Is Entropy Associated with Time's Arrow?

TL;DR

This paper explores the relationship between entropy and the Arrow of Time by introducing a generalized entropy concept based on Shannon's Measure of Information, clarifying their different behaviors over time.

Contribution

It proposes a new, general definition of entropy using Shannon's Measure of Information, distinguishing it from the traditional thermodynamic entropy and analyzing their temporal properties.

Findings

Shannon's Measure of Information (SMI) is a general concept applicable to any probability distribution.

Thermodynamic entropy is a specific case of SMI related to particle distributions.

The Boltzmann H-function is an SMI, not entropy, and can vary over time, unlike entropy.

Abstract

We start with reviewing the origin of the idea that entropy and the Second Law are associated with the Arrow of Time. We then introduced a new definition of entropy based on Shannons Measure of Information, SMI. The SMI may be defined on any probability distribution, and therefore it is a very general concept. On the other hand entropy is defined on a very special set of probability distributions. More specifically the entropy of a thermodynamic system is related the probability distribution of locations and velocities or momenta of all the particles, which maximized the Shannon Measure of Information. As such, entropy is not a function of time. We also show that the H function, as defined by Boltzmann is an SMI but not entropy. Therefore, while the H-function, as an SMI may change with time, Entropy, as a limit of the SMI does not change with time.