Manifestations of chaos in billiards: the role of mixed curvature

• URL: http://arxiv.org/abs/2501.08839v1
• Date: Wed, 15 Jan 2025 14:47:15 GMT
• Title: Manifestations of chaos in billiards: the role of mixed curvature
• Authors: Pranaya Pratik Das, Tanmayee Patra, Biplab Ganguli,
• Abstract summary: This study introduces two such billiards: a bean-shaped billiard and a peanut-shaped billiard, the latter being a variant of Cassini ovals.<n>We examine both classical and quantum dynamics of these billiards and observe a strong alignment between the two perspectives.<n>Scarring phenomena serve as a rich visual manifestation of quantum and classical correspondence, and highlight quantum suppression chaos at a local level.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The boundary of a billiard system dictates its dynamics, which can be integrable, mixed, or fully chaotic. Despite the significant implications of chaotic billiard systems with mixed curvatures, those whose boundaries feature both positive and negative curvature remain relatively under-explored. This study introduces two such billiards: a bean-shaped billiard and a peanut-shaped billiard, the latter being a variant of Cassini ovals. Unlike traditional chaotic billiards, these systems incorporate both focusing and defocusing regions along their boundaries, with no neutral segments. We examine both classical and quantum dynamics of these billiards and observe a strong alignment between the two perspectives. For classical analysis, the billiard flow diagram and billiard map reveal sensitivity to initial conditions, a hallmark of classical chaos. In the quantum domain, we use nearest-neighbour spacing distribution and spectral complexity as statistical measures to characterise chaotic behaviour. Both classical and quantum mechanical analysis are in firm agreement with each other. One of the most striking quantum phenomena we observe is the eigenfunction scarring (both scars and super-scars). Scarring phenomena serve as a rich visual manifestation of quantum and classical correspondence, and highlight quantum suppression chaos at a local level. This research contributes to a deeper understanding of chaos, especially in billiard systems with mixed curvature boundaries.

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