L-5

Goal: This is the first lecture on the dynamics of disordered systems. We will explore how disorder in various systems induces pinning up to a critical threshold. Near this threshold, the dynamics become intermittent and are dominated by large reorganizations known as avalanches.

Pinning and Depinning of a Disordered Material

In earlier lectures, we discussed how disordered systems can become trapped in deep energy states, forming a glass. Today, we will examine how such systems can also be pinned and resist external deformation. This behavior arises because disorder creates a complex energy landscape with numerous features, including minima (of varying depth), maxima, and saddle points.

When an external force is applied, it tilts this multidimensional energy landscape in a specific direction. However, local minima remain stable until a finite critical threshold is reached. Two important dynamical phase transitions are 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 to vortices in superconductors. Above a critical force, interfaces depin, their motion becomes intermittent, and a Barkhausen noise is detected.

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

Depinning tranition: the equation of motion

In the following we focus on the depinning trasition.

At zero temperature and in the overdamped regime, where  

$\rho \partial _{t}^{2}+{\frac {1}{\mu }}\partial _{t}\approx {\frac {1}{\mu }}\partial _{t}$, the equation of motion for the interface is:

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

Here we set$\mu =1$, $f$ the external force and the disorder force is $F(x,h(x,t))$. Again we can consider a gaussian force of zero mean and correlations:

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

There are usually two kinds of disorder:

• (i) Random Bond (RB): If $V(x,h)$ is short-range correlated, the area below 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 zero.
• (ii) Random Field: If 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(x,h)} behaves like 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} , then 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 correlated.

The No-Passing Rule

Interfaces obey the so-called no-passing rule. Consider two interfaces 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_\alpha(x,t)} and 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_\beta(x,t)} such 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 h_\alpha(x,t=0) < h_\beta(x,t=0)} for every 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} . In the overdamped case, 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 \alpha} will never overtake 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} .

To see why, assume for contradiction that at some time 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 t^*} , 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 \alpha} reaches 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} at a point 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.e., 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_\alpha(x^*,t^*) = h_\beta(x^*,t^*)} . At this point, it can be shown that the local velocity 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 \alpha} , denoted by 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_\alpha(x^*,t^*)} , is strictly less than the local velocity 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 \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_\beta(x^*,t^*)} .

This contradiction implies that the no-passing rule holds: 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 \alpha} cannot overtake 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} . An important consequence of the no-passing rule is that the value of the critical force 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 f_c} is independent of the initial condition. Indeed, if at a given force 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 f} the configuration 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} is a dynamically stable state, it will act as an impenetrable boundary for all configurations preceding it.

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 f = f_c} , the system possesses a single dynamically stable configuration. For 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 f > f_c} , no metastable states exist, and the system transitions into a fully moving phase.

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 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 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 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 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} }

Depinning exponents

Exponent	Observable	Mean field	d=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 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}}
$\beta$	$v_{CM}\sim |f-f_{c}|^{\beta }$	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 $m^{2}(w-h_{CM}(t))$. Increasing $w$ 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 $h_{i}$ impurities act as pinning center that trap the block around their position until a local threshold $\sigma _{i}^{th}$ 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:

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

.

• Instability: An instability occurs when $\sigma _{i}$ reaches its $\sigma _{i}^{th}$. 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...