Validity of the Boltzmann equation beyond short-range, repulsive, whole-space dynamics: the Pulvirenti–Saffirio–Simonella open problems (long-range interactions, constructive rates for attractive potentials, and bounded domains)

The theorem of Lanford (1975), extended to smooth short-range potentials by Pulvirenti–Saffirio–Simonella (2014), rigorously derives the Boltzmann equation from Newtonian many-particle dynamics in the low-density (Boltzmann–Grad) limit, but only under strong idealizations: compactly supported interactions, constructive convergence rates only for purely repulsive potentials, motion in the whole space, and a time interval that is a small fraction of the mean free time. The source paper's concluding remarks pose the natural open problems of removing the first three of these idealizations: (i) validity for long-range interactions, such as inverse power-law potentials or interaction ranges growing with the particle number; (ii) explicit constructive convergence rates for stable potentials with attractive parts, where two-body scattering times are unbounded; and (iii) validity in bounded domains with boundary conditions such as specular reflection. The fourth idealization — short-time validity, shared by all Lanford-type results of that generation — is not posed as an open problem in the concluding remarks and appears below only as background context. In the interaction-potential setting of the source, the bounded-domain component has been settled for hard spheres (Dolmaire 2023; Le Bihan 2021) and the linearized long-range problem has been resolved (Le Bihan 2026), but the nonlinear validity problem for long-range potentials, constructive rates for attractive potentials, and the potential-case extension to bounded domains all remain open.

Background

The Boltzmann equation $\partial_t f + v\cdot\nabla_x f = Q(f,f)$ describes the evolution of the one-particle distribution function $f(t,x,v)$ of a rarefied gas; the collision operator $Q$ encodes the statistics of elastic binary collisions through a kernel determined by the intermolecular force. A foundational question of statistical mechanics, and a core instance of Hilbert's sixth problem on the axiomatization of physics, is whether this equation can be rigorously derived from the deterministic Newtonian dynamics of $N$ particles. The relevant scaling is the low-density or Boltzmann–Grad limit: in three dimensions the particle diameter (or interaction range) $\varepsilon$ tends to zero while $N\varepsilon^2$ stays constant, so that each particle undergoes $O(1)$ collisions per unit time. Lanford's theorem (1975) established, for a gas of hard spheres started from chaotic (asymptotically factorized) initial data, that the first marginal of the $N$-particle distribution converges to the solution of the Boltzmann equation — but only on a time interval that is a small fraction of the mean free time (the average time between two successive collisions of a given particle), and only for hard spheres. The proof proceeds through the BBGKY hierarchy, the chain of equations relating the $j$-particle marginals of the Liouville evolution, and combines two ingredients: convergence of this hierarchy, term by term, to the Boltzmann hierarchy (propagation of molecular chaos, i.e. the persistence of statistical independence of particles before they collide), and the smallness of recollisions, i.e. of correlated collision events that would invalidate the chaotic factorization. A central difficulty in extending beyond hard spheres is that for smooth interaction potentials $\Phi$, binary encounters are not instantaneous but two-body scattering events of finite duration, related to the collision kernel through the classical differential cross-section of the potential. Pulvirenti–Saffirio–Simonella (2014) proved the validity theorem for smooth, stable, short-range potentials (compact support of size one in microscopic units) in the whole space, where stable means that the potential energy of any cluster is bounded below — a condition allowing attractive wells that prevents the collapse of particle clusters. For purely repulsive potentials they obtained an explicit (constructive) rate of convergence of the microscopic marginals to the Boltzmann solution; for general stable potentials with attractive parts they proved only qualitative convergence, because attractive wells can trap pairs into orbiting trajectories with arbitrarily long scattering times, and the required scattering-time estimate (Lemma 9 of that paper) is non-constructive. The same authors' concluding remarks state what remains open, and the first three of these limitations delimit the present problem. First, real molecules interact via inverse power-law forces (e.g. the attractive tails of the Lennard-Jones pair potential, the standard model for inert molecules, which decay as an inverse power of the intermolecular distance), for which the interaction range is infinite and the total cross-section diverges; the physically appropriate limiting equation is then the non-cutoff Boltzmann equation, the version of the equation whose angular collision kernel is not integrable, so that the cumulative effect of many grazing (small-deflection) collisions replaces instantaneous binary impacts, and it is, in their words, not clear whether it can be derived under the low-density scaling; even a sequence of potentials with range gently diverging with $N$ eluded their techniques. Second, a constructive proof with explicit rates in the attractive (merely stable) case is missing. Third, gases are confined: in a bounded box with, say, specular reflection at the wall (mirror-like elastic bouncing, with angle of incidence equal to angle of reflection), the dynamical flow is only defined almost everywhere and the geometric control of recollisions requires new arguments. In addition, as background context beyond the concluding remarks' open problems, the derivation — like all Lanford-type results of that generation — is short-time: global validity was known only in special expanding-vacuum (small-data) situations, while macroscopic physics requires times on which hydrodynamic behavior emerges. Later surveys (Gallagher–Saint-Raymond–Texier 2014; Saffirio 2016; Bodineau–Gallagher–Saint-Raymond–Simonella 2023) present exactly this list as the boundary of what is rigorously understood about the derivation of Boltzmann's equation from Newtonian mechanics for interaction potentials.

Problem Statement

Establish a rigorous derivation of the Boltzmann equation from the Newtonian dynamics of a dilute gas of $N$ particles interacting through smooth stable potentials, in the Boltzmann–Grad limit $N\varepsilon^2\to\mathrm{const}$, that removes the three idealizing hypotheses whose removal the concluding remarks of Pulvirenti–Saffirio–Simonella (2014) pose as open problems, beyond the short-range, whole-space setting in which they proved validity on a short time interval. The common scientific objective is a validity theorem for Boltzmann's equation under the conditions satisfied by real gases — whose molecules interact via long-range inverse power-law forces, may attract, and are confined — with the three requested extensions being: (P1) long-range interactions: prove (or disprove) convergence of the BBGKY marginals to the Boltzmann solution when the interaction is an inverse power-law potential (yielding the non-cutoff collision operator), or for a sequence of short-range potentials whose range grows with $N$; (P2) constructive rates: for stable potentials with attractive parts, replace the qualitative convergence by an explicit constructive error bound in $\varepsilon$; (P3) confinement: extend validity to bounded domains with boundary conditions, in particular specular reflection at the wall, where the flow is only almost everywhere defined and recollisions with the boundary require new geometric control. Each component asks exactly that the corresponding hypothesis be removed, with no hidden substitute restriction (no angular cutoff, no smallness of data, no reduced geometry) reintroduced; the hard-sphere analogues of (P3) settled in 2021–2025 do not answer the questions as posed for interaction potentials. The three components (P1)–(P3) are independently standing open problems: each can be resolved on its own, so that resolving any one is a valid partial result that leaves the remaining components open, while a full resolution of the objective requires all three.

References

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