Smooth sand piles Stefano Brocchi 1 , Paolo Massazza 2 1 Dipartimento di Sistemi e Informatica Universit` a degli Studi di Firenze - Italy 2 Dipartimento 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 Possible moves 14 11 9 6 3 1 Potential: 3 2 3 3 2 1 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 , 1 k , 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 12,9,6,3 15,10,5 ... 14,11,5 15,9,6 15,10,4,1 12,8,7,3 12,9,5,4 12,9,6,2,1 11,10,6,3 20,10 12,8,6,4 12,9,5,3,1 11,9,7,3 14,10,6 15,9,5,1 19,11 20,9,1 11,9,6,4 12,8,6,3,1 19,10,1 14,10,5,1 11,9,6,3,1 18,11,1 19,9,2 14,9,6,1 18,10,2 11,8,7,3,1 11,9,5,4,1 13,10,6,1 14,9,5,2 17,11,2 18,9,3 11,8,6,4,1 13,10,5,2 17,10,3 D=4 11,8,6,3,2 13,9,6,2 10,9,6,4,1 16,11,3 17,9,4 12,10,6,2 13,9,5,3 10,8,7,4,1 10,9,6,3,2 11,8,5,4,2 16,10,4 10,8,6,5,1 10,9,5,4,2 12,9,7,2 12,10,5,3 13,8,6,3 10,8,7,3,2 15,11,4 16,9,5 10,8,6,4,2 D=2 D=3 D=5 12,9,6,3 15,10,5 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 D a , ( D − 1) b , D c , r , with a , c ≥ 0 and b ≤ 1 These configurations can be easily computed from the number of 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 � 2 n � � 2( n − 2) � ≤ D ≤ + 1 r ( r + 1) r ( r − 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
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