L-5

Goal : This is the first lecture about the dynamics of a disordered system. We will see that different systems display pinning until a critical threshold. We will revisit Larkin arguments and discuss the spectrum of excitation of the instabilities.

Pinning and depininng of a disordered material

In the first lectures we saw that a disorder system can be trapped in deep energy states and form a glass. Today we will see that it can be also pinned and resist external deformation. Indeed disorder is at the origin of a complex energy landscape characterized by many minima (more or less deep), maxima and saddle points. An external force tilts this multidimensional landscape in a direction, but local minima survives until a finite threshold that unveils spectacular critical dynamics.

Experiments

We will discuss two examples of transition induced by pinning:

• The depinning transition: interfaces pinned by impurities are ubiquitous and range from magnetic domain walls to crack fronts, from dislocations in crystals or vortices in superconductors. Above a critical force, interfaces depin, their motion is intermittent and a Barkhausen noise is detected.

• The yielding transition: Everyday amorphous materials such as mayonnaise, toothpaste or foams display a behaviour intermediate between solid and liquid. They deform at small stress (as a solid) and flow  at large stress (as a liquid). In between we observe intermittent plastic events.

Equation of motion

We set to zero the temperature and consider the dynamics in the overdamped regime, where $\rho \partial _{t}^{2}+\eta \partial _{t}\approx \eta \partial _{t}$. In presence of an external force $f$ the equation of motion of the interface is

$\partial _{t}h(x,t)=\nabla ^{2}h+f+F(x,h(x,t)),\quad {\text{with}}\;F(x,h(x,t))=-{\frac {\delta V(x,h(x,t))}{\delta h(x,t)}}$

The disorder force $F(x,h(x,t))$ is a stochastic function:

${\overline {F(x,h)F(x',h')}}=\delta ^{d}(x-x')\Delta (h-h')$

There are usually two kind of disorder: (i) Random Bond (RB) if $V(x,h)$ is short range correlated. Hence, the area below $\Delta (h)$ is zero, (ii) Random field if $V(x,h)$ is a Brownian motion along $h$.Hence, $\Delta (h)$ is short range corraleted.

• The velocity - force characteristics

• Existence of a unique critical force $f_{c}$: no-passing rule.

• Large force behaviour shows that in the moving phase the long distance properties of the interface are described by Edwards-Wilkinson.

Scaling behaviour of the depinning transition

• The order parameter of the transition is the velocity of the center of mass of the interface. It is vanishing as

$v_{CM}\sim |f-f_{c}|^{\beta }.$

• Two point correlation function:

${\overline {v(y+x,t)v(y,t)}}\sim {\frac {e^{-x/\xi (f)}}{|x|^{\kappa }}},\quad \xi (f)\sim |f-f_{c}|^{-\nu }$

• The interface is rough at $f_{c}$

$u(bx)\sim b^{\zeta }u(x),\quad {\hat {u}}_{bq}\sim b^{\zeta -d}u_{q}$

${\overline {{\hat {u}}_{q}{\hat {u}}_{q'}}}=\delta ^{d}(q+q')S(q),\quad S(q)\sim {\frac {1}{|q|^{d+2\zeta }}}$

• The motion is intermittent with avalanches even below $f_{c}$. Their size and duration is scale free up to a cut-off:

$P(S)=S^{-\tau }f(S/S_{\max }),\quad S_{\max }\sim \xi (f)^{d+\zeta }\sim |f-f_{c}|^{-(d+\zeta )\nu }$ $P(T)=T^{-\alpha }f(T/T_{\max }),\quad T_{\max }\sim \xi (f)^{z}\sim |f-f_{c}|^{-(z)\nu }$

scaling arguemnts

Depinning exponents

Exponent	Observable	Mean field	d=1
$z$	$\ell (t)\sim t^{z}$	2	$1.43\pm 0.01$
$\zeta$	$h(bx)\sim b^{\zeta }h(x)$	0	$1.25\pm 0.01$
$\nu$	$\xi (f)\sim |f-f_{c}|^{-\nu }$	1/2	$\nu ={\frac {1}{2-\zeta }}$
$\beta$	$v_{CM}\sim |f-f_{c}|^{\beta }$	1	$\beta =\nu (z-\zeta )$
$\tau$	$P(S)\sim S^{-\tau }$	3/2	$\tau =2-{\frac {2}{d+\zeta }}$

Cellular Automata

We introduce a discrete version of the continuous equation of motion of the interface. These cellular automata share the same universality class of the original model, are very convenient for numerical studies and can be solved in the mean field limit. For simplicity we discuss the $d=1$ case.

The first step is the disretization along the $x$ direction. The line is now a collection of $L$ blocks connected by springs. The spring constant is set equal to one. Hence the velocity of each block is

$v_{i}(t)=\partial _{t}h_{i}(t)={\frac {1}{2}}(h_{i+1}(t)+h_{i-1}(t)-2h_{i}(t))+f+F_{i}(h_{i}(t)),$

To study avalanches below threshold, one as to trigger them at constant force, which is not very convenient. It is useful to replace the external force with a parabolic potential ${\frac {m^{2}}{2}}(w-h_{i}(t))^{2}$, here $m^{2}$ is the spring constant, try to bring each block at the equilibrium position $w$. The velocity of each block becomes

$v_{i}(t)=\partial _{t}h_{i}(t)=h_{i+1}(t)+h_{i-1}(t)-2h_{i}(t)+m^{2}(w-h_{i}(t))+F_{i}(h_{i}(t)),$

Note that in a stable configuration, where all the velocity are zero, the force acting on the line per unit length is $m^{2}(w-h_{CM}(t))$,

We consider a discrete version of the interface equation of motion in which the disorder can be imagined as a sequence of narrow wells. Each point of the interface is trapped in its well until it is pulled out of it and reach the next well. The obteained cellular automata is very similar too the elasto-plasstic models used for the yielding transition.

• Drive: Increasing $w$ each point of the interface is pulled with a slowly increasing force or stress:

$\sigma _{i}={\frac {1}{2}}(h_{i+1}+h_{i-1}-2h_{i})+m^{2}(w-h_{i})$

.

• Instability: An instability occurs when $\sigma _{i}$ reaches a local random threshold $\sigma _{i}^{th}$. Hence the point moves to the next well

${\begin{cases}\sigma _{i}=\sigma _{i}-\Delta \quad {\text{stress drop}}\\\\\sigma _{i\pm 1}=\sigma _{i\pm 1}+{\frac {1}{2}}{\frac {\Delta }{1+m^{2}}}\quad {\text{stress redistribution}}\\\end{cases}}$

Note that $\Delta$ is a positive random variable drwan from $g(\Delta )$.

• Avalanche: The two neighbours can be unstable... An avalanche can start...

Fully connected model (mean field)

Let's study the mean field version of the model where

• The elastic coupling is with all neighbours

$\sigma _{i}=h_{CM}-h_{i}+m^{2}(w-h_{i}),\quad$

.

• The local random threshold are all equal: $\sigma _{i}^{th}=1,\quad \forall i$. It is useful to introduce the distance from threshold

$x_{i}=1-\sigma _{i}$

Hence it is convenient to say that once a unstable point, $x_{i}<0$, is stabilized to a value $x>0$ drawn from $g(x)$ and induce a stress redistribution ${\frac {1}{L}}{\frac {x}{1+m^{2}}}$

In the limit $L\to \infty$ we define the distribution $P_{w}(x)$ and write its evolution equation.

• Drive: Changing $w\to w+dw$ gives $P_{w+dw}(x)=P_{w}(x+dw)\sim P_{w}(x)+m^{2}dw\partial _{x}P_{w}(x)$
• Instability: This shift is stable far from the origin, however for a fraction $m^{2}dwP_{w}(0)$ of the points of the interface is unstable. Due to the stress drop, their distance to instability will be $m^{2}dwP_{w}(0)g(x)$. Hence, one writes

$\partial _{w}P_{w}(x)\sim m^{2}\left[\partial _{x}P_{w}(x)+P_{w}(0)g(x)\right]$

• Stress redistribution: as a consequence all points move to the origin of

$m^{2}dwP_{w}(0){\frac {\overline {x}}{1+m^{2}}}$

where ${\overline {x}}=\int dxxg(x)$

• Avalanche: Let us call we can write

$\partial _{w}P_{w}(x)=m^{2}\left[\partial _{x}P_{w}(x)+P_{w}(0)g(x)\right]\left[1+P_{w}(0){\frac {\overline {x}}{1+m^{2}}}+(P_{w}(0){\frac {\overline {x}}{1+m^{2}}})^{2}+\ldots \right]$

and finally:

$\partial _{w}P_{w}(x)={\frac {m^{2}}{1-P_{w}(0){\frac {\overline {x}}{1+m^{2}}}}}\left[\partial _{x}P_{w}(x)+P_{w}(0)g(x)\right]$

Stationary solution

Increasing the drive the distribution converge to the fixed point:

$0=\partial _{x}P_{\text{stat}}(x)+P_{\text{stat}}(0)g(x)$

• Determne $P_{\text{stat}}(0)={\frac {1}{\overline {x}}}$ using

$1=\int _{0}^{\infty }dx\,P_{\text{stat}}(x)=-\int _{0}^{\infty }dx\,x\partial _{x}P_{\text{stat}}(x)$

• Show

$P_{\text{stat}}(x)={\frac {1}{\overline {x}}}\int _{x}^{\infty }g(z)dz$