Approach to criticality in sandpiles
A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this “density conjecture” is false when the underlying graph is any of Z2, the complet...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
American Physical Society
2011
|
| Online Access: | http://hdl.handle.net/1721.1/60853 |
| _version_ | 1826204128042287104 |
|---|---|
| author | Levine, Lionel Fey, Anne Wilson, David B. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Levine, Lionel Fey, Anne Wilson, David B. |
| author_sort | Levine, Lionel |
| collection | MIT |
| description | A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this “density conjecture” is false when the underlying graph is any of Z2, the complete graph Kn, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. In this paper, we substantiate this claim by rigorous proof and extensive simulations. We show that driven-dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. Nevertheless, we do find (and prove) a relationship between the two models: the threshold density of the fixed-energy sandpile is the point at which the driven-dissipative sandpile begins to lose a macroscopic amount of sand to the sink. |
| first_indexed | 2024-09-23T12:49:39Z |
| format | Article |
| id | mit-1721.1/60853 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:49:39Z |
| publishDate | 2011 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/608532022-10-01T11:19:55Z Approach to criticality in sandpiles Levine, Lionel Fey, Anne Wilson, David B. Massachusetts Institute of Technology. Department of Mathematics Levine, Lionel Levine, Lionel A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this “density conjecture” is false when the underlying graph is any of Z2, the complete graph Kn, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. In this paper, we substantiate this claim by rigorous proof and extensive simulations. We show that driven-dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. Nevertheless, we do find (and prove) a relationship between the two models: the threshold density of the fixed-energy sandpile is the point at which the driven-dissipative sandpile begins to lose a macroscopic amount of sand to the sink. National Science Foundation (U.S.) 2011-01-28T18:42:27Z 2011-01-28T18:42:27Z 2010-09 2010-03 Article http://purl.org/eprint/type/JournalArticle 1539-3755 1550-2376 http://hdl.handle.net/1721.1/60853 Fey, Anne, Lionel Levine, and David B. Wilson. “Approach to criticality in sandpiles.” Physical Review E 82.3 (2010): 031121. © 2010 The American Physical Society. en_US http://dx.doi.org/10.1103/PhysRevE.82.031121 Physical Review E Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Physical Society APS |
| spellingShingle | Levine, Lionel Fey, Anne Wilson, David B. Approach to criticality in sandpiles |
| title | Approach to criticality in sandpiles |
| title_full | Approach to criticality in sandpiles |
| title_fullStr | Approach to criticality in sandpiles |
| title_full_unstemmed | Approach to criticality in sandpiles |
| title_short | Approach to criticality in sandpiles |
| title_sort | approach to criticality in sandpiles |
| url | http://hdl.handle.net/1721.1/60853 |
| work_keys_str_mv | AT levinelionel approachtocriticalityinsandpiles AT feyanne approachtocriticalityinsandpiles AT wilsondavidb approachtocriticalityinsandpiles |