L-5 - Revision history

Rosso: /* Step 3: Discretization along the h direction */

2025-03-08T17:02:06Z

Step 3: Discretization along the h direction

← Older revision		Revision as of 19:02, 8 March 2025
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	<math>		<math>
	\begin{cases}		\begin{cases}
	\sigma_i = \sigma_i - \Delta \\\\		\sigma_i = \sigma_i - \Delta \\
	\sigma_{i\pm 1} = \sigma_{i\pm 1} + \frac{1}{2} \frac{\Delta}{1 + m^2} \\\\		\sigma_{i\pm 1} = \sigma_{i\pm 1} + \frac{1}{2} \frac{\Delta}{1 + m^2} \\
	\end{cases}		\end{cases}
	</math>		</math>

Rosso: /* Step 3: Discretization along the h direction */

2025-03-08T17:01:32Z

Step 3: Discretization along the h direction

← Older revision		Revision as of 19:01, 8 March 2025
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			The force <math>\sigma_i</math> is a local force, while the external force per unit length is
			<center><math> f(w) = \frac{1}{L} \sum_{i=1}^L \sigma_i = m^2(w - h_{CM})</math></center>
	The disorder potential creates a trapping force that freezes the position of the block in a narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but is set equal to 1 here for simplicity.		The disorder potential creates a trapping force that freezes the position of the block in a narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but is set equal to 1 here for simplicity.

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	<math>		<math>
	\begin{cases}		\begin{cases}
	\sigma_i = \sigma_i - \Delta ~~& \text{(stress drop)}~~ \\\\		\sigma_i = \sigma_i - \Delta \\\\
	\sigma_{i\pm 1} = \sigma_{i\pm 1} + \frac{1}{2} \frac{\Delta}{1 + m^2} ~~& \text{(stress redistribution)}~~ \\\\		\sigma_{i\pm 1} = \sigma_{i\pm 1} + \frac{1}{2} \frac{\Delta}{1 + m^2} \\\\
	\end{cases}		\end{cases}
	</math>		</math>
	</center>		</center>

	Here, <math>\Delta</math> is a positive random variable drawn from a distribution <math>g(\Delta)</math>, representing the magnitude of the ~~stress~~ drop. The distance between the wells is given by <math>(\Delta - m^2)/2</math>. After an instability, one of the two neighbors of the unstable block can become unstable. This marks the beginning of an avalanche.		Here, <math>\Delta</math> is a positive random variable drawn from a distribution <math>g(\Delta)</math>, representing the magnitude of the drop in the local force. The distance between the wells is given by <math>(\Delta - m^2)/2</math>. After an instability, one of the two neighbors of the unstable block can become unstable. This marks the beginning of an avalanche.

	Once the avalanche is over, its size <math>S</math> is a random variable defined as:		Once the avalanche is over, its size <math>S</math> is a random variable defined as:

Rosso: /* Step 3: Discretization along the h direction */

2025-03-08T16:55:33Z

Step 3: Discretization along the h direction

← Older revision		Revision as of 18:55, 8 March 2025
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	<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>		<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>



	The disorder potential creates a trapping force that freezes the position of the block in a narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but is set equal to 1 here for simplicity.		The disorder potential creates a trapping force that freezes the position of the block in a narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but is set equal to 1 here for simplicity.

Rosso: /* Step 3: Discretization along the h direction */

2025-02-15T13:26:39Z

Step 3: Discretization along the h direction

@@ Line 114: / Line 114: @@
 ====Step 3: Discretization along the  ''h'' direction====
-Finally, the most important step is to introduce the narrow-well approximation for the disorder potential. We imagine that along <math>h_i</math>, impurities act as pinning centers that trap the block of the interface around their position <math>h_i</math>. To be more precise: the force pulling the i-th  block is
+Finally, the most important step is to introduce the narrow-well approximation for the disorder potential. In this approximation, impurities act as pinning centers along trapping the block of the interface around their position <math>h_i</math>. This simplifies the analysis of the interface dynamics.
 <center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>
-The trapping force induced by the disorder froze the positin on the bloch in the narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but also set equal to unit as we do here.  Increasing <math>w</math>, each point of the interface is pulled with a slowly increasing force.   An instability occurs when
-<center><math>\sigma_i \ge \sigma_i^{th}</math></center>.
+The disorder potential creates a trapping force that freezes the position of the block in a narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but is set equal to 1 here for simplicity.
-Hence, the point moves to the next well:
-    <center><math> \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} </math></center> Here <math>\Delta</math> is a positive random variable drawn from <math>g(\Delta)</math>, you can derive that the distance between the wells is <math>(\Delta-m^2)/2</math>. After an instability on of the two neighbors of the unstable block can become unstable. This is the starting point of an avalanche. Once the avalanche is over, its size is a random variable :
+As <math>w</math> increases, each point of the interface is pulled with a slowly increasing force. An instability occurs when:
-<center><math>  S= \frac{1}{2}\sum_{i=1}^N (\Delta_i-m^2) </math> </center>
-where <math>N</math> is the number of instabilities which is a random number. Note that  replacing <math>sum_{i=1}^N \Delta_i</math> with <math>N \overline{\Delta}</math>, the size of the avalanche is proportional to <math>N</math>
+<center><math>\sigma_i \geq \sigma_i^{th}</math></center>

Rosso: /* Step 3: Discretization along the h direction */

2025-02-15T10:51:44Z

Step 3: Discretization along the h direction

← Older revision		Revision as of 12:51, 15 February 2025
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	Finally, the most important step is to introduce the narrow-well approximation for the disorder potential. We imagine that along <math>h_i</math>, impurities act as pinning centers that trap the block of the interface around their position <math>h_i</math>. To be more precise: the force pulling the i-th block is		Finally, the most important step is to introduce the narrow-well approximation for the disorder potential. We imagine that along <math>h_i</math>, impurities act as pinning centers that trap the block of the interface around their position <math>h_i</math>. To be more precise: the force pulling the i-th block is
	<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>		<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>
	The trapping force induced by the disorder froze the positin on the bloch in the narrow well up to a local threshold <math>\sigma_i^{th}</math>. Increasing <math>w</math>, each point of the interface is pulled with a slowly increasing force. ~~should be over.~~ An instability occurs when		The trapping force induced by the disorder froze the positin on the bloch in the narrow well up to a local threshold <math>\sigma_i^{th}</math>. This threshold can be random but also set equal to unit as we do here. Increasing <math>w</math>, each point of the interface is pulled with a slowly increasing force. An instability occurs when
	<center><math>\sigma_i \ge \sigma_i^{th}</math></center>.		<center><math>\sigma_i \ge \sigma_i^{th}</math></center>.
	Hence, the point moves to the next well:		Hence, the point moves to the next well:
	<center><math> \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} </math></center> ~~Note that~~ <math>\Delta</math> is a positive random variable drawn from <math>g(\Delta)</math>.		<center><math> \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} </math></center> Here <math>\Delta</math> is a positive random variable drawn from <math>g(\Delta)</math>, you can derive that the distance between the wells is <math>(\Delta-m^2)/2</math>. After an instability on of the two neighbors of the unstable block can become unstable. This is the starting point of an avalanche. Once the avalanche is over, its size is a random variable :
			<center><math> S= \frac{1}{2}\sum_{i=1}^N (\Delta_i-m^2) </math> </center>
	* Avalanche: The two neighbors can become unstable... An avalanche ~~can start...~~		where <math>N</math> is the number of instabilities which is a random number. Note that replacing <math>sum_{i=1}^N \Delta_i</math> with <math>N \overline{\Delta}</math>, the size of the avalanche is proportional to <math>N</math>

Rosso: /* Step 3: Discretization along the h direction */

2025-02-15T10:40:27Z

Step 3: Discretization along the h direction

← Older revision		Revision as of 12:40, 15 February 2025
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	====Step 3: Discretization along the ''h'' direction====		====Step 3: Discretization along the ''h'' direction====
	Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along <math>h_i</math>, impurities act as pinning centers that trap the block around their position ~~until a local threshold~~ <math>~~\sigma_i^{th}~~</math> ~~is reached~~. ~~In this limit, the block is trapped and moves to the next impurity only once the threshold is overcome.~~		Finally, the most important step is to introduce the narrow-well approximation for the disorder potential. We imagine that along <math>h_i</math>, impurities act as pinning centers that trap the block of the interface around their position <math>h_i</math>. To be more precise: the force pulling the i-th block is
	~~The obtained cellular automata can~~ be ~~described by~~ the ~~following algorithm:~~		<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>
			The trapping force induced by the disorder froze the positin on the bloch in the narrow well up to a local threshold <math>\sigma_i^{th}</math>. Increasing <math>w</math>, each point of the interface is pulled with a slowly increasing force. should be over. An instability occurs when
	* Drive: Increasing <math>w</math>, each point of the ~~interface~~ is ~~pulled with a slowly increasing force or stress:~~		<center><math>\sigma_i \ge \sigma_i^{th}</math></center>.
	<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>		Hence, the point moves to the next well:

	* Instability: An instability occurs when <math>\sigma_i</math> ~~reaches its~~ <math>\sigma_i^{th}</math>. Hence, the point moves to the next well:
	<center><math> \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} </math></center> Note that <math>\Delta</math> is a positive random variable drawn from <math>g(\Delta)</math>.		<center><math> \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} </math></center> Note that <math>\Delta</math> is a positive random variable drawn from <math>g(\Delta)</math>.

	* Avalanche: The two neighbors can become unstable... An avalanche can start...		* Avalanche: The two neighbors can become unstable... An avalanche can start...

Rosso: /* Step 2: Displacement control protocols */

2025-02-15T10:16:53Z

Step 2: Displacement control protocols

← Older revision		Revision as of 12:16, 15 February 2025
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	In this setup, the external force <math>f</math> is the control parameter. For a given <math>f</math> the interface reaches a dynamically stable configuration. An infinitesimal increase of math>f</math> triggers an avalanche. However, to gather avalanche size statistics, this approach requires finding a new dynamically stable configuration at the same <math>f</math> as before, which is computationally inefficient.		In this setup, the external force <math>f</math> is the control parameter. For a given <math>f</math> the interface reaches a dynamically stable configuration. An infinitesimal increase of math>f</math> triggers an avalanche. However, to gather avalanche size statistics, this approach requires finding a new dynamically stable configuration at the same <math>f</math> as before, which is computationally inefficient.

	====Step 2: Displacement control ~~protocols~~====		====Step 2: Displacement control protocol====

	An alternative is to control the displacement of the center of mass, <math> h_{\text{CM}}</math>, of the dynamically stable configuration. To achieve this, we introduce a parabolic potential which attracts each block to position <math>w</math> with a spring constant <math>m^2</math>. The velocity of each block becomes		An alternative is to control the displacement of the center of mass, <math> h_{\text{CM}}</math>, of the dynamically stable configuration. To achieve this, we introduce a parabolic potential which attracts each block to position <math>w</math> with a spring constant <math>m^2</math>. The velocity of each block becomes

Rosso: /* Step 3: Discretization along the h direction */

2025-02-14T20:58:46Z

Step 3: Discretization along the h direction

← Older revision		Revision as of 22:58, 14 February 2025
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	<math>f_c</math>, while at finite <math>m </math> is slightly below.		<math>f_c</math>, while at finite <math>m </math> is slightly below.

	====Step 3: Discretization along the ~~<math>~~h~~</math>~~ direction====		====Step 3: Discretization along the ''h'' direction====
	Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along <math>h_i</math> impurities act as pinning ~~center~~ that trap the block around their position until a local threshold <math> \sigma_i^{th}</math> 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.		Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along <math>h_i</math>, impurities act as pinning centers that trap the block around their position until a local threshold <math>\sigma_i^{th}</math> is reached. In this limit, the block is trapped and moves to the next impurity only once the threshold is overcome.
			The obtained cellular automata can be described by the following algorithm:

	~~The obteained cellular automata can be described by~~ the ~~following algorithm~~:		* Drive: Increasing <math>w</math>, each point of the interface is pulled with a slowly increasing force or stress:
			<center><math>\sigma_i = \frac{1}{2} (h_{i+1} + h_{i-1} - 2 h_i) + m^2(w - h_i)</math></center>

			* Instability: An instability occurs when <math>\sigma_i</math> reaches its <math>\sigma_i^{th}</math>. Hence, the point moves to the next well:
			<center><math> \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} </math></center> Note that <math>\Delta</math> is a positive random variable drawn from <math>g(\Delta)</math>.

	* ~~<Strong> Drive:</Strong> Increasing <math>w</math> each point of the interface is pulled with a slowly increasing force or stress:~~		* Avalanche: The two neighbors can become unstable... An avalanche can start...
	~~<center><math> \sigma_i= \frac 1 2 (h_{i+1}+h_{i-1} -2 h_i) + m^2(w-h_i) </math></center>.~~
	* <Strong> Instability:</Strong> An instability occurs when <math> \sigma_i </math> reaches its <math> \sigma_i^{th}</math>. Hence the point moves to the next well
	~~<center><math>~~
	~~\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}~~
	~~</math></center>~~
	~~Note that <math> \Delta</math> is a positive random variable drwan from <math> g(\Delta)</math>.~~
	* <Strong> Avalanche:~~</Strong>~~ The two ~~neighbours~~ can be unstable... An avalanche can start...

Rosso: /* Step 1: Discretization along the x direction */

2025-02-14T20:54:46Z

Step 1: Discretization along the x direction

← Older revision		Revision as of 22:54, 14 February 2025
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	Our goal is to study the avalanches observed as the critical force is approached from below. To this end, we introduce a discrete version of the interface's equation of motion. These cellular automata belong to the same universality class as the original model. They are simple to implement numerically, allow a straightforward definition of avalanches, and can be solved in the mean-field limit. For simplicity, we first discuss the <math>d=1</math> case.		Our goal is to study the avalanches observed as the critical force is approached from below. To this end, we introduce a discrete version of the interface's equation of motion. These cellular automata belong to the same universality class as the original model. They are simple to implement numerically, allow a straightforward definition of avalanches, and can be solved in the mean-field limit. For simplicity, we first discuss the <math>d=1</math> case.

	====Step 1: Discretization along the ~~<math>~~x~~</math>~~ direction====		====Step 1: Discretization along the ''x'' direction====

	The interface is represented as a collection of blocks <math> i=1, \ldots,L</math> connected by springs, where the spring constant is set to one. The velocity of the <math>i</math>-th block is given by:		The interface is represented as a collection of blocks <math> i=1, \ldots,L</math> connected by springs, where the spring constant is set to one. The velocity of the <math>i</math>-th block is given by:

Rosso: /* Step 3: Discretization along the {\displaystyle 4} direction */

2025-02-14T20:54:17Z

Step 3: Discretization along the {\displaystyle 4} direction

← Older revision		Revision as of 22:54, 14 February 2025
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	<math>f_c</math>, while at finite <math>m </math> is slightly below.		<math>f_c</math>, while at finite <math>m </math> is slightly below.

	====Step 3: Discretization along the ~~{\displaystyle 4}~~ direction====		====Step 3: Discretization along the <math>h</math> direction====
	Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along <math>h_i</math> impurities act as pinning center that trap the block around their position until a local threshold <math> \sigma_i^{th}</math> 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.		Finally, the most important step is to introduce the narrow-well approximation for the disorder. We imagine that along <math>h_i</math> impurities act as pinning center that trap the block around their position until a local threshold <math> \sigma_i^{th}</math> 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.