# Long time derivation of Boltzmann equation from hard sphere dynamics

**Authors:** Yu Deng, Zaher Hani, Xiao Ma  
**arXiv:** [2408.07818](https://arxiv.org/abs/2408.07818)  
**Date:** 2024-08-14  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2408.07818v1/

## Abstract

We provide a rigorous derivation of Boltzmann's kinetic equation from the hard-sphere system for rarefied gas, that is valid for arbitrarily long time as long as a solution to the Boltzmann equation exists. This is a crucial step in resolving Hilbert's sixth problem, and also extends Lanford's landmark theorem (1975) which justifies such derivation for sufficiently short time. The general strategy follows the paradigm introduced by the first two authors for the long-time derivation of the wave kinetic equation in wave turbulence theory. This is based on proving a long-time cumulant ansatz, which keeps memory of the full collision history of the particle system. At the heart of the matter is a reduction to combinatorial estimates for the Feynman diagrams (referred to as Collision History (CH) molecules), which are then proved by devising an elaborate cutting algorithm, a major novelty of this work.