Crossover from Selberg's type to Ruelle's type Zeta function in classical kinetics

TL;DR

This paper explores how decay rates in a chaotic billiard system transition from being described by Ruelle's Zeta function in clean conditions to Selberg's Zeta function under strong disorder, providing an interpolation formula.

Contribution

The authors introduce an interpolation formula connecting Ruelle's and Selberg's Zeta functions for decay rates in disordered chaotic billiards, extending understanding of spectral properties.

Findings

Decay rates are zeros of Ruelle's Zeta function in clean systems.

Decay rates become roots of Selberg's Zeta function under strong disorder.

An interpolation formula bridges the two Zeta functions, elucidating the transition.

Abstract

The decay rates of the density-density correlation function are computed for a chaotic billiard with some amount of disorder inside. In the case of the clean system the rates are zeros of Ruelle's Zeta function and in the limit of strong disorder they are roots of Selberg's Zeta function. We constructed the interpolation formula between two limiting Zeta functions by analogy with the case of the integrable billiards. The almost clean limit is discussed in some detail. PACS numbers: 05.20.Dd, 05.45.+b, 51.10.+y