The Mean: Axiomatics, Generalizations,Applications
John E. Gray, Andrew Vogt
TL;DR
This paper develops an axiomatic framework for the mean, explores its generalizations including Kolmogorov's, and discusses diverse applications and anomalies in physics related to the mean's properties.
Contribution
It introduces a comprehensive axiomatic approach to the mean, presents new generalizations, and clarifies its role in statistical mechanics and quantum theory.
Findings
Kolmogorov's mean based on the Weak Law of Large Numbers
Resolution of quantum anomalies related to means and variances
Examples and counterexamples illustrating mean applications
Abstract
We present an axiomatic approach to the mean and discuss generalizations of the mean, including one due to Kolmogorov based on the Weak Law of Large Numbers. We offer examples and counterexamples, describe conventional and unconventional uses of the mean in statistical mechanics, and resolve an anomaly in quantum theory concerning apparent simultaneous coexistence of means and variances of observables. These issues all arise from the familiar definition of the mean.
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Taxonomy
TopicsStatistical Mechanics and Entropy · Complex Systems and Time Series Analysis · Probability and Statistical Research