Ensemble averages, Poisson processes and Microstates

TL;DR

This paper develops a mathematical framework for ensemble averages of theories with discrete variables using Poisson processes, linking them to microscopic theories and Liouville gravity, with implications for quantum effects.

Contribution

It introduces a measure for ensemble averaging, describes it via Poisson point processes, and connects it to microscopic theories and Liouville gravity, expanding the theoretical understanding.

Findings

Ensemble averages can be modeled with Poisson point processes.

Averaging theories correspond to tracing over microscopic degrees of freedom.

Connections to Liouville gravity suggest implications for quantum effects.

Abstract

We consider ensemble averaged theories with discrete random variables. We propose a suitable measure to do the ensemble average. We also provide a mathematical description of such ensemble averages of theories in terms of Poisson point processes. Moreover, we demonstrate that averaging theories of this type has an equivalent description as tracing over parts of the microscopic degrees of freedom in a suitable continuous limit of a single microscopic theory. The results from both approaches can be identified with Liouville gravity, of which we further address some implications on the microscopic theory, including venues to look for quantum effects from the view point of the averaged theory. Generalizations to other point processes are also discussed.