Hard-ball gas as hard nut of statistical mechanics (why mathematicians missed 1/f-noise there)

TL;DR

This paper investigates the limitations of traditional mathematical approaches in statistical mechanics of hard-ball systems, emphasizing the importance of specific phase-space subsets in understanding 1/f-noise phenomena and proposing new analytical methods.

Contribution

It introduces novel approaches to modeling hard-ball systems that preserve 1/f-noise properties, challenging conventional Boltzmannian kinetics.

Findings

Identification of critical phase-space subsets affecting 1/f-noise

Demonstration of limitations in traditional Boltzmann-Grad limit approaches

Proposal of new evolution equations for probability distributions

Abstract

We continue discussion of hard-ball models of statistical mechanics, by example of random walk of hard ball immersed into equlibrium ideal gas. Our goal is to highlight decisive role of specific phase-space subsets, despite their vanishingly smaall Lebesgue measures under the Boltzmann-Grad limit. The "art of draining" such subsets in conventional mathematical constructions resulted in loss of so principal property of many-particle systems as 1/f-noise in diffusivities, mobilities and other transport and relaxation rates. We suggest new approaches to formulation and analysis of evolution equations for hierarchy of probability distribution functions of infinite hard-ball systems, thus further overcoming prejudices of Boltzmannian kinetics and mistakes of its modern adepts