© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1003.1065 · Journal of Statistical Mechanics Theory and Experiment

Fluctuations of two-time quantities and non-linear response functions

Federico Corberi, E Lippiello, Alessandro Sarracino, M Zannetti

Published Apr 1, 2010• 7 citations

Abstract

We study the fluctuations of the autocorrelation and autoresponse functions and, in particular, their variances and co-variance. In a first general part of the Article, we show the equivalence of the variance of the response function with the second-order susceptibility of a composite operator, and we derive an equilibrium fluctuation-dissipation theorem beyond-linear order relating it to the other variances. In a second part of the paper we apply the formalism to the study to non-disordered ferromagnets, in equilibrium or in the coarsening kinetics following a critical or sub-critical quench. We show numerically that the variances and the non-linear susceptibility obey scaling with respect to the coherence length $ξ$ in equilibrium, and with respect to the growing length $L(t)$ after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.

DOI: 10.1088/1742-5468/2010/04/p04003

Related papers

Out-of-equilibrium behavior of coarsening ferromagnets

Michel Pleimling

The out-of-equilibrium properties of ferromagnets quenched below the critical point are discussed in the dynamical scaling limit. The autoresponse and the autocorrelation are studied for both perfect and disordered ferromagnets. The scaling functions of the autoresponse are found to be in agreement with the predictions coming from the theory of local scale-invariance. Large finite time corrections to the scaling behavior of the autocorrelation are revealed for the disordered magnets.

Physics Procedia•Jan 1, 2010•0 citations

Fluctuations and effective temperatures in coarsening

Federico Corberi, Leticia F. Cugliandolo

We study dynamic fluctuations in non-disordered finite dimensional ferromagnetic systems quenched to the critical point and the low temperature phase. We investigate the fluctuations of two two-time quantities, called χ and C , the averages of which yield the self-linear response and correlation functions. We introduce a restricted average of the χs, summing over all configurations with a given value of C . We find that the restricted average ⟨χ⟩ C obeys a scaling form, and that the slope of the scaling function approaches the universal value of the limiting effective temperature in the long time limit and for . Our results tend to confirm the expectation that time-reparameterization invariance is not realized in coarsening systems at criticality. Finally, we discuss possible experimental tests of our proposal.

arXiv:0903.3122•Journal of Statistical Mechanics Theory and Experiment•May 15, 2009•10 citations

Response of non-equilibrium systems at criticality: ferromagnetic models in dimension two and above

Claude Godrèche, J. M. Luck

We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We perform an exact analysis of the spherical model in any dimension D >2 and numerical simulations on the two-dimensional Ising model. These systems are shown to be ageing in the long-time regime, i.e. their two-time autocorrelation and response functions, and associated fluctuation-dissipation ratio, are non-trivial scaling functions of both time variables. We show in particular that, for 1<< s (waiting time) << t (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value X ∞ , which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain X ∞ = 1-2/ D for 2< D <4 and X ∞ = ½ for D >4 (mean-field regime). For the two-dimensional Ising model we measure X ∞ ≈0.26±0.01.

arXiv:cond-mat/0001264•Journal of Physics A Mathematical and General•Dec 7, 2000•183 citations

Dissipation in nonlinear response

Laurent P. Lévy, Andrew T. Ogielski

It is shown that the measurable nonlinear susceptibilities of third and higher orders probe only a fraction of equilibrium fluctuations, and in general can give only lower bounds on the equilibrium correlation functions. This contrasts with the linear and quadratic response, which completely determines the corresponding correlations by general fluctuation–dissipation theorems. The exactly solvable one-dimensional kinetic Ising model illustrates which fluctuations give rise to dissipation; only two-magnon processes of short wavelength and opposite momenta contribute to the third-order nonlinear susceptibilities, while two- and three-magnon processes contribute to the corresponding equilibrium correlation functions.

Journal of Mathematical Physics•Mar 1, 1989•5 citations

Ageing and non-equilibrium critical phenomena in Monte Carlo simulations

Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov

It is considered the non-equilibrium critical evolution of statistical systems which displays some features, such as ageing and violation of the fluctuation-dissipation theorem. We review here some theoretical results of computations that have been obtained in recent years for universal quantities, such as the exponents determining the scaling behaviour of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the 3D pure and diluted Ising models with Glauber dynamics. We analyse an influence of critical fluctuations, different non-equilibrium initial states and presence of nonmagnetic impurities in spin systems on two-time dependence of correlation and response functions on characteristic time variables as waiting time t w and time of observation t – t w with t > t w . We discuss the obtained values of non-equilibrium exponents for autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio X ∞ . Analysis of simulation results show that the insertion of disorder leads to new values of X ∞ with X ∞ diluted > X ∞ pure .

Journal of Physics Conference Series•May 15, 2014•2 citations

Schrödinger-invariance in phase-ordering kinetics

Stoimen Stoimenov, Malte Henkel

The generic shape of the single-time and two-time correlators in non-equilibrium phase-ordering kinetics with ${z}=2$ is obtained from the co-variance of the four-point response functions. Their non-equilibrium scaling forms follow from a new non-equilibrium representation of the Schrödinger algebra.

arXiv:2511.02267•Nov 4, 2025•0 citations

We study the fluctuations of the autocorrelation and autoresponse functions and, in particular, their variances and co-variance. In a first general part of the Article, we show the equivalence of the variance of the response function with the second-order susceptibility of a composite operator, and we derive an equilibrium fluctuation-dissipation theorem beyond-linear order relating it to the other variances. In a second part of the paper we apply the formalism to the study to non-disordered ferromagnets, in equilibrium or in the coarsening kinetics following a critical or sub-critical quench. We show numerically that the variances and the non-linear susceptibility obey scaling with respect to the coherence length

ξ

in equilibrium, and with respect to the growing length

L(t)

after a quench, similarly to what is known for the autocorrelation and the autoresponse functions.

We study dynamic fluctuations in non-disordered finite dimensional ferromagnetic systems quenched to the critical point and the low temperature phase. We investigate the fluctuations of two two-time quantities, called χ and C , the averages of which yield the self-linear response and correlation functions. We introduce a restricted average of the χs, summing over all configurations with a given value of C . We find that the restricted average ⟨χ⟩ C obeys a scaling form, and that the slope of the scaling function approaches the universal value of the limiting effective temperature in the long time limit and for . Our results tend to confirm the expectation that time-reparameterization invariance is not realized in coarsening systems at criticality. Finally, we discuss possible experimental tests of our proposal.

We study the dynamics of ferromagnetic spin systems quenched from infinite temperature to their critical point. We perform an exact analysis of the spherical model in any dimension D >2 and numerical simulations on the two-dimensional Ising model. These systems are shown to be ageing in the long-time regime, i.e. their two-time autocorrelation and response functions, and associated fluctuation-dissipation ratio, are non-trivial scaling functions of both time variables. We show in particular that, for 1<< s (waiting time) << t (observation time), the fluctuation-dissipation ratio possesses a non-trivial limit value X ∞ , which appears as a dimensionless amplitude ratio, and is therefore a novel universal characteristic of non-equilibrium critical dynamics. For the spherical model, we obtain X ∞ = 1-2/ D for 2< D <4 and X ∞ = ½ for D >4 (mean-field regime). For the two-dimensional Ising model we measure X ∞ ≈0.26±0.01.

It is shown that the measurable nonlinear susceptibilities of third and higher orders probe only a fraction of equilibrium fluctuations, and in general can give only lower bounds on the equilibrium correlation functions. This contrasts with the linear and quadratic response, which completely determines the corresponding correlations by general fluctuation–dissipation theorems. The exactly solvable one-dimensional kinetic Ising model illustrates which fluctuations give rise to dissipation; only two-magnon processes of short wavelength and opposite momenta contribute to the third-order nonlinear susceptibilities, while two- and three-magnon processes contribute to the corresponding equilibrium correlation functions.

It is considered the non-equilibrium critical evolution of statistical systems which displays some features, such as ageing and violation of the fluctuation-dissipation theorem. We review here some theoretical results of computations that have been obtained in recent years for universal quantities, such as the exponents determining the scaling behaviour of dynamic response and correlation functions and the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the 3D pure and diluted Ising models with Glauber dynamics. We analyse an influence of critical fluctuations, different non-equilibrium initial states and presence of nonmagnetic impurities in spin systems on two-time dependence of correlation and response functions on characteristic time variables as waiting time t w and time of observation t – t w with t > t w . We discuss the obtained values of non-equilibrium exponents for autocorrelation and response functions and values of the universal long-time limit of the fluctuation-dissipation ratio X ∞ . Analysis of simulation results show that the insertion of disorder leads to new values of X ∞ with X ∞ diluted > X ∞ pure .

The generic shape of the single-time and two-time correlators in non-equilibrium phase-ordering kinetics with

{z}=2

is obtained from the co-variance of the four-point response functions. Their non-equilibrium scaling forms follow from a new non-equilibrium representation of the Schrödinger algebra.