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 χ 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 (d = 4) above which χ ag (t, t w ) is proportional to the defect density ρ D (t), while for d < 4 it vanishes more slowly than ρ D (t) and at d = 2 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.
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)) approximately equal to t(-a)(w) f (t/t(w)) , with a =0.26+/-0.01 , as previously found with a different method.
The exact relation between the response function R(t,t(')) and the two time correlation function C(t,t(')) is derived analytically in the one-dimensional kinetic Ising model subjected to a temperature quench. The fluctuation dissipation ratio X(t,t(')) is found to depend on time through C(t,t(')) in the time region where scaling C(t,t('))=f(t/t(')) holds. The crossover from the nontrivial form X[C(t,t('))] to X(t,t(')) identical with1 takes place as the waiting time t(w) is increased from below to above the equilibration time t(eq).
An exact analytical solution of the time-dependent Ginzburg-Landau model in the large-N limit is presented for a quench to zero temperature. Standard dynamic scaling is obeyed when the order parameter is not conserved, while a novel form of scaling, due to two (marginally different) divergent lengths and characterized by infinitely many growth exponents, is found when the order parameter is conserved. The new scaling behaviour is expected to be a pattern common to other growth phenomena.
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