Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0809.5036

Disorder and critical phenomena

B. R. Gadjiev

Published Sep 29, 2008• 1 citation
arXivHTMLPDF

Abstract

In the present paper we are discussing the influence of the fractal distribution of defects on the critical behavior of the system. We consider a case when the equation of motion for the order parameter contains influences of defects of random field and random temperature types at the same time and we show that it can lead to a change of distribution function from Gibbs distribution to Tsallis distribution or more general statistics of q-type. We are able to extend the Landau-Khalatnikov equation for the order parameter and represent it in the form of a differential equation of the fractional order. We deduce and we solve the corresponding renormalization group equation with the nonlinear dispersion law and we calculate the critical indices of the system.

DOI
10.48550/arxiv.0809.5036

Related papers

Distribution of Fluctuating Fractal Dimensions of Finite Critical Systems

六美 水戸部, Kousuke Yakubo

General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).

Journal of the Physical Society of Japan•Jul 10, 2009•3 citations

Renormalization Group, Entropy Optimization, and Nonextensivity at Criticality

A. Robledo

The scaling properties of pair correlations at criticality are reproduced through an equivalence between random walk distributions and order parameter correlations. The shift from Gaussian to fractal walks with self-similar clusters corresponds to the changeover from a Gaussian to a nontrivial fixed point with nonvanishing dimensional anomaly. We show that the renormalization group trajectories lead to fixed points of minimum entropy, and use the Tsallis entropy index $q$ to measure nonextensivity as behavior departs from Gaussian.

Physical Review Letters•Sep 20, 1999•34 citations

CRITICAL BEHAVIOUR OF 3D SYSTEMS WITH LONG-RANGE CORRELATED QUENCHED DEFECTS

V. V. Prudnikov, Andrei A. Fedorenko

A field-theoretic description of the critical behaviour of systems with quenched defects obeying a power law correlations ∼ |x | −a for large separations x is given. Directly for three-dimensional systems and different values of correlation parameter 2 ≤ a ≤ 3 a renormalization analysis of scaling function in the two-loop approximation is carried out, and the fixed points corresponding to stability of the various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by double ε, δ- expansion. The critical exponents in the two-loop approximation are calculated with the use of the Pade-Borel summation technique. PACS: 64.60.Ak; 64.60.-i; 64.60.FrIn last years, much effort has been devoted to investigation of the critical behaviour of solids containing quenched defects. In most papers considerations have been restricted to the case of point defects with small concentrations so that the defects and corresponding random fields have been assumed to be Gaussian distributed and δ- correlated. For the first time in work of Weinrib and Halperin (WH) [1] have been offered the model of the critical behaviour of disordered system in which the correlation function of the random local transition temperature g(x −y) =<< Tc(x)Tc(y)>> − << Tc(x)>> 2

•Jan 1, 1999•20 citations

Dependence of the renormalization group description of critical phenomena on the group parameter

E. V. Kholopov

We analyze the renormalization group description of the critical behavior of a system as a function of the possible choices of the truncation parameter of the group. The variation of the critical indices with temperature, where the specific heat exponent may even change sign, is associated with a movement of the fixed point of the renormalization group transformation itself as a function of the group parameter.

Soviet Journal of Low Temperature Physics•May 1, 1985•0 citations

Short-Time Critical Behavior of Systems with Random Impurities

Y. Chen, Shuo-Hong Guo, Zhibing Li

The theoretical renormalization-group approach is applied to the study of the short-time critical behavior of the Ginzburg-Landau model with long-range random impurities which have a power-like correlation r—(d—ρ. The system initially at a high temperature is firstly quenched to the critical temperature Tc and then released to an evolution with a model A dynamics. The asymptotic scaling laws are studied in the frame of a double expansion in ϵ = 4 — d and δ = ϵ + ρ with δ of the order of ϵ. The initial slip exponent θ′ for the order parameter is calculated up to the first order in ϵ = 4 — d or in ϵ1/2 corresponding to different fixed points, respectively. For d > 4, the short-time behavior of the order parameter is investigated in one loop. Two different logarithmic and exponential-logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are also solved in d = 4 dimension. Crossover between the nonrandom behavior and quenched behavior is found for n = 4 (where n is the spin dimensionality) in d = 4 dimension.

physica status solidi (b)•Feb 1, 2001•4 citations

Introduction to the Theory of Critical Phenomena: Mean Field, Fluctuations and Renormalization

Dimo I. Uzunov

The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics

•Jan 1, 1993•36 citations
In the present paper we are discussing the influence of the fractal distribution of defects on the critical behavior of the system. We consider a case when the equation of motion for the order parameter contains influences of defects of random field and random temperature types at the same time and we show that it can lead to a change of distribution function from Gibbs distribution to Tsallis distribution or more general statistics of q-type. We are able to extend the Landau-Khalatnikov equation for the order parameter and represent it in the form of a differential equation of the fractional order. We deduce and we solve the corresponding renormalization group equation with the nonlinear dispersion law and we calculate the critical indices of the system.
General form of the probability distribution function (PDF) of the rescaled fractal dimension characterizing spatial structures of local order parameters in finite but large systems near the critical point is derived. To obtain the PDF, we assume that (i) sample-to-sample fluctuations of global order parameters are described by the generalized Gumbel distribution function, which is known to be valid for many of long-range correlated systems such as critical systems and (ii) spatial structures of local order parameters are always fractal at the critical point. From the obtained PDF, it is elucidated that fluctuations of fractal dimensions do not exhibit a universal character in contrast to the case of fluctuating global order parameters. In addition, we perform numerical calculations of the PDF for two-dimensional percolation systems near the critical concentration and confirm that the analytically calculated PDF well describes numerical results. We also discuss the influence of non-fractal critical clusters neglected under the assumption (ii).
The scaling properties of pair correlations at criticality are reproduced through an equivalence between random walk distributions and order parameter correlations. The shift from Gaussian to fractal walks with self-similar clusters corresponds to the changeover from a Gaussian to a nontrivial fixed point with nonvanishing dimensional anomaly. We show that the renormalization group trajectories lead to fixed points of minimum entropy, and use the Tsallis entropy index qqq to measure nonextensivity as behavior departs from Gaussian.
A field-theoretic description of the critical behaviour of systems with quenched defects obeying a power law correlations ∼ |x | −a for large separations x is given. Directly for three-dimensional systems and different values of correlation parameter 2 ≤ a ≤ 3 a renormalization analysis of scaling function in the two-loop approximation is carried out, and the fixed points corresponding to stability of the various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by double ε, δ- expansion. The critical exponents in the two-loop approximation are calculated with the use of the Pade-Borel summation technique. PACS: 64.60.Ak; 64.60.-i; 64.60.FrIn last years, much effort has been devoted to investigation of the critical behaviour of solids containing quenched defects. In most papers considerations have been restricted to the case of point defects with small concentrations so that the defects and corresponding random fields have been assumed to be Gaussian distributed and δ- correlated. For the first time in work of Weinrib and Halperin (WH) [1] have been offered the model of the critical behaviour of disordered system in which the correlation function of the random local transition temperature g(x −y) =<< Tc(x)Tc(y)>> − << Tc(x)>> 2
The theoretical renormalization-group approach is applied to the study of the short-time critical behavior of the Ginzburg-Landau model with long-range random impurities which have a power-like correlation r—(d—ρ. The system initially at a high temperature is firstly quenched to the critical temperature Tc and then released to an evolution with a model A dynamics. The asymptotic scaling laws are studied in the frame of a double expansion in ϵ = 4 — d and δ = ϵ + ρ with δ of the order of ϵ. The initial slip exponent θ′ for the order parameter is calculated up to the first order in ϵ = 4 — d or in ϵ1/2 corresponding to different fixed points, respectively. For d > 4, the short-time behavior of the order parameter is investigated in one loop. Two different logarithmic and exponential-logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are also solved in d = 4 dimension. Crossover between the nonrandom behavior and quenched behavior is found for n = 4 (where n is the spin dimensionality) in d = 4 dimension.
The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics