Appearance of Thermal Time

TL;DR

This paper proposes a model where time is viewed as an informational and thermal entity, establishing a mathematical relationship between event measurement, temperature, and time through statistical inference and differential geometry.

Contribution

It introduces a novel perspective that time emerges from thermal and informational considerations, linking statistical inference with geometric properties of relaxation processes.

Findings

Temperature and time are statistically inferred from event measurements.

The mutual regulation between temperature and time is expressed via Fisher information.

A simple scalar curvature equation R = -1 characterizes the relaxation process.

Abstract

In this paper a viewpoint that time is an informational and thermal entity is shown. We consider a model for a simple relaxation process for which a relationship among event, time and temperature is mathematically formulated. It is then explicitly illustrated that temperature and time are statistically inferred through measurement of events. The probability distribution of the events thus provides a mutual regulation between temperature and time, which can relevantly be expressed in terms of the Fisher information metric. The two-dimensional differential geometry of temperature and time then leads us to a finding of a simple equation for the scalar curvature, R = -1, in this case of relaxation process. This basic equation, in turn, may be regarded as characterizing the nonequilibrium dynamical process and having a solution given by the Fisher information metric. The time can then be interpreted so as to appear in a thermal way.