Consider the Bak-Tang-Wiesenfeld sandpile on a grid where every site has toppling…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Linear algebra and vector spaces

Many Parts, One Behaviour, Ch Self-organised criticality and avalanches

Problem

Consider the Bak-Tang-Wiesenfeld sandpile on a grid where every site has toppling threshold

z_c=4

: whenever

z_i\ge4

, toppling site

i

removes

4

grains from

i

and adds one grain to each of its (at most four) neighbours, any grains passed across the boundary being lost to a sink. Write the toppling operator

T_i

for this fixed subtraction, defined only when

z_i\ge z_c

, so that

T_i\mathbf z=\mathbf z-\boldsymbol\Delta\mathbf e_i

, with

\boldsymbol\Delta

the toppling matrix (

4

on the diagonal,

-1

between neighbours, sink row and column deleted) and

\mathbf e_i

the standard basis vector. ((a)) Suppose sites

i\ne j

are both unstable (

z_i\ge4

and

z_j\ge4

) in some state

\mathbf z

. Show that toppling

i

cannot lower

z_j

, and hence that

T_j

remains applicable after

T_i

has been applied. ((b)) Show algebraically that

T_jT_i\mathbf z=T_iT_j\mathbf z

for such a pair of sites. ((c)) Parts (a) and (b) compare two single topplings. Now prove that the final stable state and the toppling numbers

n_k

of a whole avalanche do not depend on the order in which unstable sites are toppled. Take two legal orderings A and B, each run until every site is stable (grains lost at the boundary make each one stop). Suppose that at some moment B is about to topple a site

x

for the

(n_x+1)

th time, where

n_x

is A's total for

x

, and that this is the first moment at which B exceeds A's total at any site. Show that

x

cannot then be unstable, and conclude.

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