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

Physical Review Letters

Critical Ising Spin Dynamics near the Percolation Threshold

R. Rammal, A. D. Benoit

Published Aug 12, 1985• 49 citations
Publisher

Abstract

A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of $s$ spins is shown to scale as $\mathrm{ln}s$. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.

DOI
10.1103/physrevlett.55.649

Related papers

Critical Ising Spin Dynamics on Percolation Clusters

Christopher L. Henley

Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with ${T}_{c}=0$, and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size $L$ is proportional to $Z\mathrm{ln}L$, where $Z$ is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.

Physical Review Letters•May 6, 1985•109 citations

Breakdown of dynamic scaling at the percolation threshold

S. Jain

The critical dynamics of the two-dimensional Ising model at the bond percolation threshold is investigated by Monte Carlo simulations on a 64*64 lattice. Conventional dynamic scaling breaks down at low temperatures: the logarithm of the relaxation time depends quadratically upon the logarithm of the thermal correlation length. The coefficients of the quadratic and linear terms are 0.51 and 3.25, respectively. The results are compared with recent experimental and analytic work.

Journal of Physics A Mathematical and General•Feb 1, 1986•23 citations

Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

Vladimir V. Prudnikov, O. N. Markov

The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.

Journal of Physics A Mathematical and General•Mar 23, 1995•5 citations

Theory for the dynamics of clusters near the critical point. I. Relaxation of the Glauber kinetic Ising model

Kurt Binder, Dietrich Stauffer, Heiner Muller-Krumbhaar

A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing $l$ spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of $r$, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.

Physical review. B, Solid state•Dec 1, 1975•98 citations

Critical dynamics below the percolation threshold

S. Jain, Eduardo Lage

The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.

Journal of Physics A Mathematical and General•Dec 7, 1993•2 citations

Dimensionality dependence in the singular dynamic scaling in the dilute Ising model

Bharat B. Biswal, Debashish Chowdhury

The relaxation time \ensuremath{\tau} and the thermal correlation length ${\ensuremath{\xi}}_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling ln\ensuremath{\tau}=f(ln${\ensuremath{\xi}}_{\mathit{T}}$), where f(x)=${\mathit{Ax}}^{2}$+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.

Physical Review A•Apr 1, 1991•7 citations
A low-temperature approach for the critical Ising spin dynamics on percolation clusters is proposed. The average energy barrier to overturning a cluster of sss spins is shown to scale as lns\mathrm{ln}slns. This basic result is supported by exact numerical calculations of the energy-barrier distribution for two-and three-dimensional percolation clusters, and is argued to be a consequence of scale invariance. The resulting temperature as well as size dependence of the relaxation-time scales agrees very well with the results of our Monte Carlo simulations.
Ferromagnetically interacting Ising spins are placed on a fractal network (such as a critical percolation cluster) with Tc=0{T}_{c}=0Tc​=0, and endowed with a single-spin-flip dynamics. At low temperatures the collective dynamics is determined by thermal activation over energy barriers. The barrier to overturning of the spins in a domain of size LLL is proportional to ZlnLZ\mathrm{ln}LZlnL, where ZZZ is a new geometrical parameter characterizing the fractal. A new "singular" dynamic scaling is found in which the effective dynamic critical exponent diverges at criticality.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing lll spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of rrr, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.
The critical dynamics of the diluted two-dimensional Ising model below the percolation threshold is studied. The dynamics exhibits a crossover from a concentration-dependent behaviour for xi T >> xi p to a percolating lattice type behaviour for xi T << xi p , where xi T and xi p are the thermal and percolation correlation lengths, respectively. As a consequence, the dynamical exponents A and B are shown to be the limits of dynamical functions A(x) and B(x), respectively; here, x=ln xi p /ln xi T . Extensive Monte Carlo simulations support our predictions.
The relaxation time \ensuremath{\tau} and the thermal correlation length ${\ensuremath{\xi}}_{\mathit{T}}$ in randomly diluted Ising magnets near the percolation threshold are related by the singular dynamic scaling ln\ensuremath{\tau}=f(ln${\ensuremath{\xi}}_{\mathit{T}}$), where f(x)=Ax2{\mathit{Ax}}^{2}Ax2+Bx+C, with constants A,B,C. We investigate the dimensionality dependence of A by Monte Carlo simulation and compare our observations with the theoretical predictions.