Dynamics below the Depinning Threshold in Disordered Elastic Systems

Abstract

We study the steady-state low-temperature dynamics of an elastic line in a disordered medium below the depinning threshold. Analogously to the equilibrium dynamics, in the limit T→0, the steady state is dominated by a single configuration which is occupied with probability 1. We develop an exact algorithm to target this dominant configuration and to analyze its geometrical properties as a function of the driving force. The roughness exponent of the line at large scales is identical to the one at depinning. No length scale diverges in the steady-state regime as the depinning threshold is approached from below. We do find a divergent length, but it is associated only with the transient relaxation between metastable states

Nonsteady relaxation and critical exponents at the depinning transition

“…It is worth noting here that while h −ν can be associated with the geometrical correlation length ξ diverging in lim f → f + c in the steady state, this is not true in lim f → f − c . We can, however, find a divergent correlation length relax ∼ (f c − f ) −ν , not observed in the steady-state geometry, but associated with the deterministic part of the avalanches that are produced in the steady-state dynamics of the f < f c low-temperature creep regime [13,14]. Hence, the interpretation of Eq.…”

“…When approaching the threshold from above, the steady-state average velocity vanishes as v ∼ (f − f c ) β and the correlation length characterizing the cooperative avalanchelike motion diverges as ξ ∼ (f − f c ) −ν for f > f c with a typical diverging interevent time ξ z , where β is the velocity exponent, ν is the depinning correlation length exponent, and z is the dynamical exponent [5,[30][31][32]. At finite temperature and for f f c , the system presents an ultraslow steady-state creep motion with universal features [4,33,34] directly correlated with geometrical crossovers [13,35]. At very small temperatures the monotonic increase of the correlation length with decreasing f below f c shows that the naive analogy breaks, and that depinning must be regarded as a nonstandard phase transition [13,14].…”

“…At finite temperature and for f f c , the system presents an ultraslow steady-state creep motion with universal features [4,33,34] directly correlated with geometrical crossovers [13,35]. At very small temperatures the monotonic increase of the correlation length with decreasing f below f c shows that the naive analogy breaks, and that depinning must be regarded as a nonstandard phase transition [13,14]. The transition is then smeared out, with the velocity vanishing as v ∼ T ψ exactly at f = f c , with ψ the so-called thermal rounding exponent [36][37][38][39][40].…”

“…From a purely theoretically point of view, considerable progress has been made in recent years thanks to a fruitful interplay between the very powerful analytical [3][4][5] and numerical techniques specially developed for studying equilibrium [6,7], depinning [8][9][10][11][12], and creep [13,14] of strictly elastic manifolds. From the experimental viewpoint, on the other hand, the understanding of this particular problem is directly relevant for various experimental situations where the elastic approximation is well met, such as magnetic [15][16][17][18] or ferroelectric domain walls [19][20][21][22], contact lines in wetting [23,24], and fractures [25,26].…”

Random field Ising model swept by propagating magnetic field wave: Athermal nonequilibrium phasediagram

“…The motion of domain wall shows very interesting depinning transition at zero temperature. Due to the energy barriers created by disorder (random field), the domain wall is pinned and the motion of the domain wall remains stopped upto a critical field [8,9]. However, at any finite temperature the depinning transition is softened and the thermal fluctuation assists to overcome the energy barrier.…”