Statistical mechanical characterization of billiard systems

Abstract

Area-preserving maps play an important role in diverse fields as they are widely used for modeling complex systems. In addition, these maps provide rich observations by presenting stable orbits and chaotic behavior separately or together in the phase space depending on the control parameter. In recent years, several studies on these maps, drawing inspiration from the phase space dynamics, have shown that nonextensive statistical mechanics provides appropriate instruments to characterize these systems. In this study, we perform a rigorous numerical analysis to delve into the statistical mechanical properties of a billiard system. Our primary goal is to confirm the presence of a q-Gaussian distribution, with an estimated q value of approximately 1.935. We accomplish this by examining the probability distribution of the cumulative sum of system iterates, focusing specifically on initial conditions within the stability islands. Our findings align seamlessly with the latest research in this field. Furthermore, we show that a multi-component probability distribution containing both Gaussian and q-Gaussians describes the entire system for some parameter regions where the phase space consists of stability islands together with the chaotic sea.