Growth of long-range correlations after a quench in phase-ordering systems

We present a general framework for the time-dependent correlation functions in a phase-ordering system after a quench from the disordered phase to or below the critical point and discuss under what conditions the two-time exponents λ or ${\lambda }_{\mathit{c}}$ [characterizing the decay of local autocorrelations, 〈φ(r→,0)φ(r→,t)〉∼${\mathit{L}}^{\mathrm{-}\lambda }$ or ∼${\mathit{L}}_{\mathit{c}}^{\mathrm{-}\lambda }$ for quenches to below ${\mathit{T}}_{\mathit{c}}$ or to ${\mathit{T}}_{\mathit{c}}$, respectively, where L(t) is the correlation length at time t, and φ is the order parameter] are equal to the spatial dimension d in the conserved order parameter case. We present a few cases where exact solutions and numerical simulations suggest ${\lambda }_{\mathit{c}}$=d. The same, however, is not true for the exponent λ. We present one example, namely a deterministic conserved model in one dimension, where λ is explicitly less than d=1. This led us to study the differences and similarities between stochastic and deterministic models of coarsening. In this paper, we address this general issue with a focus in one dimension, where the two classes of models are discussed in parallel and several analytical and numerical results are derived.