L-5

Goal : This is the first lecture on the dynamics of disordered systems. We will see that in different systems, disorder induces pinning up to a critical threshold. Around this threshold, the dynamics become intermittent and are characterized by large reorganizations called avalanches.

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 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 h} .Hence, 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(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

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 v_{CM} \sim |f-f_c|^\beta. }

• Two point correlation function:

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 \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 $f_{c}$

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

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 \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:

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) = S^{-\tau} f(S/S_\max), \quad S_\max \sim \xi(f)^{d+\zeta} \sim |f-f_c|^{-(d+\zeta)\nu} } 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(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
$z$	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 \ell(t) \sim t^{z} }	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 1.43\pm 0.01}
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 \zeta}	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 h(b x) \sim b^\zeta h(x)}	0	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 1.25\pm 0.01}
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 \nu}	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 \xi(f) \sim |f-f_c|^{-\nu}}	1/2	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 \nu= \frac{1}{2-\zeta}}
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 \beta}	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 v_{CM} \sim |f-f_c|^{\beta}}	1	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 \beta= \nu(z-\zeta)}
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}	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 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 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 d=1} case.

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

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 v_i(t) =\partial_t h_i(t)= \frac12 (h_{i+1}(t)+h_{i-1}(t) -2 h_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 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 m^2} is the spring constant, try to bring each block at the equilibrium position 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} . The velocity of each block becomes

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 v_i(t) =\partial_t h_i(t)= h_{i+1}(t)+h_{i-1}(t) -2 h_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 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 m^2(w-h_{CM}(t))} . 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 } the force slowly increases if $h_{CM}$ doest not move. When an avalanche occurs the force decreses.

Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along 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 h_i} impurities act as pinning center that trap the block around their position until a local 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}} is reached. In this limit the local velocities are zero when the block is trapped and quickly move to the next impurity once the threshold is overcome. The obteained cellular automata can be described by the following algorithm:

• Drive: Increasing $w$ 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= \frac 1 2 (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 its 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} +\frac{1}{2} \frac{\Delta}{1+m^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 drwan from 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(\Delta)} .

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