Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 8

A system of billiard and its application to information-theoretic entropy

Pages: 2599 – 2610

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n8.a4

Authors

Supriyo Dutta (Department of Mathematics, National Institute of Technology Agartala, Jirania, West Tripura, India)

Partha Guha (Department of Mathematics, Khalifa University, Abu Dhabi, United Arab Emirates; and S. N. Bose National Centre for Basic Sciences, Kolkata, India)

Abstract

In this article, we define an information-theoretic entropy based on the Ihara zeta function of a graph which is called the Ihara entropy. A dynamical system consists of a billiard ball and a set of reflectors correspond to a combinatorial graph. The reflectors are represented by the vertices of the graph. Movement of the billiard ball between two reflectors is represented by the edges. The prime cycles of this graph generate the bi-infinite sequences of the corresponding symbolic dynamical system. The number of different prime cycles of a given length can be expressed in terms of the adjacency matrix of the oriented line graph. It also constructs the formal power series expansion of Ihara zeta function. Therefore, the Ihara entropy has a deep connection with the dynamical system of billiards. As an information-theoretic entropy, it fulfils the generalized Shannon–Khinchin axioms. It is a weakly decomposable entropy whose composition law is given by the Lazard formal group law.

Full Text (PDF format)

SD is thankful to Dr. Subhashish Banerjee who introduced the author to the Ihara Zeta function and its applications in quantum information theory. PG was supported by the Khalifa University of Science and Technology under grant number FSU-2021-014.

Published 5 January 2024