Global validity of the Boltzmann equation under the initial smoothness ordering in the Boltzmann–Grad limit of the Master kinetic equation

For a finite system of smooth hard spheres of diameter $\sigma$ with $N \equiv 1/\varepsilon$ particles in the Boltzmann–Grad scaling $N\sigma^{2} \sim O(1)$, the Master kinetic equation — an exact kinetic equation for the 1-body probability density function (PDF) of the finite system — yields the Boltzmann equation as its left Boltzmann–Grad (BG) limit, provided a global-in-time smoothness ordering $\delta \equiv \sigma/L_{\rho} \sim O(\varepsilon^{1/2})$ holds, where $L_{\rho}$ is the characteristic spatial scale-length of the 1-body PDF. Tessarotto, Asci, Cremaschini, Mond, Soranzo and Tironi (2017/2018) proved this implication through propositions P1–P5 but explicitly left open whether the corresponding initial ordering $\delta(t_{o}) \sim O(\varepsilon^{1/2})$ (condition (a)), i.e. smooth initial data with $L_{\rho}(t_{o}) \sim O(1)$, suffices: it is open whether the Master-equation dynamics preserves the ordering for all later times — so that the BG-limit propositions and the global-in-time validity of the Boltzmann equation continue to hold — or whether finite-$N$ dynamics can produce violations. The question is coupled to the problem of decay to kinetic equilibrium for the Master kinetic equation and to characterizing the initial data for which global validity holds.

Background

A classical monatomic gas of hard spheres is governed, at the kinetic level, by the Boltzmann equation, an evolution equation for a single-particle probability density function (PDF) in phase space, with a quadratic collision operator modeling elastic binary encounters. A foundational question of statistical mechanics, posed by Boltzmann and Grad, is whether this equation can be derived from the underlying Newtonian dynamics of the $N$-particle system when $N \to \infty$ and the hard-sphere diameter $\sigma \to 0$ in the Boltzmann–Grad (BG) scaling $N\sigma^{d-1} \sim O(1)$ in $d$ dimensions. Lanford's theorem (1975) gave a rigorous derivation, but only on a time interval that is a fraction of the mean collision time; as emphasized in the standard reference on the subject (Gallagher, Saint-Raymond and Texier, 2014, arXiv:1208.5753), this restriction reflects a genuine limitation of the uniform estimates on the BBGKY hierarchy (the chain of equations relating multi-particle PDFs), and the global-in-time validity of the Boltzmann equation remains one of the great open problems of kinetic theory.

A separate line of work (Tessarotto and collaborators, 2013–2018) developed an "ab initio" axiomatic approach to the statistical mechanics of finite hard-sphere systems, whose central object is the Master kinetic equation: an exact, non-asymptotic kinetic equation for the 1-body PDF of a finite system of $N$ smooth hard spheres undergoing unary, binary, as well as multiple instantaneous elastic collisions. The equation is written for a renormalized 1-body PDF $\bar{\rho}_{1}^{(N)}$ in which an ensemble theta function $\Theta$ (a product of step functions forbidding particle overlaps and wall penetration) is absorbed; multi-particle correlations enter through occupation coefficients $k_{s}^{(N)}$ ($s$-body integrals over products of $\bar{\rho}_{1}^{(N)}$ and $\Theta$), whose convergence to $1$ in the BG limit expresses the molecular chaos (statistical independence) of the limit. The paper by Tessarotto, Asci, Cremaschini, Mond, Soranzo and Tironi (arXiv:1705.05277; journal version in Foundations of Physics 48, 271–294, 2018) shows that the BG-limit operator does not commute with the Master-equation evolution operator, and that only the left BG-limit (the limit applied to the equation itself) produces the Boltzmann equation. It further proves five propositions: P1–P2, the BG-limits $k_{1}^{(N)} \to 1$ and $k_{s}^{(N)} \to 1$ for all $s \geq 2$; P3, convergence of the factorized $s$-body PDFs; P4, the left BG-limit of the Master kinetic equation equals the Boltzmann equation; P5, the right BG-limit does not. These are established under: (i) global existence of stochastic (i.e. smoothly differentiable, strictly positive, with finite velocity moments and finite Boltzmann–Shannon entropy) solutions of the Master kinetic equation, proven separately by the same group; and (ii) a global-in-time smoothness ordering. Specifically, writing $\varepsilon \equiv 1/N$ and requiring $N\sigma^{2} \sim O(1)$ and macroscopic scale $L_{o} \sim O(1)$, so that $\sigma \sim \varepsilon^{1/2}$, the smoothness of the 1-body PDF is quantified by the scale-lengths

$$L_{\rho}(t_{o}) = \inf_{x_{1} \in \Gamma_{1}} \left| \frac{\partial \ln \rho_{1}^{(N)}(x_{1}, t_{o})}{\partial r_{1}} \right|^{-1}, \qquad L_{\rho} = \inf_{(x_{1},t) \in \Gamma_{1} \times I} \left| \frac{\partial \ln \rho_{1}^{(N)}(x_{1}, t)}{\partial r_{1}} \right|^{-1},$$

where $\Gamma_{1}$ is the 1-particle phase space, $x_{1} = (r_{1}, v_{1})$ position and velocity, and $I$ the time interval of existence; the ordering conditions are the initial one (a) $\delta(t_{o}) \equiv \sigma/L_{\rho}(t_{o}) \sim O(\varepsilon^{1/2})$ and the global-in-time one (b) $\delta \equiv \sigma/L_{\rho} \sim O(\varepsilon^{1/2})$. Under (b) — assumed to hold for all times — propositions P1–P5 hold globally, and the left BG-limit of the Master kinetic equation coincides with the Boltzmann equation for all times, i.e. the Boltzmann equation is globally valid. The paper's concluding section states explicitly that the global validity problem "remains still unsolved" if the global assumption (b) is replaced by the initial assumption (a) alone, and raises the question of whether, and under what initial conditions — with a role played by the phenomenon of decay to kinetic equilibrium of the 1-body PDF towards a spatially homogeneous Maxwellian (the equilibrium velocity distribution of the Boltzmann equation), studied for the Master equation in companion papers — the ordering (b) is satisfied or violated by arbitrary smooth initial 1-body PDFs with $L_{\rho}(t_{o}) \sim L_{o} \sim O(1)$.

Problem Statement

In the framework of the Master kinetic equation for a finite system of $N \equiv 1/\varepsilon$ smooth hard spheres of diameter $\sigma$ in the Boltzmann–Grad scaling $N\sigma^{2} \sim O(1)$, consider stochastic initial 1-body PDFs $\rho_{1}^{(N)}(x_{1}, t_{o})$ — smoothly differentiable, strictly positive, with finite velocity moments and finite Boltzmann–Shannon entropy — whose initial characteristic scale-length satisfies $L_{\rho}(t_{o}) \sim L_{o} \sim O(\varepsilon^{0})$, so that the initial smoothness ordering

$$\delta(t_{o}) \equiv \frac{\sigma}{L_{\rho}(t_{o})} \sim O\bigl(\varepsilon^{1/2}\bigr)$$

(condition (a)) holds. The open question posed by the source is: does the time evolution governed by the Master kinetic equation preserve the smoothness ordering $\delta \equiv \sigma/L_{\rho} \sim O(\varepsilon^{1/2})$ for all later times (condition (b)), so that condition (a) alone — together with global existence of stochastic solutions — suffices for the Boltzmann–Grad limit propositions P1–P5 to hold at arbitrary times and the left BG-limit of the Master kinetic equation to coincide with the Boltzmann equation globally in time? Or can the finite-$N$ dynamics produce violations of the ordering that invalidate the derivation of the Boltzmann equation at later times? The coupled tasks are: (i) determine whether, and under what conditions on the initial data (including the role of the phenomenon of decay to kinetic equilibrium of the 1-body PDF for the Master kinetic equation), the global ordering (b) is satisfied or violated; and (ii) characterize the class of stochastic initial 1-body PDFs with $L_{\rho}(t_{o}) \sim O(1)$ for which the BG limit yields the Boltzmann equation globally in time. Task (i) is the required resolution of the open question, and a decisive answer to it settles the posed disjunction; task (ii) is required only in the event that the answer to (i) is that violations occur, since if the initial ordering (a) suffices for (b) the class sought in (ii) is the full class of admissible initial data.

References

1. M. Tessarotto, C. Asci, C. Cremaschini, M. Mond, A. Soranzo, G. Tironi, "On the Boltzmann-Grad Limit of the Master Kinetic Equation", arXiv:1705.05277 (2017), https://arxiv.org/abs/1705.05277
2. M. Tessarotto, C. Cremaschini, M. Mond, C. Asci, A. Soranzo, G. Tironi, "On the Boltzmann–Grad Limit for Smooth Hard-Sphere Systems", Foundations of Physics 48, 271–294 (2018), DOI 10.1007/s10701-018-0144-5, https://doi.org/10.1007/s10701-018-0144-5
3. Massimo Tessarotto and Claudio Cremaschini, "Macroscopic Irreversibility and Decay to Kinetic Equilibrium of the 1-Body PDF for Finite Hard-Sphere Systems", Advances in Mathematical Physics 2018, Article ID 1931308 (2018), DOI 10.1155/2018/1931308, arXiv:1812.01122, https://arxiv.org/abs/1812.01122
4. Massimo Tessarotto and Claudio Cremaschini, "Macroscopic irreversibility and decay to kinetic equilibrium for classical hard-sphere systems", arXiv:1703.04600 (2017), https://arxiv.org/abs/1703.04600
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6. Isabelle Gallagher, Laure Saint-Raymond, Benjamin Texier, "From Newton to Boltzmann: hard spheres and short-range potentials", arXiv:1208.5753 (2012); EMS Textbooks in Mathematics (2014), https://arxiv.org/abs/1208.5753
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