![Figure 4. Results for the critical state in finite samples obtained with the exact algorithm of Sec. 4.3.1. (a) Critical configuration structure factor (see definition in Eq. (6)). The fitted roughness exponent yields ζdep = 1.250 ± 0.005. (b) The finite-size critical force fc as a function of the longitudinal size L for periodic samples of size L × M with M = kLζdep . As L grows the asymptotic value of the critical threshold becomes independent on the aspect ratio k. The dashed line corresponds to the thermodynamic limit fc = 1.916±0.001. Figures are reproduced with permission from [41].](/figures/figure-4-results-for-the-critical-state-in-finite-samples-343gbqwb.webp)
![Figure 3. (a) Steady state structure factor S(q) of the line in the T → 0 limit for different forces (curves are shifted for clarity). (b) The steady state properties of the elastic string at T → 0 are determined by ℓopt (filled symbols) and by ℓav. These lengths separate regions characterized by the equilibrium exponent, the depinning exponent (gray region), and the fast flow one. The divergent length ℓrelax (open symbols) is associated only with transient dynamics. Lines are guides to the eye. Figures are adapted with permission from [31].](/figures/figure-3-a-steady-state-structure-factor-s-q-of-the-line-in-3ucmvaev.webp)
![Figure 6. (a) Structure factor Sq at a finite temperature T = 0.02 showing two regimes with different roughness exponents, ζff and ζdep, above and below the crossover length scale ℓav, respectively. (b) Velocity-force scaling function in the thermal rounding region. The disorder intensity is kept fixed and the velocity for different temperatures collapses in a single function. Figures are adapted with permission from [43].](/figures/figure-6-a-structure-factor-sq-at-a-finite-temperature-t-0-ohqhw7j7.webp)
![Figure 1. Linking transport and geometry. (a) Snapshot of a domain wall in a two dimensional ferromagnet [27]. (b) Typical velocity-force characteristics. (c) Crossover lengths ℓopt and ℓav representing the optimal excitation and the deterministic avalanches, respectively. (d) Geometric crossover diagram.](/figures/figure-1-linking-transport-and-geometry-a-snapshot-of-a-20r1dp15.webp)
![Figure 5. Evolution of γ(t) (a) and v(t) (b) for f ≈ fc, with fc the thermodynamic critical force. Two other forces, just above (f+c ) and just below (f − c ) fc are also shown. Continuous lines are fits for γ(t) and v(t) including power-law corrections to the asymptotic scaling yielding the true critical exponents. In (a) vertical dashed lines separate qualitatively different time regimes. The inset in (b) shows that corrections to scaling are power-law like. Figures are adapted with permission from [41].](/figures/figure-5-evolution-of-g-t-a-and-v-t-b-for-f-fc-with-fc-the-219j9pmv.webp)
Q2. What is the way to target the equilibrium of the elastic string?
At zero force, the geometrical properties of the elastic string can be targeted using a transfer matrix method for the discrete directed polymer model.
Q3. What is the criterion to estimate the crossover between the “mesoscopic” time regime?
A practical criterion to estimate the crossover between the “mesoscopic” time regime with corrections and the truly universal “macroscopic” time regime was found by observing that the hyperscaling relation β = ν(z − ζ) is violated in the former regime.
Q4. What is the recursive relation for the polymer ending probability?
To avoid numerical instability, all weights Zx,u at fixed x have to be divided by the largest one, which does not change the polymer ending probability [44].
Q5. What is the key property of Middleton’s theorem?
Another important property of Middleton’s theorem states that if, at an initial time, the velocities are non-negative for all points x, they will remain non-negative for all later times.
Q6. How can the authors determine the optimal power-law fit for v(t)?
At large times, when γ(t) develops a plateau, an accurate power-law fit can be performed for v(t) at the thermodynamic depinning force fc = 1.5652± 0.0003, yielding β/(νz) = 0.128± 0.003.
Q7. What is the roughness exponent for f > fc?
for f > fc the authors observe that at short length-scales ℓ < ℓav the roughness exponent is ζdep, while for ℓ > ℓav one has ζff.
Q8. What are the important phenomena for the dynamics of interfaces in random media?
At last, probably the least understood but still experimentally relevant phenomena for the dynamics of interfaces in random media are: the occurrence of “plasticity” (displayed by overhangs and bubbles [59]), the effect of internal degrees of freedom (such as the “spin phase” coupled to the position of magnetic domain walls [5,60]), and the effect of “structural relaxation” [15] in host materials.
Q9. What is the center of mass velocity for an interface of size L?
The center of mass velocity for an interface of size L is defined as,v(t) = 1LL−1 ∑x=0∂tu(x, t), (4)that, given Eq. 1 and η = c = 1, is nothing else but the spatial average of the instantaneous total forces acting on the line.
Q10. Why does the subscript opt stay from optimal?
The subscript opt stays from “optimal” because ℓopt is the size of the jump associated with the optimal barrier that the interface should overcome in order to find a new metastable state with a smaller energy.
Q11. What is the meaning of lav adn lopt?
both ℓav adn ℓopt have a “double” physical meaning: besides being roughness crossover lengths, they control the non-linear collective transport properties of the interface.
Q12. What is the typical structure factor for depinning?
At small length scales, q ≫ 1/ℓav, the structure factor shows the typical roughness regime associated to depinning, i.e. Sq ∼ q−(1+2ζdep), while at large length scales, q ≪ 1/ℓav, geometrical properties are dictated by effective thermal fluctuations induced by the disorder, i.e. Sq ∼ q−(1+2ζff), as shown in Fig. 6(a).
Q13. What is the rate of power-law divergences of important quantities?
These describe the rate of power-law divergences of important quantities as the control parameter f approaches three special states: (i) the equilibrium (f = 0); (ii) the depinning (f = fc, T = 0), and the fast-flow (f ≫ fc).
Q14. How can the authors estimate the size of the optimal excitation lopt?
The size of the optimal excitation ℓopt can be estimated by balancing the gain in energy of being pinned in a deep metastable state, Emetast(ℓ), with the gain in energy of moving the interface forward ∼ f (usaddle(ℓ)− umetast(ℓ)).
Q15. What is the equation for capturing the overdamped dynamics of an elastic interface?
(1)This equation models the overdamped dynamics of an elastic interface with a univalued scalar displacement field u(x, t), with x a vector of dimension d, such that the interface is embedded in a space of dimension D = d +