We derive for Ising spins an off-equilibrium generalization of the fluctuation dissipation theorem, which is formally identical to the one previously obtained for soft spins with Langevin dynamics [L.F. Cugliandolo, J. Kurchan, and G. Parisi, J. Phys. I 4, 1641 (1994)]. The result is quite general and holds both for dynamics with conserved and nonconserved order parameters. On the basis of this fluctuation dissipation relation, we construct an efficient numerical algorithm for the computation of the linear response function without imposing the perturbing field, which is alternative to those of Chatelain [J. Phys. A 36, 10 739 (2003)] and Ricci-Tersenghi [Phys. Rev. E 68, 065104(R) (2003)]. As applications of the new algorithm, we present very accurate data for the linear response function of the Ising chain, with conserved and nonconserved order parameter dynamics, finding that in both cases the structure is the same with a very simple physical interpretation. We also compute the integrated response function of the two-dimensional Ising model, confirming that it obeys scaling $\chi (t,t_w)\simeq t_w^{-a}f(t∕t_w)$, with $a=0.26\pm 0.01$, as previously found with a different method.

We study the statistics, in stationary conditions, of the work $W_{\tau }$ done by the active force in different systems of self-propelled particles in a time $\tau$. We show the existence of a critical value $W_{\tau }^{†}$ such that fluctuations with $W_{\tau }>W_{\tau }^{†}$ correspond to configurations where interaction between particles plays a minor role whereas those with $W_{\tau }<W_{\tau }^{†}$ represent states with single particles dragged by clusters. This twofold behavior is fully mirrored by the probability distribution $P(W_{\tau })$ of the work, which does not obey the large-deviation principle for $W_{\tau }<W_{\tau }^{†}$. This pattern of behavior can be interpreted as due to a phase transition occurring at the level of fluctuating quantities and an order parameter is correspondingly identified.

A unified derivation of the off-equilibrium fluctuation dissipation relations (FDRs) is given for Ising and continuous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDRs allows one to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second application, the important problem of the definition of an off-equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.

The behavior of a phase separating binary mixture in uniform shear flow is investigated by numerical simulations and in a renormalization group approach. Results show the simultaneous existence of domains of two characteristic scales. Stretching and cooperative ruptures of the network produce a rich interplay where the recurrent prevalence of thick and thin domains determines log-time periodic oscillations. A power-law growth $R\left(t\right)\sim t^{\alpha }$ of the average domain size, with $\alpha =4\mathrm{}/3$ and $\alpha =1\mathrm{}/3$ in the flow and shear direction, respectively, is shown to be obeyed.

The spinodal decomposition of binary mixtures in uniform shear flow is studied in the context of the time-dependent Ginzburg-Landau equation, approximated at one-loop order. We show that the structure factor obeys a generalized dynamical scaling with different growth exponents ${\alpha }_x=5\mathrm{}/4$ and ${\alpha }_y=1\mathrm{}/4$ in the flow and in the shear directions, respectively. The excess viscosity $\Delta \eta$ after reaching a maximum relaxes to zero as ${\gamma }^{-2\mathrm{}}{t}^{-3\mathrm{}/2}$, $\gamma$ being the shear rate. $\Delta \eta$ and other observables exhibit log-time periodic oscillations which can be interpreted as due to a growth mechanism where stretching and breakup of domains cyclically occur.

The basic features of the slow relaxation phenomenology arising in phase ordering processes are obtained analytically in the large-$N$ model through the exact separation of the order parameter into the sum of thermal and condensation components. The aging contribution in the response function ${\chi }_{\mathrm{ag}}{(t,t}_w)$ is found to obey a pattern of behavior, under variation of dimensionality, qualitatively similar to the one observed in Ising systems. There exists a critical dimensionality $\mathrm{}(d=4\mathrm{})\mathrm{}$ above which ${\chi }_{\mathrm{ag}}{(t,t}_w)$ is proportional to the defect density ${\rho }_D\mathrm{}\left(t\right),$ while for $d<\mathrm{}4\mathrm{}$ it vanishes more slowly than ${\rho }_D\mathrm{}\left(t\right)\mathrm{}$ and at $d=2\mathrm{}$ does not vanish. As in the Ising case, this behavior can be understood in terms of the dependence on dimensionality of the interplay between the defect density and the effective response associated to a single defect.