Latest revision as of 22:06, 1 March 2026

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 examine how such systems can also be pinned and resist external deformation.

Disorder creates a complex energy landscape with many minima, maxima, and saddle points. When an external force is applied, the landscape is tilted. Local minima remain stable until a finite threshold is reached.

This gives rise to dynamical phase transitions.

Depinning vs Yielding

Two important transitions associated with pinning are:

Depinning transition.

Interfaces pinned by impurities appear in many contexts: magnetic domain walls, crack fronts, dislocations, vortices. Above a critical force $f_{c}$ , steady motion sets in. Close to threshold, motion is intermittent and proceeds via avalanches (e.g. Barkhausen noise).

Yielding transition.

Amorphous materials (foams, emulsions, pastes) deform elastically at small stress and flow at large stress. The analogue of the depinning threshold is the yield stress $σ_{y}$ , separating solid-like from flowing behavior.

Both the critical force per unit length $f_{c}$ and the yield stress $σ_{y}$ are self-averaging quantities, analogous to the free-energy density in equilibrium disordered systems. Sample-to-sample fluctuations are universal but subleading in system size. The two transitions share many phenomenological features (threshold, intermittency, avalanches) but differ in an important respect:

Depinning obeys monotonic dynamics (no-passing rule).
Yielding generally does not, due to stress redistributions of mixed sign.

Equation of Motion for Depinning

At zero temperature, in the overdamped regime, the interface evolves as $\partial_{t} h (x, t) = \nabla^{2} h (x, t) + f + F (x, h (x, t)), F (x, h) = - \frac{δ V (x, h)}{δ h} .$

Here:

$f$ is the external driving force,
$F (x, h)$ is the quenched disorder force.

The No-Passing Rule

Consider two interfaces evolving under the same disorder: $\partial_{t} h = \nabla^{2} h + f + F (x, h) .$

Let $h_{α} (x, 0) < h_{β} (x, 0) \forall x .$

Define their difference: $δ h (x, t) = h_{β} (x, t) - h_{α} (x, t) .$

Assume that at some first contact point $(x^{*}, t^{*})$ , $δ h (x^{*}, t^{*}) = 0 .$

Subtracting the equations of motion gives $\partial_{t} δ h = \nabla^{2} δ h + F (x, h_{β}) - F (x, h_{α}) .$

At the first contact:

$δ h = 0$ ,
$\nabla^{2} δ h \geq 0$ (minimum),
the disorder force is identical because it is quenched.

One finds that the velocity of the lower interface is strictly smaller than that of the upper one: $v_{α} (x^{*}, t^{*}) < v_{β} (x^{*}, t^{*}) .$

Thus crossing is impossible and ordering is preserved.

Consequences

Metastable states are totally ordered.
The critical force $f_{c}$ is independent of initial conditions.
For $f > f_{c}$ , no metastable states survive.

This monotonic structure is specific to depinning and does not hold for yielding systems.

Outlook

The depinning transition is simpler and better understood. In the following we focus on depinning and introduce minimal cellular automata capturing its physics. We will later return to yielding and discuss how similar automata can describe plastic flow.

Cellular Automaton for Depinning

We now introduce a discrete model in the depinning universality class. Time is discrete and the interface evolves through jumps between narrow pinning wells. The model captures threshold dynamics and avalanche propagation while remaining analytically tractable.

Degrees of freedom

The interface is represented by blocks of height $h_{1}, \dots, h_{N}$ .

Elastic interactions in finite dimension

In spatial dimension $d$ , each block interacts with its nearest neighbours: $F_{i}^{e l a s t} = \frac{1}{z} \sum_{j \in n n (i)} (h_{j} - h_{i}),$ where $z$ is the coordination number.

When a block jumps forward by $Δ$ , each neighbour receives an additional stress $Δ / z$ .

Narrow-well disorder

Each block is trapped in a sequence of narrow pinning wells along the $h$ -axis. Different blocks have independent trap sequences (translationally invariant disorder).

Each well has a local depinning threshold. Disorder may affect both the threshold values and the distances between wells. Here, for simplicity, we set all thresholds equal: $f_{Y} = 1 .$ The distances $Δ > 0$ between consecutive wells are random variables drawn from a distribution $g (Δ)$ . A common choice is exponential wells: $g (Δ) = e^{- Δ} .$

Open circles: trap positions. Filled circles: instantaneous interface configuration in $d = 1$ .

Driving protocols

Two drivings will be used in the course.

Constant force

$F_{i}^{d r i v e} = F .$

Displacement control

$F_{i}^{d r i v e} = k_{0} (w - h_{i}) .$

In this page we focus on constant force. Displacement control will be introduced later to study avalanches.

Distance to instability

Define $x_{i} = f_{Y} - F_{i}^{e l a s t} - F .$

Interpretation:

$x_{i} > 0$ : block stable.
$x_{i} \leq 0$ : block unstable.

The dynamics can be written entirely in terms of the variables $x_{i}$ .

Update rule

If a block $i$ becomes unstable, it jumps to the next well: $h_{i} \to h_{i} + Δ .$

In finite dimension, this induces an elastic redistribution of stress to its neighbours. Each neighbour receives an additional stress $Δ / z$ .

Fully connected limit

In high spatial dimension, elasticity becomes mean-field: $F_{i}^{e l a s t} = h_{C M} - h_{i}, h_{C M} = \frac{1}{N} \sum_{i} h_{i} .$

When block $i$ jumps by $Δ$ : $x_{i} \to x_{i} + Δ (1 - \frac{1}{N}), x_{j \neq i} \to x_{j} - \frac{Δ}{N} .$

The jumping site tends to stabilize, while all other sites are shifted uniformly toward instability. This homogeneous redistribution of stress is the origin of avalanche propagation.

Thermodynamic limit

In the fully connected model, there is no spatial structure. All blocks are statistically equivalent.

In the thermodynamic limit $N \to \infty$ , the state of the system at time $t$ is completely characterized by the distribution $P_{t} (x),$ the probability density of distances to instability.

Define the interface velocity $v^{t} = h_{C M} (t) - h_{C M} (t - 1) .$

The evolution of $x$ for a single block is:

If $x (t) > 0$ : $x (t + 1) = x (t) - v^{t + 1} .$

If $x (t) < 0$ : $x (t + 1) = x (t) - v^{t + 1} + Δ .$

Using the update rule for stable and unstable sites separately, one obtains: $P_{t + 1} (x) = P_{t} (x + v^{t + 1}) H (x + v^{t + 1}) + \int_{0}^{\infty} d Δ P_{t} (x + v^{t + 1} - Δ) g (Δ) H (- x - v^{t + 1} + Δ) .$

This equation fully describes the dynamics of the force-controlled model.

Stationary solutions

In the stationary state the velocity becomes constant $v$ , and $P_{t} (x) \to P (x)$ .

Solving this equation in the thermodynamic limit (see exercise) yields:

Deterministic critical force

$F_{c} = 1 - \frac{\overline{Δ^{2}}}{2 \overline{Δ}} .$

The critical force is self-averaging.

Velocity–force relation

The stationary velocity satisfies the implicit quadratic relation $v^{2} + 2 v (2 F_{c} - F - 1) - 2 \overline{Δ} (F_{c} - F) = 0 .$

This equation determines the full $v - F$ characteristic curve.

LBan-IV: Difference between revisions

Latest revision as of 22:06, 1 March 2026

Contents

Pinning and Depinning of a Disordered Material

Depinning vs Yielding

Equation of Motion for Depinning

The No-Passing Rule

Consequences

Outlook

Cellular Automaton for Depinning

Degrees of freedom

Elastic interactions in finite dimension

Narrow-well disorder

Driving protocols

Distance to instability

Update rule

Fully connected limit

Thermodynamic limit

Stationary solutions

Deterministic critical force

Velocity–force relation

Navigation menu

@@ Line 1: / Line 1: @@
 = 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.
+In earlier lectures, we discussed how disordered systems can become trapped in deep energy states, forming a glass. Today we examine how such systems can also be '''pinned''' and resist external deformation.
-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
+Disorder creates a complex energy landscape with many minima, maxima, and saddle points. When an external force is applied, the landscape is tilted. Local minima remain stable until a finite threshold is reached.
-* '''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.
+This gives rise to dynamical phase transitions.
-* '''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 vs Yielding ==
-== Depinning tranition: the equation of motion ==
+Two important transitions associated with pinning are:
-In the following we focus on the depinning trasition. At zero temperature and in the overdamped regime, where
-<math>\rho \partial_t^2 + \frac{1}{\mu} \partial_t \approx \frac{1}{\mu} \partial_t</math>, the equation of motion for the interface is:
-<center><math>
-\partial_t h(x,t) = \nabla^2 h^{\text{Zhiqiang}} + 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)}
-</math></center>
-Here we set <math>\mu=1</math>, <math> f </math> the external force and the disorder force is <math>F(x,h(x,t))</math>.
-=== The No-Passing Rule ===
+* '''Depinning transition.'''
+Interfaces pinned by impurities appear in many contexts: magnetic domain walls, crack fronts, dislocations, vortices.
+Above a critical force <math>f_c</math>, steady motion sets in. Close to threshold, motion is intermittent and proceeds via avalanches (e.g. Barkhausen noise).
-Interfaces obey the so-called no-passing rule. Consider two interfaces <math>h_\alpha(x,t)</math> and <math>h_\beta(x,t)</math> such that <math>h_\alpha(x,t=0) < h_\beta(x,t=0)</math> for every <math>x</math>. In the overdamped case, <math>\alpha</math> will never overtake <math>\beta</math>.
+* '''Yielding transition.'''
+Amorphous materials (foams, emulsions, pastes) deform elastically at small stress and flow at large stress.
+The analogue of the depinning threshold is the '''yield stress''' <math>\sigma_y</math>, separating solid-like from flowing behavior.
-To see why, assume for contradiction that at some time <math>t^*</math>, <math>\alpha</math> reaches <math>\beta</math> at a point <math>x^*</math>, i.e., <math>h_\alpha(x^*,t^*) = h_\beta(x^*,t^*)</math>. At this point, it can be shown that the local velocity of <math>\alpha</math>, denoted by <math>v_\alpha(x^*,t^*)</math>, is strictly less than the local velocity of <math>\beta</math>, <math>v_\beta(x^*,t^*)</math>.
+Both the critical force per unit length <math>f_c</math> and the yield stress <math>\sigma_y</math> are '''self-averaging quantities''', analogous to the free-energy density in equilibrium disordered systems. Sample-to-sample fluctuations are universal but subleading in system size. The two transitions share many phenomenological features (threshold, intermittency, avalanches) but differ in an important respect:
-This contradiction implies that the no-passing rule holds: <math>\alpha</math> cannot overtake <math>\beta</math>.
+* Depinning obeys monotonic dynamics (no-passing rule).
-An important consequence of the no-passing rule is that the value of the critical force <math>f_c</math> is independent of the initial condition. Indeed, if at a given force <math>f</math> the configuration <math>\beta</math> is a dynamically stable state, it will act as an impenetrable boundary for all configurations preceding it.
+* Yielding generally does not, due to stress redistributions of mixed sign.
-When <math>f = f_c</math>, the system possesses a single dynamically stable configuration. For <math>f > f_c</math>, no metastable states exist, and the system transitions into a fully moving phase.
+== Equation of Motion for Depinning ==
-= Cellular Automata =
+At zero temperature, in the overdamped regime, the interface evolves as
+<math display="block">
+\partial_t h(x,t)
+= \nabla^2 h(x,t) + f + F(x,h(x,t)),
+\qquad
+F(x,h) = - \frac{\delta V(x,h)}{\delta h}.
+</math>
-We now introduce a discrete version of the interface equation of motion.
+Here:
-These cellular automata belong to the same universality class as the original model,
-and they are straightforward to implement numerically.
-For clarity, we first discuss the case of one spatial dimension, <math>d = 1</math>.
-We then extend its definition.
-==The 1D model==
+* <math>f</math> is the external driving force,
-=== Step 1: Discretization along the ''x''-direction ===
+* <math>F(x,h)</math> is the quenched disorder force.
-The interface is represented as a collection of blocks <math>i = 1, \ldots, L</math>
+== The No-Passing Rule ==
-connected by springs with spring constant set to unity.
-The velocity of the <math>i</math>-th block is given by:
-<center><math>
+Consider two interfaces evolving under the same disorder:
-v_i(t) = \partial_t h_i(t) =
+<math display="block">
-\frac{1}{2}\bigl[h_{i+1}(t) + h_{i-1}(t) - 2 h_i(t)\bigr]
+\partial_t h = \nabla^2 h + f + F(x,h).
-+ f + F_i\!\bigl(h_i(t)\bigr) .
+</math>
-</math></center>
-Here, <math>h_i(t)</math> is the position of block <math>i</math> at time <math>t</math>,
+Let
-<math>f</math> is the external driving force, and <math>F_i</math> is the quenched random pinning force.
+<math display="block">
+h_\alpha(x,0) < h_\beta(x,0) \quad \forall x.
+</math>
-=== Step 2: Discretization along the ''h''-direction ===
+Define their difference:
+<math display="block">
+\delta h(x,t) = h_\beta(x,t) - h_\alpha(x,t).
+</math>
-The key simplification is the '''narrow-well approximation''' for the disorder potential.
+Assume that at some first contact point <math>(x^*,t^*)</math>,
-In this approximation, impurities act as pinning centers, each trapping a block of the interface at discrete positions up to a local threshold force  <math>\sigma_i^{th}</math> is overcomed. The distance between two consecutive pinning centers is a positive random variable
+<math display="block">
-<math>\Delta</math>, drawn from a distribution <math>g(\Delta)</math>.
+\delta h(x^*,t^*) = 0.
+</math>
-The total force acting on block <math>i</math> is:
+Subtracting the equations of motion gives
+<math display="block">
+\partial_t \delta h
+= \nabla^2 \delta h
++ F(x,h_\beta) - F(x,h_\alpha).
+</math>
-<center><math>
+At the first contact:
-\sigma_i = \frac{1}{2} \bigl(h_{i+1} + h_{i-1} - 2 h_i \bigr) + f .
-</math></center>
-As <math>f</math> is slowly increased, each block experiences a gradually increasing pulling force.
+* <math>\delta h = 0</math>,
-An instability occurs when:
+* <math>\nabla^2 \delta h \ge 0</math> (minimum),
-<center><math>
+* the disorder force is identical because it is quenched.
-\sigma_i \geq \sigma_i^{th} ,
-</math></center>
+One finds that the velocity of the lower interface is strictly smaller than that of the upper one:
+<math display="block">
+v_\alpha(x^*,t^*) < v_\beta(x^*,t^*).
+</math>
-When this condition is met, block <math>i</math> jumps to the next available well,
+Thus crossing is impossible and ordering is preserved.
-and the forces are updated as:
-<center><math>
+=== Consequences ===
-\begin{cases}
-\sigma_i \;\to\; \sigma_i - \Delta,\\[6pt]
-\sigma_{i \pm 1} \;\to\; \sigma_{i \pm 1} + \dfrac{\Delta}{2},
-\end{cases}
-</math></center>
-After such an instability, one of the neighboring blocks may also become unstable,
+* Metastable states are totally ordered.
-initiating a chain reaction.
+* The critical force <math>f_c</math> is independent of initial conditions.
+* For <math>f > f_c</math>, no metastable states survive.
+This monotonic structure is specific to depinning and does '''not''' hold for yielding systems.
-In the narrow wells approximation, the randomness of the disordered potential reduces to two random quantities: the distance between wells <math>\Delta</math> and the threshold <math>\sigma_i^{\rm th}</math> that must be overcome for the interface to escape the trapping well.
+== Outlook ==
-The universal properties of the depinning transition remain unchanged if one of these two quantities is taken as constant. Here, we choose:
+The depinning transition is simpler and better understood. In the following we focus on depinning and introduce minimal cellular automata capturing its physics. We will later return to yielding and discuss how similar automata can describe plastic flow.
-'''Uniform thresholds:'''
-All local thresholds are taken equal to one. The only remaining random variable is <math>\Delta</math>.
-==Extensions of the 1D model ==
+= Cellular Automaton for Depinning =
-The system’s dimensionality is encoded in the elastic force acting on each block. In spatial dimension <math>d</math>, the local force on block <math>i</math> is written as a sum over its nearest neighbours:
+We now introduce a discrete model in the depinning universality class.
+Time is discrete and the interface evolves through jumps between narrow pinning wells.
+The model captures threshold dynamics and avalanche propagation while remaining analytically tractable.
-<center><math> \sigma_i = \frac{1}{z} \sum_{j \in \mathrm{nn}(i)} \bigl(h_j - h_i\bigr) + f , </math></center>
+== Degrees of freedom ==
-where <math>z</math> is the coordination number, i.e. the number of nearest neighbours of each block. The value of <math>z</math> increases with the spatial dimension (e.g. <math>z=4</math> for a square lattice in <math>d=2</math>, <math>z=6</math> in <math>d=3</math>, and so on).
+The interface is represented by blocks of height <math>h_1,\ldots,h_N</math>.
-This form of the elastic force ensures that when a block becomes unstable and advances by an amount <math>\Delta</math>, its <math>z</math> neighbours each receive an extra stress <math>\Delta/z</math>.
+== Elastic interactions in finite dimension ==
-To describe the model in the limit of high dimension, it is convenient to replace the discrete Laplacian by a fully connected elasticity, corresponding to <math>z = L</math>. In this case, the force becomes:
+In spatial dimension <math>d</math>, each block interacts with its nearest neighbours:
+<math display="block">
+F_i^{\rm elast}
+= \frac{1}{z}\sum_{j\in nn(i)} (h_j-h_i),
+</math>
+where <math>z</math> is the coordination number.
-<center><math> \sigma_i = h_{\mathrm{CM}} - h_i + f , </math></center>
+When a block jumps forward by <math>\Delta</math>, each neighbour receives an additional stress <math>\Delta/z</math>.
-where <math>h_{\mathrm{CM}} = \frac{1}{L}\sum_{j=1}^{L} h_j</math> is the center-of-mass height.
+== Narrow-well disorder ==
-In the last part of the lecture we will solve the fully connected model explicitly. However, other elastic kernels are widely studied.
+Each block is trapped in a sequence of narrow pinning wells along the <math>h</math>-axis.
+Different blocks have independent trap sequences (translationally invariant disorder).
+Each well has a local depinning threshold.
+Disorder may affect both the threshold values and the distances between wells. Here, for simplicity, we set all thresholds equal:
+<math display="block">
+f_Y = 1.
+</math>
+The distances <math>\Delta>0</math> between consecutive wells are random variables drawn from a distribution <math>g(\Delta)</math>. A common choice is exponential wells:
+<math display="block">
+g(\Delta)=e^{-\Delta}.
+</math>
+[[File:WellsFigure.png|center|200px]]
+''Open circles: trap positions.
+Filled circles: instantaneous interface configuration in <math>d=1</math>.''
-. '''Long-range depinning kernels:'''
+== Driving protocols ==
-<center><math> \sigma_i = \sum_{j \ne i} \frac{h_j - h_i}{|j-i|^{d+\alpha}} + f , </math></center>
+Two drivings will be used in the course.
-Here the sum extends over all sites, but the contribution decays with distance. The parameter <math>\alpha</math> controls the interaction range and typically lies between <math>2/d</math> and <math>2</math>. For these values, the critical exponents depend continuously on <math>\alpha</math>. For <math>\alpha \ge 2</math>, one recovers the short-range results, while for <math>\alpha \le 2/d</math> one recovers the fully connected (mean-field) behavior. Many physical systems exhibit a long-range depinning transition; for instance, a 1D crack front corresponds to <math>\alpha = 1</math>. Importantly, the transition remains a depinning transition, and in particular the no-passing rule continues to hold.
+* '''Constant force'''
+<math display="block">
+F_i^{\rm drive}=F.
+</math>
-. '''Kernels that violate the no-passing rule:'''
+* '''Displacement control'''
+<math display="block">
+F_i^{\rm drive}=k_0(w-h_i).
+</math>
-In some systems, such as the yielding transition of amorphous solids, the elastic interactions are described by Eshelby kernels. These interactions are long-ranged, anisotropic, and have a quadrupolar symmetry with zero spatial sum (the stress released in one region is redistributed so that the net force on the system remains unchanged). Such kernels break the no-passing rule and lead to qualitatively different critical behavior, which we will discuss in the conclusions of the next lecture.
+In this page we focus on constant force.
+Displacement control will be introduced later to study avalanches.
+== Distance to instability ==
+Define
+<math display="block">
+x_i = f_Y - F_i^{\rm elast} - F.
+</math>
-== Velocity-Force Caracteristics==
+Interpretation:
-We define the interface velocity at time <math>t</math>:
-<center><math> v^{t} = h_{\mathrm{CM}}(t) - h_{\mathrm{CM}}(t-1) . </math></center>
+* <math>x_i>0</math>: block stable.
+* <math>x_i\le0</math>: block unstable.
-In the fully connected model, the blocks have no spatial structure, and therefore there are no privileged interactions (such as nearest- or next-nearest-neighbor couplings).
+The dynamics can be written entirely in terms of the variables <math>x_i</math>.
-For this reason, the state of the system at time <math>t</math> is entirely characterized by the distribution of the distance to instability of a single block, <math>P_t(x)</math>.
-To derive its evolution equation, we write the dynamics for a single block:
+== Update rule ==
-<center><math> x(t+1) = 1 - F - h_{\mathrm{CM}}(t+1) + h(t+1) . </math></center>
+If a block <math>i</math> becomes unstable, it jumps to the next well:
+<math display="block">
+h_i \to h_i + \Delta.
+</math>
-We must distinguish two cases:
+In finite dimension, this induces an elastic redistribution of stress to its neighbours.
+Each neighbour receives an additional stress <math>\Delta/z</math>.
-If <math>x(t) > 0</math>:
+== Fully connected limit ==
-<center><math> x(t+1) = 1 - F - h_{\mathrm{CM}}(t) - v^{t+1} + h(t) = x(t) - v^{t+1} . </math></center>
+In high spatial dimension, elasticity becomes mean-field:
+<math display="block">
+F_i^{\rm elast}=h_{\rm CM}-h_i,
+\qquad
+h_{\rm CM}=\frac{1}{N}\sum_i h_i.
+</math>
-If <math>x(t) < 0</math>:
+When block <math>i</math> jumps by <math>\Delta</math>:
+<math display="block">
+x_i \to x_i+\Delta\Bigl(1-\frac{1}{N}\Bigr),
+\qquad
+x_{j\ne i}\to x_j-\frac{\Delta}{N}.
+</math>
-<center><math> x(t+1) = 1 - F - h_{\mathrm{CM}}(t) - v^{t+1} + h(t) + \Delta = x(t) - v^{t+1} + \Delta . </math></center>
+The jumping site tends to stabilize, while all other sites are shifted uniformly toward instability.
+This homogeneous redistribution of stress is the origin of avalanche propagation.
-Using the Heaviside function <math>H(x)</math>, the evolution equation for <math>P_{t+1}(x)</math> can be written as the sum of these two contributions:
+== Thermodynamic limit ==
-<center><math> P_{t+1}(x) = P_t(x + v^{t+1})\, H(x + v^{t+1}) \;+\; \int_0^\infty d\Delta \; P_t(x + v^{t+1} - \Delta)\, g(\Delta)\, H(-x - v^{t+1} + \Delta) . </math></center>
+In the fully connected model, there is no spatial structure.
+All blocks are statistically equivalent.
-This equation fully describes the dynamics of the system, given an initial condition <math>P_0(x)</math> and a distribution of threshold distances <math>g(\Delta)</math>.
+In the thermodynamic limit <math>N\to\infty</math>, the state of the system at time <math>t</math> is completely characterized by the distribution
+<math display="block">
+P_t(x),
+</math>
+the probability density of distances to instability.
-We are now interested in stationary solutions, which become independent of the initial condition and are characterized by a constant stationary velocity <math>v</math>.
+Define the interface velocity
-In the stationary state, the equation reads:
+<math display="block">
+v^t = h_{\rm CM}(t)-h_{\rm CM}(t-1).
+</math>
-<center><math> P(x) = P(x + v)\, H(x + v) \;+\; \int_0^\infty d\Delta \; P(x + v - \Delta)\, g(\Delta)\, H(-x - v + \Delta) . </math></center>
+The evolution of <math>x</math> for a single block is:
-From this self-consistent equation, we want to derive a relation that expresses the stationary velocity <math>v</math> as a function of the external force <math>F</math>.
+If <math>x(t)>0</math>:
-To do this, we consider the first and second moments of the left- and right-hand sides.
+<math display="block">
+x(t+1)=x(t)-v^{t+1}.
+</math>
-It is useful to verify the following identity for a generic test function <math>\phi(x)</math>:
+If <math>x(t)<0</math>:
+<math display="block">
+x(t+1)=x(t)-v^{t+1}+\Delta.
+</math>
-<center><math> \int_{-\infty}^{\infty} dx \, \phi(x) P(x) = \int_{0}^{\infty} dx \, \phi(x-v) P(x) + \int_{-\infty}^{0} dx \, P(x) \int_{0}^{\infty} d\Delta \; \phi(x-v+\Delta)\, g(\Delta) . </math></center>
+Using the update rule for stable and unstable sites separately, one obtains:
-=== First moment ===
+<math display="block">
-Using <math>\phi(x) = x</math> we obtain the equation for the first moment:
+P_{t+1}(x)
+=
+P_t(x+v^{t+1})\,H(x+v^{t+1})
++
+\int_0^\infty d\Delta\,
+P_t(x+v^{t+1}-\Delta)\,g(\Delta)\,
+H(-x-v^{t+1}+\Delta).
+</math>
-<center><math> \overline{x} = \overline{x} - v + \overline{\Delta} \int_{-\infty}^{0} P(x)\, dx , </math></center>
+This equation fully describes the dynamics of the force-controlled model.
-from which we derive the relation connecting the stationary velocity to the fraction of unstable sites:
+== Stationary solutions ==
-<center><math> v = \overline{\Delta} \int_{-\infty}^{0} P(x)\, dx . </math></center>
+In the stationary state the velocity becomes constant <math>v</math>, and <math>P_t(x)\to P(x)</math>.
-This result shows that the mean velocity is proportional to the probability of finding an unstable site, with the proportionality factor given by the average jump size <math>\overline{\Delta}</math>.
+Solving this equation in the thermodynamic limit (see exercise) yields:
-=== Second moment ===
+=== Deterministic critical force ===
-Using <math>\phi(x) = x^2</math> and <math>\overline{x} = 1-F</math>  we obtain the equation for the second moment:
-<center><math> v^2 + 2 v (1-F -\frac{\overline{\Delta^2}}{2 \overline{\Delta}}) -2 \overline{\Delta} \int_{-\infty}^0 dx \, x P(x) =0  </center></math>
+<math display="block">
+F_c
+=
+-\frac{\overline{\Delta^2}}{2\overline{\Delta}}.
+</math>
+The critical force is self-averaging.
+=== Velocity–force relation ===
+The stationary velocity satisfies the implicit quadratic relation
+<math display="block">
+v^2
++2v(2F_c-F-1)
+-2\overline{\Delta}(F_c-F)
+=0.
+</math>
+This equation determines the full <math>v\!-\!F</math> characteristic curve.

LBan-IV: Difference between revisions

Latest revision as of 22:06, 1 March 2026

Pinning and Depinning of a Disordered Material

Depinning vs Yielding

Equation of Motion for Depinning

The No-Passing Rule

Consequences

Outlook

Cellular Automaton for Depinning

Degrees of freedom

Elastic interactions in finite dimension

Narrow-well disorder

Driving protocols

Distance to instability

Update rule

Fully connected limit

Thermodynamic limit

Stationary solutions

Deterministic critical force

Velocity–force relation

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