On the thermodynamic limit for a one dimensional sandpile process
نویسندگان
چکیده
Considering the standard abelian sandpile model in one dimension we construct an in nite volume Markov process corresponding to its thermodynamic in nite volume limit The main di culty we over come is the strong non locality of the dynamics by which depending on the sand con guration changes in the height variables far away can still in uence the origin in a single updating However using similar ideas as in recent extensions of the standard Gibbs formalism for lattice spin systems we can identify a set of good con gurations on which the dynamics is e ectively local Finally we prove that ev ery con guration converges in a nite time to the unique invariant measure Up to this time the expected height increases linearly in time AMS classi cation K C
منابع مشابه
The Abelian Sandpile Model on an Infinite Tree
We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality1.
متن کاملThe infinite volume limit of dissipative Abelian sandpiles
We construct the thermodynamic limit of the stationary measures of the Bak-Tang-Wiesenfeld sandpile model with a dissipative toppling matrix (sand grains may disappear at each toppling). We prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic Markov process leaving it invariant. We show how to extend the Dhar formalism of the ‘abelian group of toppling...
متن کاملInfinite volume limits of high-dimensional sandpile models
We study the Abelian sandpile model on Z. In d ≥ 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit μ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure μ, and study ergodic properties of this process. The main techniques we use are a connection between the st...
متن کاملCalculation of Thermodynamic Properties of the Quasi-one Dimensional Liquid 3He at Finite Temperature
We have used a variational approach to calculate some thermodynamic properties of the quasi-one dimensional liquid 3He such as the energy, entropy, free energy, equation of state and heat capacity at finite temperature. We have employed the Lennard-Jones potential as the inter-atomic interaction. We have seen that the total energy increases by increasing both temperature and density....
متن کاملOrganized versus self-organized criticality in the abelian sandpile model
We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabil...
متن کامل