Global Persistence Exponent for Nonequilibrium Critical Dynamics.

probability, p(t) � t � , that the global order parameter has not changed sign in the time interval t following a quench to the critical point from a disordered state. This exponent is calculated in mean-field theory, in the n = 1 limit of the O(n) model, to first order in ǫ = 4 d, and for the 1-d Ising model. Numerical results are obtained for the 2-d Ising model. We argue that θ is a new independent exponent. For many years it was believed that critical phenomena were characterized by a set of three critical exponents, comprising two independent static exponents (other static exponents being related to these by scaling laws) and the dynamical exponent z. Then, quite recently, it was discovered that there is another dynamical exponent, the ‘non-equilibrium’ (or ‘short-time’) exponent λ, needed to describe two-time correlations in a system relaxing to the critical state from a disordered initial condition [1,2]. It is natural to ask ‘Are there any more independent critical exponents?’. In this Letter we propose such an exponent – the ‘persistence exponent’ θ associated with the probability, p(t) ∼ t −θ , that the global order parameter has not changed sign in time t following a quench to the critical point. We calculate θ in mean-field theory, in the n = ∞ limit of the O(n) model, to first order in ǫ = 4−d (d = dimension of space) and for the d = 1 Ising model. In fact, it turns out that all these results satisfy the scaling law θz = λ−d+1−η/2, which can be derived on the assumption that the dynamics of the global order parameter is a Markov process. We shall argue, however, that this process is in general non-Markovian, so that θ is in general a new, non-trivial critical exponent. The persistence exponent θ was first introduced in the context of the non-equilibrium coarsening dynamics of systems at zero temperature [3,4]. In that context it describes the power-law decay, p(t) ∼ t −θ , of the probability that the local order parameter φ(x) has not changed sign during the time interval t after the quench to T = 0. Equivalently, it gives the fraction of space in which the order parameter has not changed sign up to time t. More generally, one can consider the probability p0(t1, t2) of no sign changes between t1 and t2. Scaling considerations suggest p0(t1, t2) = f(t1/t2) ∼ (t1/t2) θ for t2 ≫ t1. Exact solutions for one-dimensional systems [4,5] indicate that, in general, θ is a new non-trivial exponent for coarsening dynamics. Recently, we have shown that even the diffusion equation exhibits a nontrivial persistence exponent, and have developed a rather accurate approximate theory for this case [6]. The diffusion equation is itself a model of ordering dynamics, via the approximate theory of Ohta, Jasnow and Kawasaki (OJK) [7], and also describes, in its essential features, the ordering kinetics of the nonconserved O(n) model in the large-n limit [8]: The exponents θ for these systems (OJK and large-n) are just those of the diffusion equation. In this Letter we introduce and calculate the analogous exponent θ for non-equilibrium critical dynamics. In this case however, one needs to consider the global, rather than the local order parameter. This is because individual degrees of freedom (‘spins’, say) are rapidly flipping so that the probability of not flipping in an interval t has an exponential tail. We shall see, however, that the probability for the global order parameter not to have flipped indeed decays as a power