Long time derivation of the Boltzmann equation from hard sphere dynamics

TL;DR

The paper proves a rigorous long-time derivation of Boltzmann's equation from the hard-sphere dynamics in the Boltzmann-Grad limit, extending Lanford's short-time result to arbitrarily long times under the existence of a Boltzmann solution. The authors introduce a novel cumulant framework and represent collision histories as layered molecules, then control their growth with a sophisticated time-layering strategy and a cutting algorithm that decomposes complex diagrams into manageable elementary pieces. The main contribution lies in proving smallness of cumulants via careful combinatorial and geometric estimates, enabling convergence of multi-particle correlations to the Boltzmann product with an explicit ε^θ rate for s up to |log ε|. The work, complemented by a companion paper linking to hydrodynamic limits, completes the kinetic limit portion of Hilbert's Sixth Problem for rarefied gases and lays the groundwork for subsequent derivations of fluid equations from particle dynamics. This framework also offers a roadmap for extensions to potentials and fluctuations, and for connecting kinetic descriptions to macroscopic hydrodynamics.

Abstract

We provide a rigorous derivation of Boltzmann's kinetic equation from the hard sphere system for rarefied gas, which is valid for arbitrarily long times, as long as the solution to the Boltzmann equation exists. This extends Lanford's landmark theorem (1975), which justifies this derivation for a sufficiently short time. In a companion paper (arXiv:2503.01800), we connect this derivation to existing literature on hydrodynamic limits. This completes the resolution of Hilbert's Sixth Problem pertaining to the derivation of fluid equations from Newton's laws, in the case of a rarefied, hard sphere gas. 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 propagating a long-time cumulant ansatz, which keeps memory of the full collision history of the relevant particles, by an important partial time expansion. The heart of the matter is proving the smallness of these cumulants in $L^1$, which can be reduced to combinatorial properties for the associated diagrams which we call molecules. These properties are then proved by devising an elaborate cutting algorithm, which is a major novelty of this work.

Figures (64)

• Figure 2: Cutting a molecule $\mathbb M$ into smaller molecules $\mathbb M_1$ and $\mathbb M_2$ (where red atoms in $\mathbb M_1$ are cut as free, and green atoms in $\mathbb M_2$ cut as fixed). The fixed ends are marked with red $\times$ symbols.
• Figure 3: An example of molecules.
• Figure 4: Admissible trajectories (Definition \ref{['def.admissible']}), where it is allowed to have $|x_i-x_j|<\varepsilon$, but two particles reaching distance $\varepsilon$ from "inside" will not collide and not change velocity, as they do not satisfy the pre-collisional condition.
• Figure 5: An illustration of topological reduction.
• Figure 6: Creating and deleting O-atoms as in Definition \ref{['def.create_delete']}. Note that each atom $\mathfrak n_j$ can be either C- or O-atoms, or absent.

Theorems & Definitions (253)

• Definition 1.1: The hard sphere dynamics Alexander
• Proposition 1.2
• proof
• Definition 1.3: The grand canonical ensemble
• Definition 1.4: The hard sphere Boltzmann equation
• Theorem 1
• Remark 1.5
• Definition 1.6: Parameters and notations
• Definition 3.1: Molecules
• Definition 3.2: Full molecules