of the mean-field approximation. The results indicate that the features of the phase diagrams can be greatly modified by changing the transverse Ising model parameters. The transition temperature of alternating superlattice is described as function of the inter- and intra-layer exchange interactions, the strength of the transverse field, the superlattice thickness and the polarizations. In addition, the effects of surface modification on finite superlattices are also studied.

Refining Mean-field Approximations by Dynamic State Truncation

2021 ◽  
Vol 5 (2) ◽  
pp. 1-30
Author(s):  
Francesca Randone ◽  
Luca Bortolussi ◽  
Mirco Tribastone
Keyword(s):  
State Space ◽  
Mean Field ◽  
Field Approximation ◽  
The State ◽  
Mean Field Approximation ◽  
Dynamic State ◽  
Field Equations ◽  
Mean Field Equations ◽  
The Mean ◽  
Mean Field Approximations

Mean-field models are an established method to analyze large stochastic systems with N interacting objects by means of simple deterministic equations that are asymptotically correct when N tends to infinity. For finite N, mean-field equations provide an approximation whose accuracy is model- and parameter-dependent. Recent research has focused on refining the approximation by computing suitable quantities associated with expansions of order $1/N$ and $1/N^2$ to the mean-field equation. In this paper we present a new method for refining mean-field approximations. It couples the master equation governing the evolution of the probability distribution of a truncation of the original state space with a mean-field approximation of a time-inhomogeneous population process that dynamically shifts the truncation across the whole state space. We provide a result of asymptotic correctness in the limit when the truncation covers the state space; for finite truncations, the equations give a correction of the mean-field approximation. We apply our method to examples from the literature to show that, even with modest truncations, it is effective in models that cannot be refined using existing techniques due to non-differentiable drifts, and that it can outperform the state of the art in challenging models that cause instability due orbit cycles in their mean-field equations.

ANATOMY OF RELATIVISTIC MEAN-FIELD APPROXIMATIONS

1992 ◽  
Vol 07 (21) ◽  
pp. 1915-1921
Author(s):  
S. CRUZ BARRIOS ◽  
M.C. NEMES
Keyword(s):  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
Many Body ◽  
Relativistic Mean Field ◽  
The Mean ◽  
Body Techniques ◽  
Set Up ◽  
Mean Field Approximations

In the present work we have set up a scheme to treat field theoretical Lagrangians in the same bases of the well known non-relativistic many-body techniques. We show here that fermions and bosons can be treated quantum mechanically in a symmetric way and obtain results for the mean field approximation.

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