Abstract
We study the numerical simulation of the shaken dynamics, a parallel Markovian dynamics for spin systems with local interaction and transition probabilities depending on the two parameters q and J that “tune” the geometry of the underlying lattice. The analysis of the mixing time of the Markov chain and the evaluation of the spin-spin correlations as functions of q and J, make it possible to determine in the (q, J) plane a phase transition curve separating the disordered phase from the ordered one. The relation between the equilibrium measure of the shaken dynamics and the Gibbs measure for the Ising model is also investigated. Finally two different coding approaches are considered for the implementation of the dynamics: a multicore CPU approach, coded in Julia, and a GPU approach coded with CUDA.
Highlights
The Gibbs sampling of lattice spin models is a major task for statistical mechanics
Once the dynamics is started from the configuration consisting of “all minus”, it will reach in a relative short time a minimizer of the free energy
We expect that the SWNE and the NW-SE correlations tend to be similar for large values of q, that is for those values of the pair (q, J ) for which the equilibrium distribution of the shaken dynamics approaches the Gibbs measure of the Ising model on the square lattice
Summary
The Gibbs sampling of lattice spin models is a major task for statistical mechanics. The related numerical techniques are mainly based on Markov chain dynamics for single and cluster spin flip [5,15,17], and can be implemented by means of random mapping representation techniques [8]. The theory of parallel Markov chains as Probabilistic Cellular Automaton (PCA) dates back to 1989 [6]. These processes are characterized by a factorized transition matrix on the configuration space, and can be simulated numerically updating all the spins using the same random map [2]. We explore the computational possibilities of this model to generalize the random sampling algorithms for Ising spin systems on a set of geometries for the two-dimensional lattices.
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