Langevin limit for a massive particle in an ideal gas of light particles: singular short-range interaction potentials and spatial dimensions $d \le 2$

The Rayleigh gas — one massive tagged particle of unit mass immersed in an ideal gas of non-interacting light particles of mass $m$, all evolving by deterministic Newtonian dynamics — is the canonical model for deriving Brownian motion from mechanics. As $m \to 0$, with the light particles' density and typical velocities diverging so that the momentum-transfer variance per unit time stays order one, the tagged particle's velocity is expected, and in settings covering collisions and smooth potentials proved, to converge to the Langevin diffusion (an Ornstein–Uhlenbeck velocity process coupled to its time integral). For smooth, compactly supported, repulsive interaction potentials, S. Liang proved this convergence in 2017 under a uniform lower bound on the initial kinetic energies of the light particles and a dimension condition that in effect forces $d \ge 6$; the author's 2021 follow-up extends this to $d \ge 3$ under the same bounded-below energy hypothesis, removing the requirement that the light particles' energies be 'high enough'. Two open directions of the same program remain: (i) extending the derivation to singular short-range repulsive potentials — repulsive, compactly supported, smooth away from the origin and diverging to infinity as $|x| \to 0$, the class motivated by the Weeks–Chandler–Andersen and truncated Lennard–Jones pair potentials (the standard molecular pair potentials: Lennard–Jones is repulsive like $r^{-12}$ at short range with an attractive $r^{-6}$ tail, and Weeks–Chandler–Andersen is its truncated, purely repulsive branch; the attractive, non-compactly-supported tail of the full Lennard–Jones potential lies outside the class treated here) — for which the force and its curvature blow up at short range, destroying the bounded-derivative estimates underlying the known proofs; and (ii) settling the smooth-potential case in spatial dimensions $d = 1, 2$, where the formal coefficient integrals are at best borderline integrable and the Langevin limit may fail or be replaced by a different scaling limit.

Background

The Rayleigh gas is a classical mechanical model of Brownian motion: one 'massive' (tagged) particle of mass normalized to 1 moves among light bath particles that do not interact with one another, only with the tagged particle, and the whole system evolves by deterministic Newtonian dynamics. The regime of interest sends the light-particle mass $m \to 0$ while their spatial density and typical velocities diverge in a coupled way (in the source papers the bath is initially a Poisson point process of intensity $m^{(d-1)/2}\rho((m/2)|v|^2, x - X_0)\,dx\,dv$ in physical velocity $v$, so that typical speeds are of order $m^{-1/2}$ and the spatial density of order $m^{-1/2}$), chosen so that each encounter transfers momentum of order $m^{1/2}$ to the tagged particle while encounters occur at rate of order $m^{-1}$; the momentum-transfer variance per unit time then stays order one. In this limit the velocity of the tagged particle is expected to obey the Langevin equation $dV = -\gamma V\,dt + \sigma\,dW_t$ with the position being its time integral — the Ornstein–Uhlenbeck process, i.e. the Gaussian Markov process solving this stochastic differential equation — so that Brownian motion is derived from Hamiltonian mechanics, a concrete instance of Hilbert's sixth problem (the axiomatization of physics). For hard-sphere (elastic-collision) interactions this was proved by Holley (1971) in one dimension and by Dürr, Goldstein and Lebowitz (1981) in three dimensions, with a half-space extension by Calderoni, Dürr and Kusuoka (1989) and a unifying survey by Szász and Tóth (1987). For smooth interaction potentials, encounters are no longer instantaneous: a light particle traverses a full two-body scattering trajectory. Kusuoka and Liang (2010) and Liang (2014) treated the case in which all light particles are 'fast' relative to the tagged one; Liang (2017) allowed slow light particles, requiring only a uniform positive lower bound $\bar{v} > 0$ on the scaled initial speeds (equivalently, kinetic energies bounded below), at the price of a dimension condition $d > 2(1+\|h''\|_\infty)(-h''(0))^{-1/2} + 1$ — where $U(x) = h(|x|)$ is the spherically symmetric, compactly supported, smooth repulsive potential — which is always at least 6, so no physical dimension is covered. The proof approximates each encounter by the two-body problem with the tagged particle held fixed (the freezing approximation), controls the error through bounds on $\nabla^2 U$ (via $\|h''\|_\infty$), bounds the logarithmic divergence of the interaction duration in the transverse offset of the incoming trajectory, decomposes the velocity increment into a martingale plus remainder terms, and identifies the limit through the generator $L = \frac{1}{2}\sum_{kl} a_{kl}\,\partial^2/\partial V_k\partial V_l + \sum_{kl} b_{kl} V_l\,\partial/\partial V_k + \sum_k V_k\,\partial/\partial X_k$, where the diffusion coefficients $a_{kl}$ and drift coefficients $b_{kl}$ are explicit integrals of the two-body scattering map and its linear response against the far-field velocity distribution and the incoming-trajectory flux measure. The author's 2021 follow-up (Liang 2021) covers, per its published abstract, dimensions $d \ge 3$ 'without any assumption requiring that the initial energies of the environmental particles should be restricted to be high enough' — the uniform lower bound on the initial energies is retained, and it is the 'high enough' restriction itself that is removed. Adjacent rigorous results treat collision models or different scalings: Bodineau, Gallagher and Saint-Raymond (2018) derived an Ornstein–Uhlenbeck limit for a massive convex body in a rarified gas of hard spheres via Lanford-type Boltzmann–Grad arguments (the Boltzmann–Grad limit being the regime of vanishing particle size with diverging density at fixed mean free path); Matthies, Stone and Theil (2018), Matthies and Syntaka (2024) and Fougères (2024, 2026) derive linear-Boltzmann-type limits — and, with fat-tailed velocity distributions, fractional diffusion — for Rayleigh-gas models at comparable masses in the Boltzmann–Grad regime, not the $m \to 0$ Langevin regime; Boldrighini, Frigio and Tognetti (2002) provide one-dimensional numerical evidence for the low-mass limit, and Bálint, Tóth and Tóth (2007) analyze the one-dimensional zero-mass limit of tagged-particle diffusion for the hard-core gas. The singular-potential case and the smooth-potential Langevin limit in dimensions $d = 1, 2$ are not covered by the results above.

Problem Statement

Consider the Rayleigh-gas system of Liang (2017): in $\mathbb{R}^d$, a massive particle with mass normalized to 1 and initial data $(X_0, V_0)$ moves among light particles of mass $m$, initially distributed as a Poisson point process with intensity $\tilde\lambda_m(dx\,dv) = m^{(d-1)/2}\rho((m/2)|v|^2, x - X_0)\,dx\,dv$, where $\rho$ is bounded, has finite moments of order three, and converges (in its spatial argument, as $|x - X_0| \to \infty$) to a far-field density $\rho_0$; light particles do not interact with each other; each (massive, light) pair interacts through a potential $U$; all particles evolve by Newton's laws. Write $(X^{(m)}, V^{(m)})$ for the massive particle's position and velocity. It is known that when $U$ is smooth, compactly supported, spherically symmetric and repulsive, and the initial light-particle kinetic energies are uniformly bounded below, $(X^{(m)}, V^{(m)})$ converges weakly as $m \to 0$ to the diffusion with generator $L = \frac{1}{2}\sum_{kl} a_{kl}\,\partial^2/\partial V_k\partial V_l + \sum_{kl} b_{kl} V_l\,\partial/\partial V_k + \sum_k V_k\,\partial/\partial X_k$, where $a_{kl}$ and $b_{kl}$ are explicit integrals of the two-body scattering data of $U$ against the far-field density $\rho_0$ and the incoming-trajectory flux measure — proved in high dimensions (effectively $d \ge 6$) by Liang (2017), and for $d \ge 3$ by Liang (2021) under the same bounded-below energy hypothesis, removing the requirement that the initial energies be 'high enough'. Two components of this program remain open, as independent directions; a resolution of either component on its own resolves the task. Component 1 (singular potentials): for interaction potentials $U$ with a singular short-range repulsive part — repulsive, compactly supported, smooth away from the origin and diverging to infinity as $|x| \to 0$, the class motivated by the Weeks–Chandler–Andersen and truncated Lennard–Jones pair potentials — establish, in at least the physical dimension $d = 3$ and under the environment assumptions of Liang (2021) (initial light-particle kinetic energies uniformly bounded below, with no requirement that they be 'high enough', together with the stated moment and tail assumptions on $\rho_0$), weak convergence of $(X^{(m)}, V^{(m)})$ as $m \to 0$ to the Langevin diffusion with coefficients given by the corresponding two-body scattering-data integrals (well defined for such repulsive singular potentials, such as the truncated repulsive branch of Lennard–Jones, i.e. Weeks–Chandler–Andersen type). Attractive, non-compactly-supported tails (full Lennard–Jones type) lie outside this class. Any solution must include a-priori well-posedness of the Newtonian dynamics (absence of collision singularities, finiteness of total impulses) for the chosen potential class. Component 2 (low dimensions): for the smooth compactly supported repulsive potentials already treated, under the assumptions of Liang (2021) (initial light-particle kinetic energies uniformly bounded below, with no requirement that they be 'high enough'), in both dimensions $d = 1$ and $d = 2$, prove the diffusion limit or prove that it fails there (for example, divergence of the formal coefficient integrals), and where it fails identify the correct scaling limit if one exists; a mixed outcome, with the limit holding in one dimension and failing in the other, resolves this component provided each of $d = 1$ and $d = 2$ is settled. The two components are distinct open directions of a single program, whose scientific target is a deterministic-mechanical derivation of Langevin dynamics for the massive-particle/light-gas system under the interaction potentials and spatial dimensions of physical interest, with friction and diffusion coefficients expressed through microscopic scattering data (in Component 2, alternatively, a rigorous determination, dimension by dimension for $d = 1$ and $d = 2$, of whether and why the Langevin picture fails there and of the correct scaling limit where one exists).

References

1. S. Liang, 'A Mechanical Model of Brownian Motion for One Massive Particle Including Slow Light Particles', Journal of Statistical Physics 170(2), 286–350 (2018; online December 2017). DOI: 10.1007/s10955-017-1934-4, https://doi.org/10.1007/s10955-017-1934-4
2. S. Liang, 'A mechanical model of Brownian motion for one massive particle including low energy light particles in dimension $d \ge 3$', Random Operators and Stochastic Equations 29(3), 203–235 (2021). DOI: 10.1515/rose-2021-2062, https://doi.org/10.1515/rose-2021-2062
3. D. Dürr, S. Goldstein, J. L. Lebowitz, 'A mechanical model of Brownian motion', Communications in Mathematical Physics 78(4), 507–530 (1981). DOI: 10.1007/BF02046762, https://doi.org/10.1007/BF02046762
4. R. Holley, 'The motion of a heavy particle in an infinite one dimensional gas of hard spheres', Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 17(3), 181–219 (1971). DOI: 10.1007/BF00536757, https://doi.org/10.1007/BF00536757
5. D. Szász, B. Tóth, 'Towards a unified dynamical theory of the Brownian particle in an ideal gas', Communications in Mathematical Physics 111(1), 41–62 (1987). DOI: 10.1007/BF01239014, https://doi.org/10.1007/BF01239014
6. P. Calderoni, D. Dürr, S. Kusuoka, 'A mechanical model of Brownian motion in half-space', Journal of Statistical Physics 55(3–4), 649–693 (1989). DOI: 10.1007/BF01041603, https://doi.org/10.1007/BF01041603
7. S. Kusuoka, S. Liang, 'A classical mechanical model of Brownian motion with plural particles', Reviews in Mathematical Physics 22(7), 733–838 (2010). DOI: 10.1142/S0129055X10004077, https://doi.org/10.1142/S0129055X10004077
8. S. Liang, 'A mechanical model of Brownian motion with uniform motion area', Journal of Mathematical Sciences, University of Tokyo 21(2), 235–334 (2014). zbMATH: 6537529, https://zbmath.org/6537529, http://hdl.handle.net/2261/59057
9. T. Bodineau, I. Gallagher, L. Saint-Raymond, 'Derivation of an Ornstein–Uhlenbeck process for a massive particle in a rarified gas of particles', Annales Henri Poincaré 19(6), 1647–1709 (2018). DOI: 10.1007/s00023-018-0674-6, https://doi.org/10.1007/s00023-018-0674-6
10. K. Matthies, G. Stone, F. Theil, 'The derivation of the linear Boltzmann equation from a Rayleigh gas particle model', Kinetic and Related Models 11(2), 1–41 (2018). DOI: 10.3934/krm.2018008, https://doi.org/10.3934/krm.2018008
11. K. Matthies, T. Syntaka, 'Fractional diffusion as the limit of a short range potential Rayleigh gas', arXiv:2405.19025 (2024). https://arxiv.org/abs/2405.19025
12. F. Fougères, 'On the derivation of the linear Boltzmann equation from the nonideal Rayleigh gas. Adaptive pruning and improvement of the convergence rate', Journal of Statistical Physics 191, Article 136 (2024). DOI: 10.1007/s10955-024-03353-1, https://doi.org/10.1007/s10955-024-03353-1
13. F. Fougères, 'About a nonideal Rayleigh gas mixture model', arXiv:2605.19694 (2026). https://arxiv.org/abs/2605.19694
14. R. Erban, R. A. Van Gorder, 'Langevin dynamics for a heavy particle immersed within a flow of light particles', Proceedings of the Royal Society A 480, 20230851 (2024). DOI: 10.1098/rspa.2023.0851, https://doi.org/10.1098/rspa.2023.0851
15. C. Boldrighini, S. Frigio, D. Tognetti, 'Numerical evidence for the low-mass behavior of the one-dimensional Rayleigh gas with local interaction', Journal of Statistical Physics 108(3–4), 703–712 (2002). DOI: 10.1023/A:1015738209984, https://doi.org/10.1023/A:1015738209984
16. P. Bálint, B. Tóth, P. Tóth, 'On the zero mass limit of tagged particle diffusion in the 1-d Rayleigh-gas', Journal of Statistical Physics 127(4), 657–675 (2007). DOI: 10.1007/s10955-007-9304-2, https://doi.org/10.1007/s10955-007-9304-2
17. D. M. Heyes, H. Okumura, 'Some physical properties of the Weeks–Chandler–Andersen fluid', Molecular Simulation 32(1), 45–50 (2006). DOI: 10.1080/08927020500529442, https://doi.org/10.1080/08927020500529442
18. C. Kim, G. E. Karniadakis, 'Brownian motion of a Rayleigh particle confined in a channel: a generalized Langevin equation approach', Journal of Statistical Physics 158, 1100–1125 (2015; online December 2014). DOI: 10.1007/s10955-014-1160-2, https://doi.org/10.1007/s10955-014-1160-2
19. S. Albeverio, 'Detlef Dürr's path from mechanics of the Brownian motion to the mechanics of the quantum world: a personal point of view', in Fundamental Theories of Physics, Springer, 11–21 (2024). DOI: 10.1007/978-3-031-45434-9_2, https://doi.org/10.1007/978-3-031-45434-9_2
20. S. Liang, 'Stochastic Newton equations in the strong potential limit for the multi-dimensional case', Stochastics and Dynamics 23(5), 2350043 (2023). DOI: 10.1142/S0219493723500430, https://doi.org/10.1142/S0219493723500430
21. S. Liang, KAKENHI Grant 17K05290, 'A classical mechanical model of diffusion process' (2017–2022), Japan Society for the Promotion of Science, https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17K05290/
22. S. Liang, KAKENHI Grant 21K03302, 'Derivation by classical mechanical systems with singularity of diffusion processes' (拡散過程の特異性がある古典力学系による導出, 2021–2025; discontinued FY2024), Japan Society for the Promotion of Science, https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21K03302/