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 sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size L like ∼Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ peak in the thermodynamic limit L→∞. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.
Abstract In this paper, we review recent results on sample-to-sample fluctuations in a critical Ising model with quenched random ferromagnetic couplings. In particular, in terms of the renormalized replica Ginzburg–Landau Hamiltonian in dimensions D < 4 an explicit expression for the probability distribution function (PDF) of the critical free energy fluctuations is derived. Next, using known fixed-point values for the renormalized coupling parameters the universal curve for such PDF in the dimension D = 3 is obtained. For the specific case of the two-dimensional Ising model, using replica calculations in the renormalization group framework, we derive explicit expressions for the PDF of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energy is Gaussian, and its typical sample-to-sample fluctuations as well as its average value scale with the system size L like ~ Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ-peak in the thermodynamic limit L → ∞.
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.
We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity
P in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of
P,
VP, which characterizes these fluctuations is calculated as a function of the systems' linear size
l, focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice [Phys. Rev. 64, 178 (1943)]. Using extensive Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality all thermodynamic quantities which were examined (energy, magnetization, specific heat, susceptibility) are strongly self-averaging, that is ${V}_{P}\ensuremath{\sim}{l}^{\ensuremath{-}d}$ (where
d=2 is the dimension). At criticality though, the results indicate that the magnetization
M and the susceptibility $\ensuremath{\chi}$ are non-self-averaging, i.e., $\frac{{V}_{\ensuremath{\chi}}}{{\ensuremath{\chi}}^{2}}, \frac{{V}_{M}}{{M}^{2}}\ensuremath{\nrightarrow}0$. The energy
E at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ with
0<yv<d. Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self-averaging. A phenomenological theory of finite size scaling for disordered systems is developed, based on physical considerations similar to those leading to the Harris criterion. Its main prediction is that when the specific heat exponent $\ensuremath{\alpha}<0$ ($\ensuremath{\alpha}$ of the disordered model) then, for a quantity
P which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance
VP will scale asymptotically as ${l}^{2\ensuremath{\rho}+\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}}$. The theory is not applicable in the asymptotic limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$) to the bond-disordered Ashkin-Teller model where $\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}={0}_{+}$. Nonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for ${V}_{\ensuremath{\chi}}$ and
VE. The theory may also be compatible with the data for the variance of the magnetization
VM and the variance of the specific heat
VC, but evidence for this is less convincing.