30.04.2019 – Preprint online
The preprint of our new article on families of sandpile monomorphisms between each NxM rectangular domain and all s(N+1)-1 x s(M+1)-1 domains went online today. Check it out at https://arxiv.org/abs/1904.12209.
Sandpile monomorphism and harmonic dynamics
Video demonstrating the successive action of four harmonic fields on the abelian sandpile identity.
More videos
Background
The abelian sandpile model is a cellular automaton defined on a domain of the standard square lattice. It serves as the archetypical model for self-organized criticality, and important process in physics, biology, geology and other fields. The recurrent configurations of the sandpile model form an abelian group, the sandpile group. This website summarizes our results on this group.
Harmonic sandpile dynamics
The identity of the sandpile group shows smooth 1-periodic dynamics induced by harmonic fields. To analyze these harmonic dynamics, we provide an open-source software solution called InterPile to interactively explore and manipulate sandpiles on various domains, as well as to generate movies showing the dynamics of the sandpile identity or any other recurrent configuration induced by arbitrary harmonic functions and for arbitrarily defined domains.
Sandpile monomorphisms
The concept of criticality is strongly related to scale-invariance, i.e. certain properties of a critical system, like the distribution of avalanche sizes for the sandpile model, remain qualitatively the same under spatial-temporal scaling. We recently showed that also the sandpile group itself is susceptible to such scaling relationships in the form of families of monomorphisms from each sandpile group on an NxM rectangular domain to all sandpile groups on s(N+1)-1xs(M+1)-1 domains.
Videos and Software
To access the videos showing the harmonic sandpile dynamics, visit our Videos section. To test the sandpile monomorphisms or create own videos of custom harmonic sandpile dynamics, download InterPile.
Learn more
To learn more about the harmonic sandpile dynamics, read our PNAS paper on this topic. For the sandpile monomorphisms, we refer to our recent preprint.
News
28.04.2019 – New beta and upcoming paper
We released beta 4 of Interpile 1.1, just in time for our upcoming paper. The main new functionality is that the families of sandpile monomorphisms from every NxM domain to all s(N+1)-1xs(M+1)-1 domains are now completely implemented and bug free (hopefully, at least, that's why it's still only a beta). What are these sandpile monomorphisms we are talking about? Check the theoretical background in our upcoming paper "Sandpile monomorphisms and harmonic functions" from Moritz Lang and Mikhail Shkolnikov, which should be available at arXive at Tuesday, 04/30/2019. We keep you updated as soon as the paper is online, and we have the full reference!
27.02.2019 – New beta and more press
We released beta 3 of Interpile 1.1, which now includes the first glimpse on some new functionality: unique coordinates identifying recurrent configurations, as well as group homomorphisms between sandpile groups on certain NxN domains. All that is however yet pretty preliminary, extremely slow and potentially even buggy...well, it's only a first glimpse. Furthermore, there was some more press coverage of our PNAS paper in derstandart.at, science.orf.at, and phys.org.
14.02.2019 – Press coverage
The APA (Austria Presse Agentur) has covered the release of our PNAS paper in a small article. Check it out under https://science.apa.at/site/natur_und_technik/detail.html?key=SCI_20190214_SCI39391351446907992 (in German).
06.02.2019 – Paper out
Our paper on the harmonic sandpile dynamics is now officially online at PNAS, and can be accessed under https://www.pnas.org/content/early/2019/02/05/1812015116.abstract.
26.07.2018 – First Domain Transformation Movies
With the first beta of Interpile v1.1, we also uploaded the first movies of a new category, to which we refer as domain transformation. In these movies, the domain is e.g. slowly transformed from a square into a circle. Enjoy!
24.07.2018 – First InterPile Release
After a lot of testing, we now finally released the first version of Interpile (v1.0). There are only minor changes as compared to the last beta, and mainly bugs were fixed.
12.07.2018 – First Domain Scaling Movies and new Release of InterPile
The first videos are online showing configurations corresponding to the sandpile identity dynamics induced by a given harmonic fields at the same or scaled times for increasing domain sizes. This visualizes both the capability of our approach for renormalization, as well as the super-position of different time scales. We also released a new beta of InterPile (beta 3), with which these videos can be easily generated. Feel free to send us your best videos!
10.07.2018 – Video of fifth order harmonics
We uploaded a bunch of additional videos, including the first video showing fifth-order harmonic sandpile identity dynamics. The other videos mainly show the sandpile dynamics for lower order harmonics when starting with some interesting fractal patterns found in the videos of certain harmonics.
02.07.2018 – Even more videos
We uploaded even more videos, most of which are showing the effect when combining different harmonic fields. We also created a "promotional video" to illustrate how the different dynamics play together, called A roundtrip through the sandpile group, which can be seen on the start page of this website.
29.06.2018 – Manuscript and more videos
The preprint of our manuscript just went online, and can be accessed under https://arxiv.org/abs/1806.10823. We also uploaded several additional videos of sandpile identity dynamics, which can be accessed in the videos section of this webpage.
28.06.2018 – First videos online
The first videos showing the sandpile identity dynamics induced by harmonic fields for square and circular domains are online. Videos showing the dynamics on other domains will follow soon!
20.06.2018 – New Homepage
In preparation for the publication of the preprint of our research on the harmonic dynamics of the Abelian sandpile identity, we created this homepage. The current version of the open-source software InterPile used to generate these dynamics is already available for download. Videos showing the sandpile identity dynamics induced by different harmonics will be made available soon.
