Fluctuation-dissipation relations and field-free algorithms for the computation of response functions

We discuss the relation between the fluctuation-dissipation relation derived by Chatelain and Ricci-Tersenghi [C. Chatelain, J. Phys. A 36, 10739 (2003); F. Ricci-Tersenghi, Phys. Rev. E 68, 065104(R) (2003)] and that by Lippiello-Corberi-Zannetti [E. Lippiello, F. Corberi, and M. Zannetti, Phys. Rev. E 71, 036104 (2005)]. In order to do that, we rederive the fluctuation-dissipation relation for systems of discrete variables evolving in discrete time via a stochastic nonequilibrium Markov process. The calculation is carried out in a general formalism comprising the Chatelain, Ricci-Tersenghi, result and that by Lippiello-Corberi-Zannetti as special cases. The applicability, generality, and experimental feasibility of the two approaches are thoroughly discussed. Extending the analytical calculation to the variance of the response function, we show the advantage of field-free numerical methods with respect to the standard method, where the perturbation is applied. We also show that the signal-to-noise ratio is better (by a factor $\sqrt{2}$) in the algorithm of Lippiello-Corberi-Zannetti with respect to that of Chatelain-Ricci Tersenghi.

Figure 1

(Color online) Integrated autoresponse function $\chi \left(n,m\right)$ (left panel) and its variance (right panel) in the three-dimensional Ising model quenched from $T=\infty$ to $T_c=4.5115$. Left panel: different curves correspond to computations performed with the SM (using two different values of $h$) and with the field-free algorithms of CRT and LCZ, as indicated in the key. In the right panel, the behavior of the variances of the response function computed with the different methods is shown (continuous lines with heavy symbols, see key). The terms ${\Delta }_0^{\left(\chi ,CRT\right)}$ and ${\Delta }_0^{\left(\chi ,LCZ\right)}$ are also plotted (dashed lines).

Figure 2

(Color online) As in Fig. 1, for the three-dimensional Ising model quenched from infinite temperature to $T=3<T_c$. Left panel: different algorithms yield the same result except the SM with the largest value of $h$, which is always affected by nonlinear effects.

Figure 3

(Color online) Difference ${\Delta }^{\left(\chi \right)}\left(n,m\right)-{\Delta }_0^{\left(\chi \right)}\left(n,m\right)$ for the three algorithms in the Ising model quenched to $T=3$. The dashed orange line has a slope of 0.5.

Figure 4

(Color online) The integrated autoresponse function $\chi \left(n,m\right)$ (insets) and its variance (main) in the three-dimensional EA model quenched from $T=\infty$ to $T_c\simeq 1.2$ (left panel) or to $T=1<T_c$. Insets: different curves correspond to computations performed with the SM and two different values of $h$ and with the field-free algorithms of CRT and LCZ, as indicated in the key. In the main part of the two panels, the behavior of the variances of the response computed with the different methods is shown (continuous lines with heavy symbols, see key). The terms ${\Delta }_0^{\left(\chi ,CRT\right)}$ and ${\Delta }_0^{\left(\chi ,LCZ\right)}$ are also plotted (dashed lines).

Figure 5

(Color online) Results of the numerical computation of the integrated autoresponse function $\chi \left(n,m\right)$ for the one-dimensional FA model with $m={10}^5$. The response functions computed with the SM (red filled symbols) and with the LCZ algorithm (continuous black lines) agree for the three temperatures. In the SM, the external field is $h=0.01T$, satisfying linearity. The equilibrium value (green empty symbols) is also plotted for each temperature.