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International Journal of Modern Physics A

STATISTICS ON NETWORKS

A. P. Balachandran, Elisa Ercolessi

Published Jul 30, 1992• 11 citations
Publisher

Abstract

One-dimensional networks are excellent examples of topological spaces which are not manifolds and which admit interesting quantum physics. They are of importance in the free electron model for molecules and crystals. They also occur as fabricated mesoscopic networks. We review single-particle quantum physics on networks from topological and operator theoretic points of view and then initiate study of identical particles on networks. It is established that the available statistics on networks in its range and complexity rival the richness of statistical options in two dimensions. We treat two-particle statistics on simple networks with detail, taking care to cover operator theoretic issues pertaining to multiple connectivity and also those due to basic topological differences between a generic network and a manifold.

DOI
10.1142/s0217751x9200209x

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One-dimensional networks are excellent examples of topological spaces which are not manifolds and which admit interesting quantum physics. They are of importance in the free electron model for molecules and crystals. They also occur as fabricated mesoscopic networks. We review single-particle quantum physics on networks from topological and operator theoretic points of view and then initiate study of identical particles on networks. It is established that the available statistics on networks in its range and complexity rival the richness of statistical options in two dimensions. We treat two-particle statistics on simple networks with detail, taking care to cover operator theoretic issues pertaining to multiple connectivity and also those due to basic topological differences between a generic network and a manifold.
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We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrödinger equations describing open quantum systems.
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