144 research outputs found
Exact solutions for a mean-field Abelian sandpile
We introduce a model for a sandpile, with N sites, critical height N and each
site connected to every other site. It is thus a mean-field model in the
spin-glass sense. We find an exact solution for the steady state probability
distribution of avalanche sizes, and discuss its asymptotics for large N.Comment: 10 pages, LaTe
Stress Propagation and Arching in Static Sandpiles
We present a new approach to the modelling of stress propagation in static
granular media, focussing on the conical sandpile constructed from a point
source. We view the medium as consisting of cohesionless hard particles held up
by static frictional forces; these are subject to microscopic indeterminacy
which corresponds macroscopically to the fact that the equations of stress
continuity are incomplete -- no strain variable can be defined. We propose that
in general the continuity equations should be closed by means of a constitutive
relation (or relations) between different components of the (mesoscopically
averaged) stress tensor. The primary constitutive relation relates radial and
vertical shear and normal stresses (in two dimensions, this is all one needs).
We argue that the constitutive relation(s) should be local, and should encode
the construction history of the pile: this history determines the organization
of the grains at a mesoscopic scale, and thereby the local relationship between
stresses. To the accuracy of published experiments, the pattern of stresses
beneath a pile shows a scaling between piles of different heights (RSF scaling)
which severely limits the form the constitutive relation can take ...Comment: 38 pages, 24 Postscript figures, LATEX, minor misspellings corrected,
Journal de Physique I, Ref. Nr. 6.1125, accepte
Computational Complexity of Avalanches in the Kadanoff two-dimensional Sandpile Model
15 pagesIn this paper we prove that the avalanche problem for Kadanoff sandpile model (KSPM) is P-complete for two-dimensions. Our proof is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with configurations in KSPM. The proof is also related to the known prediction problem for sandpile which is in NC for one-dimensional sandpiles and is P-complete for dimension 3 or greater. The computational complexity of the prediction problem remains open for two-dimensional sandpiles
Computational universality of fungal sandpile automata
Hyphae within the mycelia of the ascomycetous fungi are compartmentalised by
septa. Each septum has a pore that allows for inter-compartmental and
inter-hyphal streaming of cytosol and even organelles. The compartments,
however, have special organelles, Woronin bodies, that can plug the pores. When
the pores are blocked, no flow of cytoplasm takes place. Inspired by the
controllable compartmentalisation within the mycelium of the ascomycetous fungi
we designed two-dimensional fungal automata. A fungal automaton is a cellular
automaton where communication between neighbouring cells can be blocked on
demand. We demonstrate computational universality of the fungal automata by
implementing sandpile cellular automata circuits there. We reduce the Monotone
Circuit Value Problem to the Fungal Automaton Prediction Problem. We construct
families of wires, cross-overs and gates to prove that the fungal automata are
P-complete
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