Non–equilibrium relaxation of an elastic string in random media

We study the relaxation of an elastic string in a two dimensional pinning landscape using Langevin dynamics simulations. The relaxation of a line, initially flat, is characterized by a growing length, L(t), separating the equilibrated short length scales from the flat long distance geometry that keep memory of the initial condition. We find that, in the long time limit, L(t) has a non–algebraic growth, consistent with thermally activated jumps over barriers with power law scaling, U(L) ∼ L . Understanding the dynamics of elastic manifolds in random media has been the focus of intense activities both on the theoretical and experimental side [1]. Because of the competition between disorder and elasticity, glassy properties arise leading to divergent barriers separating metastable states. The creep, i.e. the steady–state response to a small external force, has been studied theoretically [2,3] and observed experimentally [4]. Much less is known about the nonstationary relaxation to equilibrium. Theoretical attempts to tackle this problem have been made using mean field and renormalization group approaches [5,6], while numerical simulations have been mostly restricted to 2–dimensional systems [7]. Direct applications of these results to one dimensional domain walls are however difficult. In this paper we study, numerically, the slow non-stationary motion of an elastic string in a two dimensional random medium, relaxing from a flat initial configuration. The string obeys the equation of motion γ∂tu(z, t) = c∂ 2 zu(z, t)+Fp(u, z)+ η(z, t), where γ is the friction coefficient and c the elastic constant, Fp(u, z) the pinning force for a random bond disorder and η(z, t) is the thermal noise. The details of the numerical method are given in [8]. Fig. 1(a) shows the typical relaxation of a string and Fig. 1(b) the evolution of the structure factor S(q, t) which characterize the geometry of the line [3, 8]. At short length scales the line has reached equilibrium and it is characterized by the well known roughness exponent ζ = 2/3 [9]. At large length scales a plateau still keeps memory of the initial flat condition. A unique crossover length L(t) can be defined. Its evolution is shown in Fig. 1(c). Two main scenarios have been proposed to describe the long time growth of L(t). The first one relies on phenomenological scaling arguments, based on creep. At low temperatures the relaxation is dominated by the energy barriers U(L) that must be overcomed in order to equilibrate the system up to a length scale L. Using the Arrhenius thermal activation law we can thus express the relaxation time t(L) ∼ exp[βU(L)]. Even if the exact numerical determination of U(L) is an NP-complete problem it is usually conjectured that the typical barriers of the energy landscape scale, asymptotically with L, the same way as the free energy fluctuations: