Breakthrough derivation of the Boltzmann equation 

22/02/2026

A breakthrough in mathematical physics resolved a major open problem by rigorously deriving the Boltzmann equation from the dynamics of individual atoms

In a landmark paper, Long time derivation of the Boltzmann equation from hard sphere dynamics, mathematicians Yu Deng, Zaher Hani, and Xiao Ma rigorously derived the Boltzmann equation describing the behavior of a gas at the mesoscopic scale, directly from Newton''''''''''''''''s laws applied to individual spherical (hard) atoms. This work extends a 1975 theorem by Oscar Lanford, which proved such a connection for short-time. 

A recent article from Quanta Magazine article straightforwardly describes this achievement. A key idea consists in using techniques from the study of wave interactions to create a "cutting algorithm" managing the complexity of possible collision histories among particles. This is then used to prove that complicated, recurring collisions are extraordinarily rare. This work appears to give insight into the problem of understanding how the arrow of time seen at the macroscopic scale emerges from time-reversible laws ruling the microscopic world. This result, together with previous work, solves an aspect of Hilbert''''''''''''''''s Sixth Problem.