Local scale-invariance in ageing phenomena

Many materials quenched into their ordered phase undergo ageing and there show dynamical scaling. For any given dynamical exponent z, this can be extended to a new form of local scale-invariance which acts as a dynamical symmetry. The scaling functions of the two-time correlation and response functions of ferromagnets with a non-conserved order parameter are determined. These results are in agreement with analytical and numerical studies of various models, especially the kinetic Glauber-Ising model in 2 and 3 dimensions. PACS: 05.70.Ln, 74.40.Gb, 64.60.Ht Ageing in its most general sense refers to the change of material properties as a function of time. In particular, physical ageing occurs when the underlying microscopic processes are reversible while on the other hand, biological systems age because of irreversible chemical reactions going on within them. Historically, ageing phenomena were first observed in glassy systems, see [1], but it is of interest to study them in systems without disorder. These should be conceptually simpler and therefore allow for a better understanding. Insights gained this way may become useful for a later study of glassy systems. 1 Phenomenology of ageing In describing the phenomenology of ageing system, we shall refer throughout to simple ferromagnets, see [2,3,4,5] for reviews. We consider systems which undergo a second-order equilibrium phase transition at a critical temperature Tc > 0 and we shall assume throughout that the dynamics admits no macroscopic conservation law. Initially, the system is prepared in some initial state (typically one considers an initial temperature Tini = ∞). The system is brought out of equilibrium by quenching it to a final temperature T ≤ Tc. Then T is fixed and the system’s temporal evolution is studied. It turns out that the relaxation back to global equilibrium is very slow (e.g. algebraic in time) with a formally infinite relaxation time for all T ≤ Tc. Let φ(t, r) denote the timeand space-dependent order parameter and consider the two-time correlation and (linear) response functions C(t, s; r) = 〈φ(t, r)φ(s,0)〉 , R(t, s; r) = δ〈φ(t, r)〉 δh(s,0) ∣∣∣∣ h=0 (1)