Up-Hill Diffusion Creating Density Gradient - What is the Proper Entropy?

TL;DR

This paper explores how topological constraints on space influence entropy measures, proposing a modified entropy concept that aligns with thermodynamics and explaining phenomena like up-hill diffusion in magnetospheres.

Contribution

It introduces a new perspective on entropy based on topological constraints and invariant measures, extending statistical mechanics beyond traditional energy conservation.

Findings

Modified entropy measure aligns with the second law of thermodynamics.

Application to magnetosphere explains up-hill diffusion phenomena.

Demonstrates the importance of coordinate-invariant measures in statistical mechanics.

Abstract

It is always some constraint that yields any nontrivial structure from statistical averages. As epitomized by the Boltzmann distribution, the energy conservation is often the principal constraint acting on mechanical systems. Here, we investigate a different type: the topological constraint imposed on `space'. Such constraint emerges from the null space of the Poisson operator linking energy gradient to phase space velocity, and appears as an adiabatic invariant altering the preserved phase space volume at the core of statistical mechanics. The correct measure of entropy, built on the distorted invariant measure, behaves consistently with the second law of thermodynamics. The opposite behavior (decreasing entropy and negative entropy production) arises in arbitrary coordinates. An ensamble of rotating rigid bodies is worked out. The theory is then applied to up-hill diffusion in a magnetosphere.