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

Journal of Physics A Mathematical and General

Non-equilibrium dynamics of the Ising model for T less-than/equal-toTc

K. Humayun, Alan J. Bray

Published Apr 21, 1991• 107 citations
Publisher

Abstract

The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.

DOI
10.1088/0305-4470/24/8/030

Related papers

Dynamical scaling and decay of correlations for spinodal decomposition at Tc

Francis J. Alexander, David A. Huse, Steven A. Janowsky

We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to ${\mathit{T}}_{\mathit{c}}$ from random initial states. As in the case of subcritical quenches, we observe dynamical scaling of the equal time order parameter correlation function G(r,t,t), but here the nonequilibrium correlation length grows as L(t)\ensuremath{\sim}${\mathit{t}}^{1/\mathit{z}}$, where z=4/15 is the usual dynamic critical exponent for this system. Moreover, we find dynamical scaling for correlations with the initial condition G(r,0,t). The temporal decay of autocorrelations at ${\mathit{T}}_{\mathit{c}}$ yields a critical exponent ${\ensuremath{\lambda}}_{\mathit{c}}$=1.96\ifmmode\pm\else\textpm\fi{}0.05 for the nonequilibrium dynamics: G(r,0,t)\ensuremath{\approxeq}${\mathit{L}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$(t)G\ifmmode \tilde{}\else \~{}\fi{}(r/L(t)).

Physical review. B, Condensed matter•Jul 1, 1994•16 citations

Growth of order in vector spin systems; scaling and universality

Alan J. Bray, K. Humayun

The growth of order in vector spin systems with nonconserved order parameter ('model A') is considered following an instantaneous quench from infinite to zero temperature. The results of numerical simulations in spatial dimension d=2 and spin dimension 2n5 are presented. For n4, a scaling regime (where a characteristic length scale L(t) emerges) is entered for sufficiently long times, with L(t)t 1 2/. The autocorrelation function. A(t) decays with time as A(t)t -(d- lambda ) 2/, and the exponent lambda (n) agrees well with the predictions of the 1/n-expansion. The cases n=2 and 3 are more complicated, due to the non-trivial role played by topological singularities, i.e. vortices (n=2) and Polyakov solitons, (n=3). For n4, universal amplitudes and scaling functions characterizing the energy relaxation and the equal-time correlation function are identified. It is argued that for d3, where an ordered phase exists at low temperature, such universal quantities characterize the entire ordered phase.

Journal of Physics A Mathematical and General•Dec 21, 1990•46 citations

Coarsening in theq-state Potts model and the Ising model with globally conserved magnetization

Clément Sire, Satya N. Majumdar

We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}${\mathit{t}}^{1/2}$. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.

arXiv:cond-mat/9503163•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 1, 1995•78 citations

Study of the nonequilibrium critical quenching and the annealing dynamics for the long-range Ising model in one dimension

D. E. Rodríguez, M. A. Bab, Ezequiel Vicente Albano

Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1/rd + σ, with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated interaction ranges. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with renormalization group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different to the RG results probably due to the fact that it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.

Journal of Statistical Mechanics Theory and Experiment•Sep 14, 2011•6 citations

FINITE-SIZE SCALING ANALYSIS OF THE DECAY OF AN UNSTABLE STATE DURING A FIRST ORDER PHASE TRANSITION

Jorge Viñals, David Jasnow

We extend standard finite-size scaling methods to study the dynamical evolution of an unstable state far from equilibrium as the system undergoes a first order phase transition. We suggest that the nonequilibrium structure factor S(q, t, L), at late times and for large enough lattice sizes, scales as S(q, t, L)=LdF(qL, t1/x/L). L is the linear dimension of the system and 1/x is the domain growth exponent. We obtain x=2 in the case of the kinetic Ising model with a nonconserved order parameter. For a critical quench in a system with conserved order parameter, scaling of the peak of the structure factor gives 1/x≈0.27. Higher wavenumbers, however, are more consistent with x=3.

Modern Physics Letters B•Apr 1, 1988•4 citations

Universal short-time dynamics in the Kosterlitz-Thouless phase

Peter Czerner, Uwe Ritschel

We study the short-time dynamics of systems that develop quasi-long-range order after a quench to the Kosterlitz-Thouless phase. With the working hypothesis that the universal short-time behavior, previously found in Ising-like systems, also occurs in the Kosterlitz-Thouless phase, we explore the scaling behavior of thermodynamic variables during the relaxational process following the quench. As a concrete example, we investigate the two-dimensional six-state clock model by Monte Carlo simulation. The exponents governing the magnetization, the second moment, and the autocorrelation function are calculated. From them, by means of scaling relations, estimates for the equilibrium exponents $z$ and $\ensuremath{\eta}$ are derived. In particular, our estimates for the temperature-dependent anomalous dimension $\ensuremath{\eta}$ that governs the static correlation function are consistent with existing analytical and numerical results, and, thus, confirm our working hypothesis.

arXiv:cond-mat/9510078•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 1996•20 citations
The growth of order in Ising models with nonconserved order parameter is considered for quenches to final temperatures T f =0 and T f =T c . The results of numerical simulations in spatial dimension d=2 are presented. In all cases a scaling regime is entered for sufficiently long times, where the characteristic length scale is the 'domain size' L(t) approximately t 1 2/, for T f =0, and the 'nonequilibrium correlation length', xi (t) approximately t 1 z/, for T f =T c . The equal-time correlation function has the expected scaling forms f(r/L(t)) and r -(d-2+ eta ) f c (r/ xi (t)) for T f =0 and T c respectively. The scaling function f c (x) has interesting short-distance behaviour which is elucidated using scaling arguments and by in - and 1/n-expansions. The T=0 scaling function f(x) depends on whether the spin correlations present in the initial conditions are of long or short range, as does the exponent which describes the decay of the autocorrelation function, A(t)=((S i (t))S i (O)) approximately L(t) - . Results for a quench from the equilibrium critical state to T f =0 are consistent with theoretical predictions.
We have investigated the nonequilibrium critical dynamics of a two-dimensional, spin-exchange (i.e., conserved magnetization), kinetic Ising model following quenches to Tc{\mathit{T}}_{\mathit{c}}Tc​ from random initial states. As in the case of subcritical quenches, we observe dynamical scaling of the equal time order parameter correlation function G(r,t,t), but here the nonequilibrium correlation length grows as L(t)\ensuremath{\sim}t1/z{\mathit{t}}^{1/\mathit{z}}t1/z, where z=4/15 is the usual dynamic critical exponent for this system. Moreover, we find dynamical scaling for correlations with the initial condition G(r,0,t). The temporal decay of autocorrelations at Tc{\mathit{T}}_{\mathit{c}}Tc​ yields a critical exponent ${\ensuremath{\lambda}}_{\mathit{c}}$=1.96\ifmmode\pm\else\textpm\fi{}0.05 for the nonequilibrium dynamics: G(r,0,t)\ensuremath{\approxeq}${\mathit{L}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$(t)G\ifmmode \tilde{}\else \~{}\fi{}(r/L(t)).
The growth of order in vector spin systems with nonconserved order parameter ('model A') is considered following an instantaneous quench from infinite to zero temperature. The results of numerical simulations in spatial dimension d=2 and spin dimension 2n5 are presented. For n4, a scaling regime (where a characteristic length scale L(t) emerges) is entered for sufficiently long times, with L(t)t 1 2/. The autocorrelation function. A(t) decays with time as A(t)t -(d- lambda ) 2/, and the exponent lambda (n) agrees well with the predictions of the 1/n-expansion. The cases n=2 and 3 are more complicated, due to the non-trivial role played by topological singularities, i.e. vortices (n=2) and Polyakov solitons, (n=3). For n4, universal amplitudes and scaling functions characterizing the energy relaxation and the equal-time correlation function are identified. It is argued that for d3, where an ordered phase exists at low temperature, such universal quantities characterize the entire ordered phase.
We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}t1/2{\mathit{t}}^{1/2}t1/2. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1/rd + σ, with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated interaction ranges. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with renormalization group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different to the RG results probably due to the fact that it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.
We extend standard finite-size scaling methods to study the dynamical evolution of an unstable state far from equilibrium as the system undergoes a first order phase transition. We suggest that the nonequilibrium structure factor S(q, t, L), at late times and for large enough lattice sizes, scales as S(q, t, L)=LdF(qL, t1/x/L). L is the linear dimension of the system and 1/x is the domain growth exponent. We obtain x=2 in the case of the kinetic Ising model with a nonconserved order parameter. For a critical quench in a system with conserved order parameter, scaling of the peak of the structure factor gives 1/x≈0.27. Higher wavenumbers, however, are more consistent with x=3.
We study the short-time dynamics of systems that develop quasi-long-range order after a quench to the Kosterlitz-Thouless phase. With the working hypothesis that the universal short-time behavior, previously found in Ising-like systems, also occurs in the Kosterlitz-Thouless phase, we explore the scaling behavior of thermodynamic variables during the relaxational process following the quench. As a concrete example, we investigate the two-dimensional six-state clock model by Monte Carlo simulation. The exponents governing the magnetization, the second moment, and the autocorrelation function are calculated. From them, by means of scaling relations, estimates for the equilibrium exponents zzz and $\ensuremath{\eta}$ are derived. In particular, our estimates for the temperature-dependent anomalous dimension $\ensuremath{\eta}$ that governs the static correlation function are consistent with existing analytical and numerical results, and, thus, confirm our working hypothesis.