LBan-V - Revision history

Rosso: /* First generation */

2026-03-03T08:07:30Z

First generation

← Older revision		Revision as of 10:07, 3 March 2026
Line 60:		Line 60:
	During the shift, the center of mass has not yet moved.		During the shift, the center of mass has not yet moved.

	* Stable ~~sites~~ (<math>x_i > k_0dw</math>):		* Stable blocks (<math>x_i > k_0dw</math>):
	<math display="block">		<math display="block">
	x_i^{t=1} = x_i - k_0dw.		x_i^{t=1} = x_i - k_0dw.
	</math>		</math>

	* ~~Sites~~ with <math>0 < x_i < k_0dw</math> ~~become~~ unstable.		* Blocks with <math>0 < x_i < k_0dw</math> are unstable. Since <math>dw</math> is infinitesimal, their fraction is

	Since <math>dw</math> is infinitesimal, their fraction is
	<math display="block">		<math display="block">
	P_w(0)\,k_0dw.		P_w(0)\,k_0dw.
	</math>		</math>

	~~These unstable blocks jump and stabilize at~~
	~~<math display="block">~~
	~~x_i^{t=1} = \Delta (1+k_0).~~
	~~</math>~~

	~~Here <math>~~\Delta~~</math> is a random variable~~ drawn from ~~the distribution <math>~~g(\Delta)~~</math>~~.
			Unstable blocks first relax to ($x_i=0$) and are
			then reinjected with a random kick $\Delta$ drawn from $g(\Delta)$,
			leading to
			\[
			x_i^{t=1} = (1+k_0)\Delta.
			\]
	Let <math>p(x)\,dx</math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.		Let <math>p(x)\,dx</math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
	By change of variables we have		By change of variables we have

Rosso: /* First generation */

2026-03-03T07:54:07Z

First generation

← Older revision		Revision as of 09:54, 3 March 2026
Line 90:		Line 90:
	<math display="block">		<math display="block">
	p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).		p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).
	\]		</math>

	=== Second generation ===		=== Second generation ===

Rosso: /* First generation */

2026-03-03T07:53:27Z

First generation

← Older revision		Revision as of 09:53, 3 March 2026
Line 78:		Line 78:

	Here <math>\Delta</math> is a random variable drawn from the distribution <math>g(\Delta)</math>.		Here <math>\Delta</math> is a random variable drawn from the distribution <math>g(\Delta)</math>.
	Let <math>p(x)\,dx<math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.		Let <math>p(x)\,dx</math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
	By change of variables we have		By change of variables we have
	<math display="block">		<math display="block">

Rosso: /* First generation */

2026-03-03T07:52:54Z

First generation

← Older revision		Revision as of 09:52, 3 March 2026
Line 72:		Line 72:
	</math>		</math>

	~~They~~ jump and stabilize at		These unstable blocks jump and stabilize at
	<math display="block">		<math display="block">
	x_i^{t=1} = \Delta(1+k_0).		x_i^{t=1} = \Delta (1+k_0).
	</math>		</math>

			Here <math>\Delta</math> is a random variable drawn from the distribution <math>g(\Delta)</math>.
			Let <math>p(x)\,dx<math> be the probability that an unstable block stabilizes in the interval <math>(x, x+dx)</math>.
			By change of variables we have
			<math display="block">
			p(x)\,dx = g(\Delta)\,d\Delta.
			</math>
			Since <math>x = (1+k_0)\Delta</math>, it follows that <math>\Delta = \frac{x}{1+k_0}</math> and
			<math display="block">
			d\Delta = \frac{dx}{1+k_0}.
			</math>
			Therefore,
			<math display="block">
			p(x) = \frac{1}{1+k_0}\, g\!\left(\frac{x}{1+k_0}\right).
			\]

	=== Second generation ===		=== Second generation ===

Rosso: /* Derivation of the Evolution Equation */

2026-03-02T14:47:25Z

Derivation of the Evolution Equation

← Older revision		Revision as of 16:47, 2 March 2026
Line 51:		Line 51:
	We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:		We shift the parabola by <math>w \to w + \mathrm{d}w</math>. Before the shift:
	<math display="block">		<math display="block">
	x_i(w) = 1 - k_0(w - h_i(w)) + (h_{CM}(w) - h_i(w)).		x_i(w) = 1 - k_0(w - h_i(w)) - (h_{CM}(w) - h_i(w)).
	</math>		</math>

Rosso: /* Higher generations */

2026-03-02T10:17:47Z

Higher generations

← Older revision		Revision as of 12:17, 2 March 2026
Line 112:		Line 112:
	The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.		The quantity <math>\overline{\Delta}P_w(0)</math> plays the role of a '''branching ratio''': it measures the average number of sites destabilized by one instability.

	~~We obtain~~		For stable sites
	<math display="block">		<math display="block">
	~~x \to x~~ - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.		x_i(w+dw) = x_i(w) - \frac{k_0}{1-\overline{\Delta}P_w(0)}\,dw.
	</math>		</math>
	~~and~~ a fraction		while unstable sites are reinjected at a random location <math>\Delta(1+k_0)</math>. The fraction of unstable sites is
	<math display="block">		<math display="block">
	\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw		\frac{P_w(0)}{1-\overline{\Delta}P_w(0)}k_0dw
	</math>		</math>
	is ~~reinjected at a random location <math>\Delta(1+k_0)</math>.~~		is

	This yields		This yields
	<math display="block">		<math display="block">

Rosso: /* Quasi-Static Protocol and Avalanche Definition */

2026-03-02T09:31:05Z

Quasi-Static Protocol and Avalanche Definition

← Older revision		Revision as of 11:31, 2 March 2026
Line 21:		Line 21:
	== Quasi-Static Protocol and Avalanche Definition ==		== Quasi-Static Protocol and Avalanche Definition ==

	To study avalanches, the position <math>w</math> is increased '''quasi-statically'''~~: it is shifted by an infinitesimal amount <math>w \to w + \mathrm{d}w</math>~~ so that the block closest to its instability threshold reaches it,		To study avalanches, the position <math>w</math> is increased '''quasi-statically''' so that the block closest to its instability threshold reaches it,
	<math display="block">x_i = 0.</math>		<math display="block">x_i = 0.</math>

Rosso: /* Avalanches at the Depinning Transition */

2026-03-02T09:19:36Z

Avalanches at the Depinning Transition

← Older revision		Revision as of 11:19, 2 March 2026
Line 17:		Line 17:
	</math>		</math>

	As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>L \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.		As <math>w</math> is increased quasistatically, the force <math>F(w)</math> would increase if <math>h_{CM}</math> were fixed. When an avalanche takes place, <math>h_{CM}</math> jumps forward and <math>F(w)</math> suddenly decreases. However, in the steady state and in the thermodynamic limit <math>N \to \infty</math>, the force recovers a well-defined value. In the limit <math>k_0 \to 0</math>, this force tends to the critical depinning force <math>F_c</math>; at finite <math>k_0</math> it lies slightly below <math>F_c</math>.

	== Quasi-Static Protocol and Avalanche Definition ==		== Quasi-Static Protocol and Avalanche Definition ==
Line 30:		Line 30:
	\begin{cases}		\begin{cases}
	x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]		x_i = 0 \;\longrightarrow\; x_i = \Delta (1 + k_0), \\[6pt]
	x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{L} \quad (j \neq i).		x_j \;\longrightarrow\; x_j - \dfrac{\Delta}{N} \quad (j \neq i).
	\end{cases}		\end{cases}
	</math>		</math>
Line 175:		Line 175:
	From order statistics:		From order statistics:
	<math display="block">		<math display="block">
	\int_0^{x_1}P_w(t)dt=\~~frac1L~~.		\int_0^{x_1}P_w(t)dt=\frac1N.
	</math>		</math>

	Thus		Thus
	<math display="block">		<math display="block">
	x_n \sim \frac{n}{~~LP_w~~(0)}.		x_n \sim \frac{n}{NP_w(0)}.
	</math>		</math>

	Each instability gives kicks <math>\Delta/L</math>.		Each instability gives kicks <math>\Delta/N</math>.

	Compare mean kick and mean gap:		Compare mean kick and mean gap:
	<math display="block">		<math display="block">
	\frac{\overline{\Delta}}{L}		\frac{\overline{\Delta}}{N}
	\quad \text{vs}\quad		\quad \text{vs}\quad
	\frac{1}{~~LP_w~~(0)}.		\frac{1}{NP_w(0)}.
	</math>		</math>

Line 207:		Line 207:
	Define the random increments		Define the random increments
	<math display="block">		<math display="block">
	\eta_1 = \frac{\Delta_1}{L}- x_1,		\eta_1 = \frac{\Delta_1}{N}- x_1,
	\quad		\quad
	\eta_2 = \frac{\Delta_2}{L}- (x_2-x_1),		\eta_2 = \frac{\Delta_2}{N}- (x_2-x_1),
	\quad		\quad
	\eta_3 = \frac{\Delta_3}{L}- (x_3-x_2),		\eta_3 = \frac{\Delta_3}{N}- (x_3-x_2),
	\ldots		\ldots
	</math>		</math>
Line 223:		Line 223:
	\overline{\eta}		\overline{\eta}
	=		=
	\frac{\overline{\Delta}}{L}		\frac{\overline{\Delta}}{N}
	-		-
	\frac{1}{L P_w(0)}.		\frac{1}{N P_w(0)}.
	</math>		</math>

Line 295:		Line 295:
	=		=
	-		-
	k_0\,\frac{\overline{\Delta}}{L}.		k_0\,\frac{\overline{\Delta}}{N}.
	</math>		</math>

Rosso at 20:08, 1 March 2026

2026-03-01T20:08:49Z

Show changes

Rosso: /* Second generation */

2026-02-28T17:27:29Z

Second generation

← Older revision		Revision as of 19:27, 28 February 2026
Line 123:		Line 123:
	</center>		</center>

	* Newly unstable fraction:		* the fraction of unstable sites grows Newly unstable fraction:

	<center>		<center>
	<math>		<math>
	\left(1+\overline{\Delta}P_w(0)\right)P_w(0)k_0dw.		P_w(0)k_0dw + \left(\overline{\Delta}P_w(0)\right)P_w(0)k_0dw .
	</math>		</math>
	</center>		</center>