LBan-IV - Revision history

Rosso at 20:06, 1 March 2026

2026-03-01T20:06:33Z

2026-02-25T15:43:15Z

Narrow-well disorder

← Older revision		Revision as of 17:43, 25 February 2026
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	Each well has a local depinning threshold.		Each well has a local depinning threshold.
	Disorder may affect both the threshold values and the distances between wells.		Disorder may affect both the threshold values and the distances between wells. Here, for simplicity, we set all thresholds equal:
	Here, for simplicity, we set all thresholds equal:

	<center>		<center>
	<math>		<math>
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	</math>		</math>
	</center>		</center>
			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:
	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:

	<center>		<center>
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	</center>		</center>

	[[File:WellsFigure.png\|center\|~~300px~~]]		[[File:WellsFigure.png\|center\|200px]]

	''Open circles: trap positions.		''Open circles: trap positions.
	Filled circles: instantaneous interface configuration in <math>d=1</math>.''		Filled circles: instantaneous interface configuration in <math>d=1</math>.''

2026-02-25T15:42:34Z

Narrow-well disorder

← Older revision		Revision as of 17:42, 25 February 2026
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	</center>		</center>

	[[File:WellsFigure.png\|center\|~~600px~~]]		[[File:WellsFigure.png\|center\|300px]]

	''Open circles: trap positions.		''Open circles: trap positions.

2026-02-25T15:04:00Z

The No-Passing Rule

← Older revision		Revision as of 17:04, 25 February 2026
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	<center>		<center>
	<math>		<math>
	\~~Delta~~(x,t) = h_\beta(x,t) - h_\alpha(x,t).		\delta h(x,t) = h_\beta(x,t) - h_\alpha(x,t).
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\~~Delta~~(x^,t^) = 0.		\delta h(x^,t^) = 0.
	</math>		</math>
	</center>		</center>
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	<center>		<center>
	<math>		<math>
	\partial_t \~~Delta~~		\partial_t \delta h
	= \nabla^2 \~~Delta~~		= \nabla^2 \delta h
	+ F(x,h_\beta) - F(x,h_\alpha).		+ F(x,h_\beta) - F(x,h_\alpha).
	</math>		</math>
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	At the first contact:		At the first contact:

	* <math>\~~Delta~~ = 0</math>,		* <math>\delta h = 0</math>,
	* <math>\nabla^2 \~~Delta~~ \ge 0</math> (minimum),		* <math>\nabla^2 \delta h \ge 0</math> (minimum),
	* the disorder force is identical because it is quenched.		* the disorder force is identical because it is quenched.

2026-02-25T15:00:30Z

Cellular Automata

2026-02-25T08:33:00Z

2025-09-17T10:31:09Z

Velocity-Force Caracteristics

← Older revision		Revision as of 12:31, 17 September 2025
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	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).		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).
	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>.		For this reason, the state of the system at time <math>t</math> is entirely characterized by the distance to instability of a single block:
			<center><math> x(t) = 1 - F - h_{\mathrm{CM}}(t) + h(t) . </math></center>
			Our goal is to determine their distribution, <math>P_t(x)</math>.

	To derive ~~its~~ evolution equation, we write the dynamics for a single block:		To derive the evolution equation of <math>P_t(x)</math>, we write the dynamics for a single block:

	<center><math> x(t+1) = 1 - F - h_{\mathrm{CM}}(t+1) + h(t+1) . </math></center>		<center><math> x(t+1) = 1 - F - h_{\mathrm{CM}}(t+1) + h(t+1) . </math></center>

2025-09-16T19:09:40Z

Depinning tranition: the equation of motion

← Older revision		Revision as of 21:09, 16 September 2025
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	== Depinning tranition: the equation of motion ==		== Depinning tranition: the equation of motion ==
	In the following we focus on the depinning trasition. At zero temperature and in the overdamped regime~~, where~~		In the following we focus on the depinning trasition. At zero temperature and in the overdamped regime, the equation of motion for the interface is:
	~~<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>		<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)}		\partial_t h(x,t) = \nabla^2 h(x,t) + 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>		</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>.		Here <math> f </math> the external force and the disorder force is <math>F(x,h(x,t))</math>.

	=== The No-Passing Rule ===		=== The No-Passing Rule ===

2025-09-16T19:06:13Z

Extensions of the 1D model

← Older revision		Revision as of 21:06, 16 September 2025
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	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 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.



	== Velocity-Force Caracteristics==		== Velocity-Force Caracteristics==

2025-09-08T17:07:09Z

Second moment

← Older revision		Revision as of 19:07, 8 September 2025
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	<center><math> v^2 + 2 v (1-F -\frac{\overline{\Delta^2}}{2 \overline{\Delta}}) -2 \overline{\Delta} (1-F-\frac{\overline{\Delta^2}}{2 \overline{\Delta}}) =0 </math></center>		<center><math> v^2 + 2 v (1-F -\frac{\overline{\Delta^2}}{ \overline{\Delta}}) -2 \overline{\Delta} (1-F-\frac{\overline{\Delta^2}}{2 \overline{\Delta}}) =0 </math></center>
			Let's define


			<center><math> F_c= 1 -\frac{\overline{\Delta^2}}{ 2\overline{\Delta}} </math></center>

			we can write


			<center><math> v^2 + 2 v (2F_c -F -1) -2 \overline{\Delta} (F_c-F) =0 </math></center>