The derivation of the Liouville equation from the Schrodinger equation and its implications

TL;DR

This paper tackles the problem of deriving the Boltzmann equation from quantum mechanics by presenting a direct, mathematically rigorous route from the Schrödinger equation. The authors use a smooth-envelope decomposition of the wavefunction to obtain a Liouville-type evolution for a two-variable density and show that, in the appropriate limits, this yields the non-collision part of the Boltzmann equation; they then derive the collision integral from a quantum transition-rate matrix consistent with Fermi's golden rule, yielding the full Boltzmann equation $ abla f$ evolution with a collision term $ ext{St}(f)$. A key insight is that the non-collision dynamics arise in the long-wavelength (low-frequency) limit, while collisions originate from short-wavelength, rapid transitions, connecting reversible quantum dynamics to irreversible kinetic behavior via non-coherence. The work thus links quantum mechanics, decoherence concepts, and classical kinetic theory, providing a unified framework for transport in quasiparticle systems and clarifying the microscopic origin of irreversibility; it also discusses limitations (e.g., long-range interactions and infrared divergences) and outlines avenues for separating sharp vs smooth potentials to extend the approach.

Abstract

We present a new way of deriving classical mechanics from quantum mechanics. A key feature of the method is its compatibility with the standard approach used to derive transition rates between quantum states due to interactions. We apply the developed method to derive the main formulas of physical kinetics. We observe that, through the Liouville equation, we can deduce the non-collision part of the Boltzmann equation, and that, through the matrix of transition rates, we can deduce the collision integral. As a final result of the manuscript, we derive the Boltzmann equation from the Schrödinger equation as a single piece of formal mathematical manipulation, without any non-rigorous plausible reasoning used to glue together its different parts.