L-6 - Revision history

Rosso: /* Bienaymé Galton Watson process */

2025-09-09T07:54:00Z

Bienaymé Galton Watson process

← Older revision		Revision as of 09:54, 9 September 2025
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	<center> <math> S_{\max} \sim m^{-4}</math> </center>		<center> <math> S_{\max} \sim m^{-4}</math> </center>

	== Bienaymé Galton Watson process==		= Bienaymé Galton Watson process=

Rosso: /* Avalanches or instability? */

2025-03-09T08:21:40Z

Avalanches or instability?

← Older revision		Revision as of 10:21, 9 March 2025
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	=== Avalanches or instability?===		=== Avalanches or instability?===
	We consider an avalanche starting from a single unstable site <math> x_0=0 </math> and the sequence of sites more close to instabitity <math> x_1< x_2<x_3\ldots </math>. For each unstable block, all the blocks receive a random kick:		We consider an avalanche starting from a single unstable site <math> x_0=0 </math> and the sequence of sites more close to instabitity <math> x_1< x_2<x_3\ldots </math>. For each unstable block, all the blocks receive a random kick:
	<center><math> \frac{\Delta_1}{(1+m^2)L}, \frac{\Delta_2}{(1+m^2)L}, \frac{\Delta_3}{(1+m^2)L}, \ldots</math></center>		<center><math> \frac{\Delta_1}{(1+m^2)L},\quad \frac{\Delta_2}{(1+m^2)L}, \quad \frac{\Delta_3}{(1+m^2)L}, \ldots</math></center>
	with <math> \Delta_1,\Delta_2,\Delta_3, \~~lodts~~ </math> drwan from <math> g(\Delta) </math> Are these kick able to destabilize other blocks?		with <math> \Delta_1,\Delta_2,\Delta_3, \ldots </math> drwan from <math> g(\Delta) </math> Are these kick able to destabilize other blocks?

Rosso: /* Avalanches or instability? */

2025-03-09T08:20:01Z

Avalanches or instability?

← Older revision		Revision as of 10:20, 9 March 2025
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	=== Avalanches or instability?===		=== Avalanches or instability?===
	We consider an avalanche starting from a single unstable site <math> x_0=0 </math> and the sequence of sites more close to instabitity <math> x_1< x_2<x_3\ldots </math>. For each unstable block, all the blocks receive a random kick:		We consider an avalanche starting from a single unstable site <math> x_0=0 </math> and the sequence of sites more close to instabitity <math> x_1< x_2<x_3\ldots </math>. For each unstable block, all the blocks receive a random kick:
	<center><math> \frac{\Delta_1}{(1+m^2)L}, \frac{\Delta_2}{(1+m^2)L}, \frac{\Delta_3}{(1+m^2)L}, \~~lodts~~</math></center>		<center><math> \frac{\Delta_1}{(1+m^2)L}, \frac{\Delta_2}{(1+m^2)L}, \frac{\Delta_3}{(1+m^2)L}, \ldots</math></center>
	with <math> \Delta_1,\Delta_2,\Delta_3, \lodts </math> drwan from <math> g(\Delta) </math> Are these kick able to destabilize other blocks?		with <math> \Delta_1,\Delta_2,\Delta_3, \lodts </math> drwan from <math> g(\Delta) </math> Are these kick able to destabilize other blocks?

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	<center><math> x_1 \sim \frac{1}{L P_w(0)}, \; x_2 \sim \frac{2}{L P_w(0)}, \; x_3 \sim \frac{3}{L P_w(0)}, \ldots </math></center>		<center><math> x_1 \sim \frac{1}{L P_w(0)}, \; x_2 \sim \frac{2}{L P_w(0)}, \; x_3 \sim \frac{3}{L P_w(0)}, \ldots </math></center>
	Hence we need to compare the mean value of the kick with the mean gap between nearest unstable sites:		Hence we need to compare the mean value of the kick with the mean gap between nearest unstable sites:
	<center><math> \frac{\overline{\Delta}}{(1+m^2)L} \quad \text{versus }\quad 1 /( P_w(0) L) </math></center>		<center><math> \frac{\overline{\Delta}}{(1+m^2)L} \quad \text{versus }\quad \frac{1}{ P_w(0) L} </math></center>
	Note that <math>L </math> simplifies. We expect three possibilities:		Note that <math>L </math> simplifies. We expect three possibilities:
	* if the mean kick is smaller than the mean gap the system is subcritical and avalanches quickly stops.		* if the mean kick is smaller than the mean gap the system is subcritical and avalanches quickly stops.

Rosso: /* Avalanches or instability? */

2025-03-09T08:18:55Z

Avalanches or instability?

← Older revision		Revision as of 10:18, 9 March 2025
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	=== Avalanches or instability?===		=== Avalanches or instability?===
			We consider an avalanche starting from a single unstable site <math> x_0=0 </math> and the sequence of sites more close to instabitity <math> x_1< x_2<x_3\ldots </math>. For each unstable block, all the blocks receive a random kick:
			<center><math> \frac{\Delta_1}{(1+m^2)L}, \frac{\Delta_2}{(1+m^2)L}, \frac{\Delta_3}{(1+m^2)L}, \lodts</math></center>
			with <math> \Delta_1,\Delta_2,\Delta_3, \lodts </math> drwan from <math> g(\Delta) </math> Are these kick able to destabilize other blocks?

	Given the initial condition and <math> w </math>, the state of the system is described by <math> P_w(x) </math>. ~~For each unstable block, all~~ the ~~blocks receive a kick. The mean value of~~ the ~~kick is~~
	~~<center><math> x_{\text{kick}}= \frac{\overline{\Delta}}{(1+m^2)L} </math></center>~~		Given the initial condition and <math> w </math>, the state of the system is described by <math> P_w(x) </math>. From the extreme values theory we know the equation setting the average position of the most unstable block is
	~~Is this kick able to destabilize another block? The~~ equation setting the average position of the most unstable block is
	<center><math> \int_0^{x_1} P_w(t) dt =\frac{1}{L} </math></center>		<center><math> \int_0^{x_1} P_w(t) dt =\frac{1}{L} </math></center>
	Hence, for large systems we have		Hence, for large systems we have
	<center><math> x_1 \sim \frac{1}{L P_w(0)}, \; x_2 \sim \frac{2}{L P_w(0)}, \; x_3 \sim \frac{3}{L P_w(0)}, \ldots </math></center>		<center><math> x_1 \sim \frac{1}{L P_w(0)}, \; x_2 \sim \frac{2}{L P_w(0)}, \; x_3 \sim \frac{3}{L P_w(0)}, \ldots </math></center>
	~~We expect three possibilities:~~		Hence we need to compare the mean value of the kick with the mean gap between nearest unstable sites:
	* if the mean kick, <math>\~~sim~~ \overline{\Delta}/(1+m^2) </math> ~~is smaller than the mean gap~~ <math>~~\sim 1 /P_w(0)~~</math>, the system is subcritical and avalanches quickly stops.		<center><math> \frac{\overline{\Delta}}{(1+m^2)L} \quad \text{versus }\quad 1 /( P_w(0) L) </math></center>
	* if the mean kick~~, <math>\sim \overline{\Delta}/(1+m^2) </math>~~ is equal to the mean gap ~~<math>\sim 1 /P_w(0)</math>,~~ the system is critical and avalanches are power law distributed		Note that <math>L </math> simplifies. We expect three possibilities:
	* if the mean kick~~, <math>\sim \overline{\Delta}/(1+m^2) </math>~~ is larger of the mean gap ~~<math>\sim 1 /P_w(0)</math>,~~ the system is super-critical and avalanches are unstable.		* if the mean kick is smaller than the mean gap the system is subcritical and avalanches quickly stops.
			* if the mean kick is equal to the mean gap the system is critical and avalanches are power law distributed
			* if the mean kick is larger of the mean gap the system is super-critical and avalanches are unstable.

	Note that in the stationary regime the system is subcritical when <math>m>0 </math> and critical for <math>m=0 </math>		Note that in the stationary regime the ratio between mean kick and mean gap is <math> 1/(1+m^2) </math>. Hence, the system is subcritical when <math>m>0 </math> and critical for <math>m=0 </math>


	====Mapping to the Brownian motion====		====Mapping to the Brownian motion====

Rosso: /* Dynamics */

2025-03-08T19:04:03Z

Dynamics

← Older revision		Revision as of 21:04, 8 March 2025
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	* <Strong> Stabilization : </Strong> A fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable. The stabilization induces the change <math>m^2 d w P_w(0) \to m^2 d w P_w(0) g(x) </math>. Hence, one writes		* <Strong> Stabilization : </Strong> A fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable. The stabilization induces the change <math>m^2 d w P_w(0) \to m^2 d w P_w(0) g(x) </math>. Hence, one writes
	<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>		<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>
			The stabilization of the unstable blocks induce a drop of the force per unit length
			<center><math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math> </center> \

			* <Strong> Redistribution </Strong> This drop is (partially) compensated by the redistribution. The force acting on all points is increased:
	* <Strong> Redistribution </Strong~~> The total drop on the force acting on the unstable blocks <math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math~~> ~~per unit length.~~ This drop is partially compensated by the redistribution. The force acting on all points is increased:
	<center> <math> x_i \to x_i - m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center>		<center> <math> x_i \to x_i - m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center>
	Again, most of the distribution will be driven to instability while a fraction of the blocks become unstable... we can write		Again, most of the distribution will be driven to instability while a fraction of the blocks become unstable... we can write

Rosso: /* Fully connected (mean field) model for the cellular automaton */

2025-03-08T18:49:57Z

Fully connected (mean field) model for the cellular automaton

← Older revision		Revision as of 20:49, 8 March 2025
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	<center><math> x_i = 1-\sigma_i		<center><math> x_i = 1-\sigma_i
	</math></center>		</math></center>
	~~Our goal~~ is ~~thus to determine their distribution~~ <math>~~P_w(x)~~</math>~~, given their intial distribution,~~ <math>~~P_0~~(x)</math>~~, and a value of <math> w~~ </~~math~~>.		The instability occurs when a block is at <math> x_i =0 </math> and is followed by its stabilization and a redistribution on all the blocks :

			<center>
			<math>
			\begin{cases}
			x_i=0 \to x_i= \Delta & (stabilization) \\
			x_{j} \to x_j- \frac{1}{L} \frac{\Delta}{1 + m^2} & (redistribution) \\
			\end{cases}
			</math>
			</center>


	=== Dynamics ===		=== Dynamics ===
	Let's ~~rewrite~~ the dynamics ~~with the new variables~~		Our goal is thus to determine the distribution <math>P_w(x)</math> of all blocks, given their intial distribution, <math>P_0(x)</math>, and a value of <math> w </math>.
			Let's decompose in steps the dynamics

	* <Strong> Drive:</Strong> Increasing <math>w \to w + dw</math> each ~~point~~ decreases its distance to threshold		* <Strong> Drive:</Strong> Increasing <math>w \to w + dw</math> each block decreases its distance to threshold
	<center><math> x_i \to x_i - m^2 dw </math></center>.		<center><math> x_i \to x_i - m^2 dw </math></center>.
	As a consequence		As a consequence
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	* <Strong> ~~Instability~~ : </Strong> ~~The instability occurs when a point is at <math> x_i =0 </math> and is followed by a stabilization:~~		* <Strong> Stabilization : </Strong> A fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable. The stabilization induces the change <math>m^2 d w P_w(0) \to m^2 d w P_w(0) g(x) </math>. Hence, one writes
	~~<center> <math> x_i =0 \to x_i = \Delta </math></center> with <math>\Delta </math> drawn from <math> g(\Delta) </math>.~~
	~~In our case,~~ fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable. The stabilization induces the change <math> m^2 d w P_w(0) g(x) </math>. Hence, one writes
	<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>		<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>


	* <Strong> Redistribution </Strong> ~~The stabilization of a single block induces a redistribution where all blocks approach threshold.~~		* <Strong> Redistribution </Strong> The total drop on the force acting on the unstable blocks <math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math> per unit length. This drop is partially compensated by the redistribution. The force acting on all points is increased:
	~~<center><math>~~		<center> <math> x_i \to x_i - m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center>
	~~x_i \to = x_i - \frac{1}{L} \frac{\Delta}{1+m^2}~~		Again, most of the distribution will be driven to instability while a fraction of the blocks become unstable... we can write
	~~</math></center>~~
	The total drop on the force acting on the unstable blocks <math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math> per unit length.
	This drop in partially compensated by the redistribution. The force acting on all points is increased of
	<center> <math> m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center>
	Again, most of the distribution will be driven to instability while a ~~franction~~ of the blocks become unstable... we can write
	<center> <math>\partial_w P_{w}(x) = m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] \left[ 1+P_w(0) \frac{\overline{\Delta}}{1+m^2} + (P_w(0) \frac{\overline{\Delta}}{1+m^2})^2 +\ldots\right] </math> </center>		<center> <math>\partial_w P_{w}(x) = m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] \left[ 1+P_w(0) \frac{\overline{\Delta}}{1+m^2} + (P_w(0) \frac{\overline{\Delta}}{1+m^2})^2 +\ldots\right] </math> </center>
	and finally:		and finally:

Rosso at 18:18, 8 March 2025

2025-03-08T18:18:39Z

← Older revision		Revision as of 20:18, 8 March 2025
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	* <Strong> Instability : </Strong> The instability occurs when a point is at <math> x_i =0 </math>~~. Then, the point~~ is ~~stabilized~~:		* <Strong> Instability : </Strong> The instability occurs when a point is at <math> x_i =0 </math> and is followed by a stabilization:
	<center> <math> x_i =0 \to x_i = \Delta </math></center> with <math>\Delta </math> drawn from <math> g(\Delta) </math>		<center> <math> x_i =0 \to x_i = \Delta </math></center> with <math>\Delta </math> drawn from <math> g(\Delta) </math>.
	~~The~~ fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable ~~and the~~ stabilization induces the change <math> m^2 d w P_w(0) g(x) </math>. Hence, one writes		In our case, fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable. The stabilization induces the change <math> m^2 d w P_w(0) g(x) </math>. Hence, one writes
	<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>		<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>


	* <Strong> Redistribution </Strong> The ~~stress drop~~ of a single block induces a ~~stress~~ redistribution where all blocks approach threshold.		* <Strong> Redistribution </Strong> The stabilization of a single block induces a redistribution where all blocks approach threshold.
	<center><math>		<center><math>
	x_i \to = x_i - \frac{1}{L} \frac{\Delta}{1+m^2}		x_i \to = x_i - \frac{1}{L} \frac{\Delta}{1+m^2}
	</math></center>		</math></center>
	The total ~~stress~~ drop is <math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math> ~~hence~~ all points ~~move to the origin~~ of		The total drop on the force acting on the unstable blocks <math> m^2 d w P_w(0) \int d x x g(x) = m^2 d w P_w(0) \overline{\Delta} </math> per unit length.
			This drop in partially compensated by the redistribution. The force acting on all points is increased of
	<center> <math> m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center>		<center> <math> m^2 dw P_w(0) \frac{\overline{\Delta}}{1+m^2} </math> </center>
	~~part~~ of ~~them shifts, part~~ of ~~them~~ become unstable... we can write		Again, most of the distribution will be driven to instability while a franction of the blocks become unstable... we can write
	<center> <math>\partial_w P_{w}(x) = m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] \left[ 1+P_w(0) \frac{\overline{\Delta}}{1+m^2} + (P_w(0) \frac{\overline{\Delta}}{1+m^2})^2 +\ldots\right] </math> </center>		<center> <math>\partial_w P_{w}(x) = m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] \left[ 1+P_w(0) \frac{\overline{\Delta}}{1+m^2} + (P_w(0) \frac{\overline{\Delta}}{1+m^2})^2 +\ldots\right] </math> </center>
	and finally:		and finally:

Rosso: /* Dynamics */

2025-03-08T17:41:23Z

Dynamics

← Older revision		Revision as of 19:41, 8 March 2025
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	* <Strong> Instability 1: ~~Stress drop~~ </Strong> The instability occurs when a point is at <math> x_i =0 </math>. Then, the point is stabilized ~~(stress drop)~~:		* <Strong> Instability : </Strong> The instability occurs when a point is at <math> x_i =0 </math>. Then, the point is stabilized:
	<center> <math> x_i =0 \to x_i = \Delta </math></center>		<center> <math> x_i =0 \to x_i = \Delta </math></center> with <math>\Delta </math> drawn from <math> g(\Delta) </math>
	~~Increasing~~ <math>w \~~to w + dw~~</math>, a fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable~~. Due to~~ the ~~stress drop, their distance to threshold becomes~~ <math> m^2 d w P_w(0) g(x) </math>. Hence, one writes		The fraction <math> m^2 d w P_w(0) </math> of the blocks is unstable and the stabilization induces the change <math> m^2 d w P_w(0) g(x) </math>. Hence, one writes
	<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>		<center> <math> \partial_w P_{w}(x) \sim m^2 \left[\partial_x P_w(x) + P_w(0) g(x) \right] </math> </center>


	* <Strong> ~~Instability 2: Stress redistribution~~ </Strong> The stress drop of a single block induces a stress redistribution where all blocks approach threshold.		* <Strong> Redistribution </Strong> The stress drop of a single block induces a stress redistribution where all blocks approach threshold.
	<center><math>		<center><math>
	x_i \to = x_i - \frac{1}{L} \frac{\Delta}{1+m^2}		x_i \to = x_i - \frac{1}{L} \frac{\Delta}{1+m^2}

Rosso: /* Bienaymé Galton Watson process */

2025-02-15T18:04:39Z

Bienaymé Galton Watson process

← Older revision		Revision as of 20:04, 15 February 2025
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	</math></center>. Note that the normalization imposes <math> Q_0(t)=1 </math>.		</math></center>. Note that the normalization imposes <math> Q_0(t)=1 </math>.

	* <Strong> ~~Backward approach~~</Strong>: ~~derive~~ the following equation for <math> Q_s(t)</math>		* <Strong> Evolution equation</Strong>: Consider the evolution up to the time <math> t+dt</math> as a first evolution from <math> 0</math> to <math> dt</math> and a following evolution from <math> dt</math> to <math> t+ dt</math>. Derive the following equation for <math> Q_s(t)</math>
	<center> <math> Q_s(t+dt) = (1-(a+b) d t) e^{-s dt} Q_s(t) +a dt + b dt Q_s^2(t) +O(dt^2) </math></center>		<center> <math> Q_s(t+dt) = (1-(a+b) d t) e^{-s dt} Q_s(t) +a dt + b dt Q_s^2(t) +O(dt^2) </math></center>
	which gives		which gives

Rosso: /* Bienaymé Galton Watson process */

2025-02-15T16:38:29Z

Bienaymé Galton Watson process

← Older revision		Revision as of 18:38, 15 February 2025
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	A time <math> t=0 </math> appears as infected individual which dies with a rate <math> a </math> and branches with a rate <math> b </math>. On average, each infection generates <math> R_0 = b/a </math> new		A time <math> t=0 </math> appears as infected individual which dies with a rate <math> a </math> and branches with a rate <math> b </math>. On average, each infection generates in average <math> R_0 = b/a </math> new
	ones. Real epidemics corresponds to <math> R_0>1 </math>.		ones. Real epidemics corresponds to <math> R_0>1 </math>.