Smooth sand piles Stefano Brocchi 1 , Paolo Massazza 2 1 Dipartimento - - PowerPoint PPT Presentation

Smooth sand piles

Stefano Brocchi1, Paolo Massazza2

1Dipartimento di Sistemi e Informatica

Universit` a degli Studi di Firenze - Italy

2Dipartimento di Scienze Teoriche e Applicate

Universit` a degli Studi dell’Insubria Varese Italy

17 September, 2012 Reachability Problems 2012

Stefano Brocchi, Paolo Massazza Smooth sand piles

Smooth sand piles

We propose a new variant of the Sand Pile Model (SPM) called Smooth Sand Pile Model (SmSPM) The sand pile model (SPM) has been introduced by Goles and Kiwi [92] The model studies how a pile of sand grains evolves and eventually gets to a stable configuration Several physical situations are described, as avalanches and landslides

Stefano Brocchi, Paolo Massazza Smooth sand piles

The sand pile model

Sandpiles are represented by integer partitions, where each value represents the number of grains in a column A grain can fall from a column to the next if the difference in height is greater than 1

(8,6,5,2,1) (8,6,4,3,1)

Stefano Brocchi, Paolo Massazza Smooth sand piles

The sand pile model

The model starts with all of the grains in the first column After a series of configuration, the model eventually evolves to a stable situation said fixed point (14) (9,4,1) (7,4,2,1) (4,4,3,2,1)

Stefano Brocchi, Paolo Massazza Smooth sand piles

A new proposal

Many variations of the original Sand Pile Model exist and have been extensively studied

Ice piles, bidimensional sand piles, symmetrical sand piles, Kadanoff sand pile model...

What are the motivations for a new model ?

Stefano Brocchi, Paolo Massazza Smooth sand piles

Sand pile model: limitations

In our opinion, there are two main limitations of SPM As a first issue, some configurations that can be reached in the model may be unreasonable under the physical point of view

Stefano Brocchi, Paolo Massazza Smooth sand piles

Sand pile model: limitations

The natural extension of the model in two dimensions seems troublesome In this case, the pile is represented by a matrix of integers To deal with plane partitions, we should impose that the number of grains is decreasing along rows and columns The bidimensional evolution rule forbids moves that would be natural in a real physical model

Stefano Brocchi, Paolo Massazza Smooth sand piles

Bidimensional fixed points

We expect that fixed points should be smooth (as the sand pile to the left in figure) while many of them result not to be so (right in figure)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Smooth bidimensional sand piles

In two dimensions, our model has only nearly pyramidal fixed points with minimal potential Further, we could describe also sand piles with different starting points that do not represent plane partitions The study of the bidimensional model seems not trivial

Even representing a strong motivation for our work, the expansion in two dimension has been left as a theme for future research

Stefano Brocchi, Paolo Massazza Smooth sand piles

The idea of the model

Our idea: the grains move faster where the slope of the pile is higher

Slow grains Fast grains

By consequence, the slope of the pile is naturally balanced

Stefano Brocchi, Paolo Massazza Smooth sand piles

Moves

It is useful to represent a sandpile with its potentials, i.e. the height difference between adjacent columns A move can occur only where the potential is maximal

14 11 9 6 3 1 3 2 3 3 2 1 Potential: Possible moves

Stefano Brocchi, Paolo Massazza Smooth sand piles

Sequences of maximal potentials

A basic property: the maximal potential increases when a move occurs in a sequence of at least two maximal potentials The next moves will leave the potential equal until eventually the potential will decrease again The final configuration will not depend on which of the positions initially started moving

Stefano Brocchi, Paolo Massazza Smooth sand piles

Evolution of the model: example

An example of the evolution of a model:

Potentials:

(2,2,2,2,2)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Evolution of the model: example

An example of the evolution of a model:

Potentials:

(2,2,2,2,2) (2,3,0,3,2)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Evolution of the model: example

An example of the evolution of a model:

Potentials:

(2,2,2,2,2) (2,3,0,3,2) (2,3,1,1,3)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Evolution of the model: example

An example of the evolution of a model:

Potentials:

(2,2,2,2,2) (2,3,0,3,2) (2,3,1,1,3) (2,3,1,2,1,1)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Evolution of the model: example

An example of the evolution of a model:

Potentials:

(2,2,2,2,2) (2,3,0,3,2) (2,3,1,1,3) (2,3,1,2,1,1) (3,1,2,2,1,1)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Evolution of the model: example

An example of the evolution of a model:

Potentials:

(2,2,2,2,2) (2,3,0,3,2) (2,3,1,1,3) (2,3,1,2,1,1) (3,1,2,2,1,1) (1,2,2,2,1,1)

Stefano Brocchi, Paolo Massazza Smooth sand piles

A unique move

We chose to simplify the model, merging all of these moves in a unique move The grain falls from the first position of the sequence of maximal potentials, and finishes after the last position

Potentials:

(2,2,2,2,2) (2,3,0,3,2) (2,3,1,1,3) (2,3,1,2,1,1) (3,1,2,2,1,1) (1,2,2,2,1,1)

Stefano Brocchi, Paolo Massazza Smooth sand piles

SmSPM and SmSPM*

We call this model SmSPM, while the model including all possible moves where the potential is maximal is said SmSPM* SmSPM results to be simpler to characterize, and the reachable states can be defined in a compact form We could obtain the configurations of SmSPM* by adding the intermediate configurations between those of SmSPM

Stefano Brocchi, Paolo Massazza Smooth sand piles

Reachable states

Theorem A linear partition with maximal potential D is reachable in SSPM if and only if:

• 1. Apart from the last position, no potential is smaller than D − 2
• 2. The pattern D − 2, (D − 1)k, D − 2 (k ≥ 0) does not appear in

the sequence of potentials Point 2 is a generalization of the forbidden sequences of potential in the standard SPM, that cannot contain 0, 1k, 0

Stefano Brocchi, Paolo Massazza Smooth sand piles

The poset

The configurations of SmSPM can be organized in a poset, where s ≤ t if there is a sequence of moves from s to t The poset is also a lattice SmSPM ⊂ SPM

Stefano Brocchi, Paolo Massazza Smooth sand piles

Bottlenecks

For each maximal potential D there is at most one configuration that is maximal in the poset Such configuration is said to be a bottleneck

Stefano Brocchi, Paolo Massazza Smooth sand piles

SmSPM poset: an example

In figure, the poset for 30 grains (in gray: SmSPM*)

30 20,10 ... 19,11 20,9,1 19,10,1 15,10,5 15,9,6 15,10,4,1 14,11,5 14,10,6 15,9,5,1 14,10,5,1 14,9,6,1 13,10,6,1 14,9,5,2 13,10,5,2 13,9,6,2 12,10,6,2 13,9,5,3 12,10,5,3 12,9,6,3 12,9,7,2 13,8,6,3 18,11,1 19,9,2 18,10,2 17,11,2 18,9,3 17,10,3 16,11,3 17,9,4 16,10,4 15,11,4 16,9,5 15,10,5 12,9,6,3 11,10,6,3 12,8,7,3 12,9,5,4 12,9,6,2,1 11,9,7,3 12,8,6,4 12,9,5,3,1 11,9,6,4 12,8,6,3,1 11,9,6,3,1 11,8,7,3,1 11,9,5,4,1 11,8,6,4,1 10,9,6,4,1 11,8,6,3,2 10,8,7,4,1 10,9,6,3,2 11,8,5,4,2 10,8,6,5,1 10,8,7,3,2 10,9,5,4,2 10,8,6,4,2

D=5 D=4 D=3 D=2

Stefano Brocchi, Paolo Massazza Smooth sand piles

Bottlenecks: main results

Bottlenecks divide the poset in smaller posets The sequence of potentials of a bottleneck is of the form Da, (D − 1)b, Dc, r, with a, c ≥ 0 and b ≤ 1 These configurations can be easily computed from the number

• f grains n

Every evolution of the model must go through all of the bottlenecks

Stefano Brocchi, Paolo Massazza Smooth sand piles

Other properties

Thanks to the existance of bottlenecks, in our poset for a configuration s with maximum potential P(s) it stands P(s) > P(t) → s < t The maximum potential D of a sandpile of length r > 1 and n grains is bounded by

• 2n

r(r + 1)

• ≤ D ≤

2(n − 2) r(r − 1)

• + 1

Stefano Brocchi, Paolo Massazza Smooth sand piles

Future works

In future works, we plan to study more properties of the model and eventually an efficient exhaustive generation algorithm Also, it would be interesting to merge the definition of our move with the one of the Kadanoff model, in order to study an alternative scenario We also plan to study the properties of the bidimensional model, for which our model seems to obtain interesting results

Stefano Brocchi, Paolo Massazza Smooth sand piles

Online simulator

A simulator is available online at

http://www.dsi.unifi.it/users/brocchi/smoothsandpiles.php http://www.dsi.unifi.it/users/brocchi/2dimsmoothsandpiles.php (2

dimensions)

Stefano Brocchi, Paolo Massazza Smooth sand piles

Thanks

Thanks for your attention

Stefano Brocchi, Paolo Massazza Smooth sand piles