Special Relativistic Liouville Equation Completed

TL;DR

This paper completes the formulation of a special relativistic Liouville equation by deriving a Lorentz-invariant universal time from entropy, enabling a consistent statistical mechanics framework for relativistic systems.

Contribution

It introduces a Lorentz-invariant universal time derived from entropy, completing the relativistic Liouville equation and simplifying the relativistic ideal gas partition function.

Findings

The relativistic Liouville equation is now well-defined with entropy-based universal time.

The derived partition function for the relativistic ideal gas is simpler than previous models.

The approach aligns the second law of thermodynamics with relativistic invariance.

Abstract

In two previous papers, the author raised the possibility of a special relativistic Liouville equation. The conclusion then was yes, such an equation is possible in 8N phase space if a Lorentz-invariant Universal (LiU) time can be defined for all the degrees of freedom. Without this LiU time, the existence of a special relativistic Liouville equation is empty and may just be a waste of time. In this paper, I propose and argue that the LiU time follows from entropy, which should not be surprising given the second law of thermodynamics and the fact that regardless of how temperature and heat transform under Lorentz transformation, the entropy is invariant. Thus, it is natural to define LiU time from entropy.This now completes the existence of a special relativistic Liouville equation, which will determine the Gibbs distribution, the starting point of classical statistical mechanical description of physical systems. To illustrate the formalism, the partition function for the relativistic ideal gas is derived. The result is much simpler than the Juttner gas result.