Escaping from cycles through a glass transition

Details Escaping from cycles through a glass transition Escaping from cycles through a glass transition

2003-01-10

arxiv.org/abs/cond-mat/0301147v2

Phys. Rev. E 68, 016104 (2003)

Physics

Condensed Matter

Disordered Systems and Neural Networks

11 pages, 4 figures

Scientific paper

10.1103/PhysRevE.68.016104

A random walk is performed over a disordered media composed of $N$ sites random and uniformly distributed inside a $d$-dimensional hypercube. The walker cannot remain in the same site and hops to one of its $n$ neighboring sites with a transition probability that depends on the distance $D$ between sites according to a cost function $E(D)$. The stochasticity level is parametrized by a formal temperature $T$. In the case $T = 0$, the walk is deterministic and ergodicity is broken: the phase space is divided in a ${\cal O}(N)$ number of attractor basins of two-cycles that trap the walker. For $d = 1$, analytic results indicate the existence of a glass transition at $T_1 = 1/2$ as $N \to \infty$. Below $T_1$, the average trapping time in two-cycles diverges and out-of-equilibrium behavior appears. Similar glass transitions occur in higher dimensions choosing a proper cost function. We also present some results for the statistics of distances for Poisson spatial point processes.

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