Towards the BBGKY hierarchy: a scheme beyond the Boltzmann equation and application to a weakly confined QCD gas

TL;DR

This paper develops a novel scheme to go beyond the Boltzmann equation by considering the full BBGKY hierarchy, and applies it to analyze spectra in a weakly confined QCD gas, revealing insights into operator spectra at different energy scales.

Contribution

It introduces a scheme that accounts for the entire BBGKY hierarchy without truncation, extending kinetic theory beyond traditional approximations.

Findings

The scheme can explicitly analyze the second level of the hierarchy.

Higher levels capture operator spectra in the ultra-violet regime.

Comparison with holographic duality results provides new insights.

Abstract

In classical kinetic theory, the BBGKY hierarchy is an infinite chain of integro-differential equations that describes the full time-reversal-invariant (Liouville) system of interacting (quasi)-particles in terms of $N$-particle distribution functions. In this work, instead of truncating the hierarchy at the lowest level, as is done by the Boltzmann equation, we develop a scheme similar to the relaxation time approximation that is in principle able to account for the entire chain of equations. We then explicitly investigate its truncation at the second level of the BBGKY hierarchy and, within this scheme, study the spectra of conserved operator correlation functions in a gas of weakly confined hadrons. We also discuss how these higher levels account for parts of the operator spectra `deeper in the ultra-violet regime' and compare them to known results derived from the holographic duality.