Non-Trivial Algebraic Decay in a Soluble Model of Coarsening

A non-trivial exponent /3 characterising non-equilibrium coarsening processes is calculated in a soluble model. For a spin model, the exponent describes how the fraction po of spins which have never flipped (or, equivalently, the fraction of space which has never been traversed by a domain wall) depends on the characteristic domain scale L: p, - I/-'. For the one-dimensional time-dependent Ginzburg-Landau equation at zero temperature we show that the critical exponent /3 is the zero of a transcendental equation, and find ,8 = 0.824 924 12 ... . Coarsening phenomena are rather common in physics. A typical example is the non- equilibrium evolution of the ordered domains that form when a system is thermally quenched from a homogeneous phase into a two-phase region(l). Other examples include grain growth (2), soap froths (3), and breath figures (4). A common feature of these phenomena is the scale-invariant morphology that develops at late times: the structure at different times is statistically similar apart from an overall change of scale, i.e. the system is described by a single, time-dependent length scale Ut). To fix our ideas, consider one of the simplest such systems-the d = 1 Ising model, with Glauber dynamics, evolved from a random initial condition at temperature T = 0. The behaviour of this system is well understood. The domain walls behave as independent random walkers. When two domain walls meet they annihilate. The average domain size (1) grows as t '1'. The equal-time (5,6) and two-time (5) spin-spin correlation functions can be exactly calculated.