Phase Transitions within the Fully Developed Chaotic Regime

Abstract

In this contribution we report on a special type of phase transitions, which arises in a particular one-dimensional fully developed chaotic (FDC) map, the bungalow-tent map f _a (x), depending on a control variable a. Phase-transition-like behaviour can be observed in the Lyapounov spectrum ⋋(a) for a sequence of critical values of the parameter a.¹ The phase-transition-like phenomena can be understood in terms of a particular symbolic dynamics description of the chaotic process, which is equivalent to a statistical description of a phase transition in a half-infinite spin chain. The (long-range) interactions between the spins are implicitly given by the probabilities of the spin configurations. Exact expressions for these probabilities can be derived due to the special choice of the symbolic dynamics. This enables us to find exact analytical expressions for the correlation function and the critical exponents near the critical value of a. It thus turns out that the bungalow-tent map provides an exactly solvable model for a phase transition in one dimension. More details about the (rather lengthy) derivations will be published soon.²