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

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.

Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz–Thouless phase

“…The results for the critical cases considered here (except possibly from the quenched clock model) are clearly different from what has been found in sub-critical quenches of simple coarsening systems [11] where the moments associated to the correlation scale differently from the ones where the fluctuations associated to the linear response enter. The analysis of the second-order momenta for a ferromagnet quenched to its critical point computed numerically in [11] and analytically in the spherical model in [10] unveil a behavior analogous to the one found in this paper and the scaling found by these authors conforms to the form given in Eq. (106).…”

“…expected on the basis of critical scaling arguments [11], and confirmed in [10] in the spherical model. The same scaling is obeyed also by V (2,1) k=0 and V (2,2) k=0 , with the same large-t/t ′ behavior of the scaling functions, since we have proven in Sec.…”

“…Similar considerations have led to use this same fluctuating quantity in ferromagnetic systems in [11]. Furthermore, it must be recalled that composite operators formed with the field φ( x, t)ξ( x ′ , t ′ )/(2T ) are directly related to higher order response functions [9,11,12,15], a property analogous to the one discussed above for the fluctuating part of the correlation function. This allows one to define ratios between composite operators of the fluctuating parts of response and correlation (see Sec.…”

“…Traditionally these systems have been characterized in terms of the space and time dependence of two-point correlation functions and responses (see, e.g., the review articles [4][5][6]). It has been only relatively recently that interest has moved to the study of fluctuations of these quantities and, in particular, their characterization via higher order correlation and response functions [7][8][9][10][11][12]. These properties should, obviously, give us a more detailed description of the behavior of these highly non-trivial processes.…”

How soon after a zero-temperature quench is the fate of the Ising model sealed?

“…For the system sizes commonly used in the literature L 100 − 1000, t p 10 − 30, and percolation is very quickly attained. It is important to notice that, since the equilibration time remains at least t eq ∼ L 2 t p [1], or even longer due to blocked states [10,11], after stable percolation of an ordered cluster is established at t p the systems are still far from equilibrium, as shown by the correlation and response functions that continue to relax well beyond this time-scale [19,20]. The new feature provided by our study is that the scaling properties are modified by the extra time-scale t p .…”