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 focus on zero temperature and on the overdamped regime. 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 characteristic

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

• Large force behaviour

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:

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

• 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:

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

To work below threshold it is useful to disretize the equation of motion and introduce the following protocol

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

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	$\beta =\nu (z-\zeta )$
$\tau$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(S) \sim S^{-\tau}}	3/2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau= 2 -\frac{2}{d+\zeta}}

Cellular Automata

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} each point of the interface is pulled with a slowly increasing force or stress:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i= (h_{i+1}+h_{i-1} -2 h_i) + m^2(w-h_i) }

.

• Instability: An instability occurs when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i } reaches a local random threshold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i^{th}} . Hence the point moves to the next well

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} \sigma_i=\sigma_i -\Delta \quad \text{stress drop}\\ \\ \sigma_{i\pm 1}=\sigma_{i\pm 1} +(\Delta-m^2)/2 \quad \text{stress redistribution}\\ \end{cases} }

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} is a positive random variable.

• 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i= h_{CM} - h_i + m^2(w-h_i), \quad }

.

• The local random threshold are all equal: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i^{th}=1, \quad \forall i } . It is useful to introduce the distance from threshold

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i = 1-\sigma_i }

Hence it is convenient to say that once a point is unstable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i<0 } the stress drop to a value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>0 } drawn from a distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) }

In the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\to \infty} we define the distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_w(x)} and write its evolution equation. Changing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w\to w+dw} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{w+dw}(x) = P_w(x+dw) \sim m^2 dw \partial_x P_w(x)}

This shift is stable far from the origin, however for a fraction $m^{2}dwP_{w}(0)$ of the points of the interface the shift is unstable. Their distance to unstability is redistibuted to $m^{2}dwP_{w}(0)g(x)$. Hence one writes

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