Objective molecular dynamics (OMD) exploits a group of symmetries to simulate macroscopic fluid motion with a small representative set of particles, and its Boltzmann-Grad scaling leads formally to a homo-energetic Boltzmann equation: a spatially homogeneous kinetic equation with an additional time-dependent hyperbolic (drift) term −L(t)w⋅∇wg-L(t)w\cdot\nabla_w g, where L(t)=A(I+tA)−1L(t)=A(I+tA)^{-1} encodes the imposed affine deformation. James, Qi, and Wang derived, by formal Hilbert and Chapman-Enskog expansions in the small Knudsen number ε\varepsilon, a reduced Euler system of ODEs for the density ρ(t)\rho(t) and temperature θ(t)\theta(t), and a first-order Navier-Stokes correction featuring a viscosity μ(θ)\mu(\theta) given by an explicit quadrature. The open problem is to make these expansions rigorous: prove that solutions of the scaled homo-energetic equation converge as ε→0\varepsilon\to0, with quantitative error control (ideally global in time), to the Maxwellian profile governed by the reduced Euler ODEs, and justify the Chapman-Enskog correction and the viscosity formula μ(θ)\mu(\theta). No existing rigorous result for homoenergetic flows establishes this hydrodynamic limit: known rigorous results cover well-posedness and long-time asymptotics in special regimes.

Background

Objective molecular dynamics (OMD) is a reduced-order particle simulation method in which only a small fundamental cell of particles is propagated by Newton's laws while all other particles are generated by a symmetry group (isometries composed with an affine flow); it was introduced through R. D. James's theory of objective structures and is used to simulate macroscopic fluid motion atomistically. For the simplest translation-group case, the particle dynamics carries a constant matrix A∈R3×3A\in\mathbb{R}^{3\times3} such that the macroscopic velocity field is u(t,x)=A(I+tA)−1xu(t,x)=A(I+tA)^{-1}x; write L(t)=A(I+tA)−1L(t)=A(I+tA)^{-1}. James, Qi, and Wang (2024) connected this microscopic picture to kinetic theory: under a Boltzmann-Grad scaling (particle number N→∞N\to\infty, interaction range ε→0\varepsilon\to0 with Nε2N\varepsilon^2 fixed), the single-particle distribution g(t,w)g(t,w) in the deviation variable w=v−L(t)xw=v-L(t)x formally satisfies the homo-energetic Boltzmann equation, a spatially homogeneous equation whose left-hand side contains, besides the time derivative, the hyperbolic transport term −L(t)w⋅∇wg-L(t)w\cdot\nabla_w g representing the imposed shear/dilatation. The term 'homo-energetic' (Cercignani) refers to solutions of the full Boltzmann equation of the special affine form f(t,x,v)=g(t,v−L(t)x)f(t,x,v)=g(t,v-L(t)x); the collision operator Q(g,g)Q(g,g) is the standard quadratic Boltzmann collision operator for, e.g., hard spheres or hard potentials with angular cutoff. Existence of homo-energetic solutions for shear flow goes back to Cercignani (1989).

James–Qi–Wang then insert a Knudsen number ε\varepsilon (the ratio of mean free path to macroscopic scale, realized by rescaling the interaction) and consider

∂tgε(t,w)−(L(t)w)⋅∇wgε=1ε Q(gε,gε).\partial_t g^\varepsilon(t,w)-\big(L(t)w\big)\cdot\nabla_w g^\varepsilon=\frac{1}{\varepsilon}\,Q\big(g^\varepsilon,g^\varepsilon\big).

Seeking gεg^\varepsilon as a power series g0+εg1+⋯g_0+\varepsilon g_1+\cdots (Hilbert expansion), the order ε−1\varepsilon^{-1} relation Q(g0,g0)=0Q(g_0,g_0)=0 forces g0g_0 to be the Maxwellian (Gaussian equilibrium) M[ρ,θ](w)=ρ(2πθ)−3/2e−∣w∣2/(2θ)M[\rho,\theta](w)=\rho(2\pi\theta)^{-3/2}e^{-|w|^2/(2\theta)}, and the next order yields a closed system of ODEs for the density and temperature,

∂tρ=−Tr(L) ρ,∂tθ=−23 Tr(L) θ,\partial_t\rho=-\mathrm{Tr}(L)\,\rho,\qquad \partial_t\theta=-\tfrac{2}{3}\,\mathrm{Tr}(L)\,\theta,

a reduced compressible Euler system (the spatial homogeneity inherited from the OMD symmetry reduces the PDEs to ODEs). Keeping the first correction in a Chapman-Enskog expansion (the classical gradient-driven correction to local equilibrium) gives, to order ε\varepsilon, the viscous-heating term

∂tθ+23 Tr(L) θ=ε μ(θ) 12[Tr(L2)+L:L−23(Tr L)2],\partial_t\theta+\tfrac{2}{3}\,\mathrm{Tr}(L)\,\theta=\varepsilon\,\mu(\theta)\,\tfrac{1}{2}\Big[\mathrm{Tr}\big(L^2\big)+L{:}L-\tfrac{2}{3}\big(\mathrm{Tr}\,L\big)^{2}\Big],

where L:L=∑ijLijLijL{:}L=\sum_{ij}L_{ij}L_{ij} and the viscosity μ(θ)\mu(\theta) is given by an explicit quadrature over the collision kernel (formula (74) of James–Qi–Wang); this is the reduced compressible Navier-Stokes system for the OMD class and matches the continuum OMD equations used in atomistic fluid simulation.

For the classical Boltzmann equation, the analog of this passage — the hydrodynamic limit as the Knudsen number tends to zero — is a central chapter of Hilbert's sixth problem: Caflisch (1980) rigorously justified the compressible Euler limit for smooth solutions, and incompressible limits are also understood, while convergence to the compressible Navier-Stokes system with validated viscosity remains open even there. For the homo-energetic equation the standard frameworks do not apply directly, because the time-dependent drift L(t)w⋅∇wL(t)w\cdot\nabla_w is of order one while the collision operator is amplified by 1/ε1/\varepsilon: the local Maxwellian is not invariant under the transport part, the deformation matrix breaks the rotational symmetry used in classical proofs, and the viscous-heating source produces unbounded temperature growth. James–Qi–Wang state explicitly that this kinetic-to-macroscopic route is established 'only in a heuristic manner' and that the undertaking is challenging due to the hyperbolic term; their Hilbert and Chapman-Enskog derivations are formal power series without convergence or error estimates. Related rigorous theory exists only for different asymptotics: the long-time (t→∞t\to\infty) analysis of homoenergetic solutions by James–Nota–Velázquez in the collision-dominated and hyperbolic-dominated regimes (the latter producing non-Maxwellian tails), Kepka's long-time stability of high-temperature Maxwellians for hard potentials, and Duan–Liu's well-posedness and small-shear-strength expansion for uniform shear flow with hard potentials.

Problem Statement

For a constant matrix A∈R3×3A\in\mathbb{R}^{3\times3} (with I+tAI+tA invertible for t≥0t\ge0) set L(t)=A(I+tA)−1L(t)=A(I+tA)^{-1}, and let gεg^\varepsilon solve the scaled homo-energetic Boltzmann equation of objective molecular dynamics

∂tgε(t,w)−(L(t)w)⋅∇wgε=1ε Q(gε,gε),gε∣t=0=ginε,\partial_t g^\varepsilon(t,w)-\big(L(t)w\big)\cdot\nabla_w g^\varepsilon=\frac{1}{\varepsilon}\,Q\big(g^\varepsilon,g^\varepsilon\big),\qquad g^\varepsilon\big|_{t=0}=g^{\varepsilon}_{\mathrm{in}},

with QQ the Boltzmann collision operator for hard spheres (or, more generally, hard potentials with angular cutoff) and well-prepared initial data ginεg^{\varepsilon}_{\mathrm{in}} close to a Maxwellian M[ρ0,θ0]M[\rho_0,\theta_0]. Prove, with quantitative error bounds:

  1. (Reduced Euler limit) As ε→0\varepsilon\to0, the solution converges, with a quantitative rate on finite time intervals and, where feasible, uniformly for all t≥0t\ge0, to the local Maxwellian M[ρ(t),θ(t)](w)M[\rho(t),\theta(t)](w), where (ρ,θ)(\rho,\theta) solve the reduced Euler ODEs ∂tρ=−Tr(L) ρ\partial_t\rho=-\mathrm{Tr}(L)\,\rho, ∂tθ=−23 Tr(L) θ\partial_t\theta=-\tfrac{2}{3}\,\mathrm{Tr}(L)\,\theta with initial data (ρ0,θ0)(\rho_0,\theta_0).

  2. (Chapman–Enskog correction and viscosity validation) Justify the first-order correction, i.e. prove that the temperature evolves according to ∂tθ+23Tr(L)θ=ε μ(θ) 12[Tr(L2)+L:L−23(TrL)2]+o(ε)\partial_t\theta+\tfrac{2}{3}\mathrm{Tr}(L)\theta=\varepsilon\,\mu(\theta)\,\tfrac{1}{2}\big[\mathrm{Tr}(L^2)+L{:}L-\tfrac{2}{3}(\mathrm{Tr}L)^2\big]+o(\varepsilon) with the viscosity μ(θ)\mu(\theta) given by the quadrature formula (74) of James–Qi–Wang applied to the chosen collision kernel, thereby validating the reduced Navier–Stokes system and its transport coefficient for the OMD class of affine flows.

References

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