Mean-field theories for depinning and their experimental signatures

Mean-field theory is an approximation replacing an extended system by a few variables. For depinning of elastic manifolds, these are the position $u$ of its center of mass and the statistics of the forces $F\left(u\right)$. There are two proposals how to model the latter: as a random walk (ABBM model), or as uncorrelated forces at integer $u$ (discretized particle model, DPM). While for many experiments the ABBM model (in the literature misleadingly equated with mean-field theory) makes quantitatively correct predictions for the distributions of velocities, or avalanche size and duration, the microscopic disorder force-force correlations cannot grow linearly, and thus unboundedly as a random walk, with distance. Even the effective (renormalized) disorder forces which do so at small distances are bounded at large distances. To describe both regimes, we model forces as an Ornstein-Uhlenbeck process. The latter has the statistics of a random walk at small scales, and is uncorrelated at large scales. By connecting to results in both limits, we solve the model largely analytically, allowing us to describe in all regimes the distributions of velocity, avalanche size, and duration. To establish experimental signatures of this transition, we study the response function, and the correlation function of position $u$, velocity $\dot{u}$, and forces $F$ under slow driving with velocity $v>0$. While at $v=0$ force or position correlations have a cusp at the origin and then decay at least exponentially fast to zero, this cusp is rounded at a finite driving velocity. We give a detailed analytic analysis for this rounding by velocity, which allows us, given experimental data, to extract the timescale of the response function, and to reconstruct the force-force correlator at $v=0$. The latter is the central object of the field theory, and as such contains detailed information about the universality class in question. We test our predictions by careful numerical simulations extending over up to ten orders in magnitude.

Figure 1

Avalanche-size distribution $P\left(S\right)$ for a particle with forces given by Eq. (5). The theoretical curves are the kicked ABBM model as given by Eq. (46) (cyan dotted), and the DPM as given by Eq. (62) (blue dashed). Improved numerical solver (INS), $m^2={10}^{-4}$, $\delta w=1$, $\delta t={10}^{-4}$, $S_m:=〈S^2〉/\left(2〈S〉\right)=2408.89$, ${\rho }_m=2329.95$, $N={10}^8$.

Figure 2

Geometric solution to find $u_w$ as a function of $w$ for the DPM model of [33].

Figure 3

$\mathrm{\Delta }\left(w\right)=e^{-w}$ (blue solid), and the finite-velocity correlator (25) (red) for $\tau v=0.3$, compared to the boundary-layer approximation (27) (green dotted).

Figure 4

$u_w-w$ for $\delta t={10}^{-4}$, $v=0.1$, $m^2=0.1$, INS. See Appendix pp2 for implementation details. Red points are data points equally spaced in time, the green line between them is an interpolation.

Figure 5

$\dot{u}\left(w\right)-v$ with the same parameters as in Fig. 4.

Figure 6

$F\left(w\right)$ with the same parameters as in Fig. 4.

Figure 7

The critical force $f_c$ as a function of $m$. Thick black curve: the results of the integral (68). The small-$m$ expansion is in blue, cyan, green, and yellow. The data points are from our numerical simulation. (DNS, $\delta u={10}^{-2}$, $\delta t={10}^{-4}$, $N=3\times {10}^6$.)

Figure 8

$P\left(\dot{u}\right)$ for $m^2={10}^{-3}$ (orange solid line). The blue dashed curve is given by Eq. (73); the gray dotted one by the same formula, replacing $m^{-2}\to 0.39m^{-2}$, yields a perfect fit. The cyan dotted line is the result for ABBM, Eq. (44). (INS, $\delta t={10}^{-3}$, $v=0.1$, $N=5\times {10}^8$.)

Figure 9

The response function for the discretized force model (needle disorder) for $m^2={10}^{-2}$ (blue). The red dashed curve is an exponential function with timescale ${\tau }_m$ as given by Eq. (64). Inset: the same for $lnR\left(t\right)$. (DNS, $\delta t={10}^{-4}$, $\delta w=0.1$.)

Figure 10

The response function for the Ornstein-Uhlenbeck process: the analytical result is the black dashed curve. Numerical results from bottom to top are for $m^2={10}^{-2}$ (blue), $m^2={10}^{-3}$ (orange), $m^2={10}^{-4}$ (cyan), and $m^2={10}^{-5}$ (red). DNS, $\delta t={10}^{-4}$, $N={10}^8$. The two larger masses have $\delta w=0.1$ and $\delta u={10}^{-6}$, the two smaller ones $\delta w=1$ and $\delta u={10}^{-4}$. Inset: The measured timescale $\langle t\rangle$ (dots) compared to the prediction for ${\tau }_m$ in Eq. (64).

Figure 11

The function ${\delta }_{\mathrm{BL}}\left(t\right):=(\tau {\partial }_t+1)R\left(t\right)$ for the measured response function with $m^2={10}^{-2}$ (blue), $m^2={10}^{-3}$ (orange), $m^2={10}^{-4}$ (cyan). Parameters as in Fig. 10, except $N=2\times {10}^8$ for $m^2={10}^{-4}$.

Figure 12

Inverse of the measured timescale ${〈t〉}_v:={\int }_0^{\infty }dtt{R}_v\left(t\right)$, as a function of $v$. The data points are given with $99\%$ confidence intervals (green). Superimposed are the two analytical curves (81), valid for $v\le f_{\mathrm{c}}$, and (82), valid for $v\ge f_{\mathrm{c}}$. (DNS, $\delta t={10}^{-4}$, $\delta u={10}^{-6}$ for $m^2=0.01$, and $\delta u={10}^{-4}$ for $m^2={10}^{-4}$.)

Figure 13

Main plot: The response function rescaled with the theoretically predicted timescale ${\tau }_m$. Velocities from top to bottom are $v=0$ (red), $v=0.3$ (blue), $v=0.5$ (green), $v=1$ (violet), $v=1.5$ (brown), $v=2$ (black), and $v=3\approx f_{\mathrm{c}}$ (magenta). Inset: Scaling collapse for the velocities $v\le 2$, using the measured timescale $〈t〉$ instead of ${\tau }_m$ for the rescaling. (DNS, $\delta u={10}^{-6}$, $\delta t={10}^{-4}$, $N={10}^7$.)

Figure 14

$P\left(T\right)$ measured numerically via INS for $m^2={10}^{-3}$, ($\delta w=1$, $\delta t={10}^{-3}$, $N=5\times {10}^8$). This is compared to the theory prediction (48) for ABBM (cyan dotted), and (66) for DPM (blue dashed).

Figure 15

The measured ${\mathrm{\Delta }}_v$ (cyan), ${\mathrm{\Delta }}_F$ (orange dashed), and ${\mathrm{\Delta }}_{\dot{u}}$ (purple dotted) for $m^2=0.01$ at $v=0.5$. The inset shows that the difference ${\mathrm{\Delta }}_F-{\mathrm{\Delta }}_v-{\mathrm{\Delta }}_{\dot{u}}$ vanishes. (DNS, $\delta u=0.01$, $\delta t={10}^{-4}$, $N={10}^8$.)

Figure 16

Numerical test of Eq. (88) for $m^2={10}^{-2}$, $v=0.5$. Plotted are ${\mathrm{\Delta }}_{\dot{u}}\left(w\right)$ (cyan dotted) and $-\frac{v^2}{m^4}{\mathrm{\Delta }}_v^{″}\left(w\right)$ (red dashed). The inset shows the difference. (DNS, $\delta u=0.01$, $\delta t={10}^{-4}$, $N={10}^8$.)

Figure 17

Convergence of the measured $\tilde{\mathrm{\Delta }}\left(w\right)$ for $m^2={10}^{-2}$ (gray dot-dashed), $m^2={10}^{-3}$ (cyan dotted), and $m^2={10}^{-4}$ (red dashed). This is compared to three theoretical curves: an exponential function with the same slope at $w=0$ (green, top curve), Eq. (92) (black, middle curve), and the one-loop functional renormalization group result (blue, bottom curve); see Eqs. (4.5) and (4.11) of [23]. The inset shows the difference of the measured $\tilde{\mathrm{\Delta }}\left(w\right)$ at $m^2={10}^{-4}$ and the theoretical curves in their respective colors, together with error bars at a $99\%$ confidence level. (DNS, $\delta u=0.01$, $\delta t={10}^{-4}$, $N={10}^8$, $\delta w=0.1$.).

Figure 18

The numerically measured ${\mathrm{\Delta }}_v\left(w\right)$ for $v=0.5$ (red dashed, with $95\%$-confidence intervals shaded red) as defined in Eq. (18). For $w>1000$ it agrees well with the theoretical prediction (91) (black solid). The inset shows the same curves, superimposed with the theoretical prediction (91) folded according to Eq. (20) with the numerically measured response function (green), lying underneath ${\mathrm{\Delta }}_v\left(w\right)$ obtained numerically (red dashed). (DNS, $\delta u=0.01$, $\delta t={10}^{-4}$, $N={10}^7$.)

Figure 19

The measured amplitude of the rounding $v^{-1}\left[{\mathrm{\Delta }}_{v=0}\left(0\right)-{\mathrm{\Delta }}_v\left(0\right)\right]$ for $m^2=0.01$ and $v=0.1,0.3,0.5,1,2,3.5,4$; $f_{\mathrm{c}}\simeq 3.035$. Error bars (green) are included in the fit and represent $99\%$ confidence intervals on the data. We compare several curves. In black is shown the prediction from Eq. (22) with $\tau ={\tau }_m$, and ${\mathrm{\Delta }}^{\prime }\left(0^{+}\right)$ as given by Eq. (91). In blue is the prediction from Eq. (22) with $\tau =〈t〉$, and ${\mathrm{\Delta }}^{\prime }\left(0^{+}\right)$ as measured in the simulation, accompanied by error bars in green. In red is a weighted fit to the velocities $v\le 0.5$, verifying Eq. (22) with a relative precision of $3\times {10}^{-3}$. Note the transition at $v=f_c$. (DNS, $\delta u=0.01$, $\delta t={10}^{-4}$, $N={10}^8$.)

Figure 20

Measurement of ${\mathrm{\Delta }}_v\left(w\right)$ for $v=0$ (solid blue, top), $v=0.1$ (solid red, middle), and $v=0.3$ (solid green, bottom). We show unfolding via a boundary layer analysis, using $\tau :=〈t〉=14.08$ as measured for the response function, both for $v=0.1$ (black dots), prolonged to zero (red dashed), and $v=0.3$ (black dots), prolonged to zero (green dashed). The variable $\tilde{w}=\sqrt{w^2+{\left(\tau v\right)}^2}$. (DNS, $\delta t={10}^{-4}$, $\delta u={10}^{-2}$, $N={10}^8$.)

Figure 21

Measurement of ${\mathrm{\Delta }}_v\left(w\right)$ for $v=0$ (solid blue, top), $v=0.1$ (solid red, middle), and $v=0.3$ (solid green, bottom). We show unfolding via the differential equation (34) using $\tau :=〈t〉=14.08$ as measured for the response function, both for $v=0.1$ (red dashed) and $v=0.3$ (green dotted). (DNS, $\delta t={10}^{-4}$, $\delta u={10}^{-2}$, $N={10}^8$.)