Boltzmann Grad Research Articles

Periodic Lorentz gas with small scatterers

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size rho may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive sqrt{nlog n} scaling (i) for fixed infinite horizon configurations—letting first nrightarrow infty and then rho rightarrow 0—studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first rho rightarrow 0 and then nrightarrow infty —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .

Rates of convergence for the superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas

In this article, we obtain the rates of convergence for superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas, which is one of the fundamental models for studying diffusions in deterministic systems. In their seminal work, Marklof and Strömbergsson (2011) proved the Boltzmann–Grad limit of the periodic Lorentz gas, following which Marklof and Tóth (2016) established a superdiffusive central limit theorem in large time for the Boltzmann–Grad limit. Based on their work, we apply Stein’s method to derive the convergence rates for the superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas. The convergence rate in Wasserstein distance is obtained for the discrete-time displacement, while the result for the Berry–Essen type bound is presented for the continuous-time displacement.

Efficient algorithm for simulating particles in true quasiperiodic environments

We introduce an algorithm based on generalized dual method (GDM) to efficiently study the dynamics of a particle in quasiperiodic environments without the need to use periodic approximations or to save the information of the vertices that make up the quasiperiodic lattice. We show that the computation time and the memory required to find the tile in which a particle is located as a function of the distance R to the center of symmetry remains constant in our algorithm, while using the GDM directly both quantities go like R 2. This allows us to perform realistic simulations with low consumption of computational resources. The algorithm can be used to study any quasiperiodic lattice that can be produced by the cut-and-project method. Using this algorithm, we have calculated the free path length distribution in quasiperiodic Lorentz gases reproducing previous results and for systems with high symmetries at the Boltzmann–Grad limit. We have found for the Boltzmann–Grad limit, that the distribution of free paths depends on the rank r of the quasiperiodic system and not on its symmetry. The distribution as a function of the free path length l appears to be a combination of exponential decay and a power-law behavior. The latter seems to become important only for probabilities less than , showing an exponential decaying free-path length distribution for r → ∞, similar to what is observed in disordered systems.

Invariance Principle for the Random Wind-Tree Process

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler.

Quantum Transport in a Crystal with Short-Range Interactions: The Boltzmann\u2013Grad Limit

We study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann–Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry–Tabor conjecture in quantum chaos.

On the derivation of the wave kinetic equation for NLS

AbstractA fundamental question in wave turbulence theory is to understand how the wave kinetic equation describes the long-time dynamics of its associated nonlinear dispersive equation. Formal derivations in the physics literature, dating back to the work of Peierls in 1928, suggest that such a kinetic description should hold (for well-prepared random data) at a largekinetic time scale$T_{\mathrm {kin}} \gg 1$and in a limiting regime where the sizeLof the domain goes to infinity and the strength$\alpha $of the nonlinearity goes to$0$(weak nonlinearity). For the cubic nonlinear Schrödinger equation,$T_{\mathrm {kin}}=O\left (\alpha ^{-2}\right )$and$\alpha $is related to the conserved mass$\lambda $of the solution via$\alpha =\lambda ^2 L^{-d}$.In this paper, we study the rigorous justification of this monumental statement and show that the answer seems to depend on the particularscaling lawin which the$(\alpha , L)$limit is taken, in a spirit similar to how the Boltzmann–Grad scaling law is imposed in the derivation of Boltzmann’s equation. In particular, there appear to betwofavourable scaling laws: when$\alpha $approaches$0$like$L^{-\varepsilon +}$or like$L^{-1-\frac {\varepsilon }{2}+}$(for arbitrary small$\varepsilon $), we exhibit the wave kinetic equation up to time scales$O(T_{\mathrm {kin}}L^{-\varepsilon })$, by showing that the relevant Feynman-diagram expansions converge absolutely (as a sum over paired trees). For the other scaling laws, we justify the onset of the kinetic description at time scales$T_*\ll T_{\mathrm {kin}}$and identify specific interactions that become very large for times beyond$T_*$. In particular, the relevant tree expansion diverges absolutely there. In light of those interactions, extending the kinetic description beyond$T_*$toward$T_{\mathrm {kin}}$for such scaling laws seems to require new methods and ideas.

Is the Boltzmann Equation Reversible? A Large Deviation Perspective on the Irreversibility Paradox

We consider the kinetic theory of dilute gases in the Boltzmann–Grad limit. We propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann molecular chaos hypothesis (Stosszahlansatz), we derive a large deviation rate function, or action, that describes the stochastic process for the empirical distribution. The quasipotential for this action is the negative of the entropy when the conservation laws are verified, as should be expected. While the Boltzmann equation appears as the most probable evolution, corresponding to a law of large numbers, the action describes a genuine reversible stochastic process for the empirical distribution, in agreement with the microscopic reversibility. As a consequence, this large deviation perspective gives the expected meaning to the Boltzmann equation and explains its irreversibility as the natural consequence of limiting the physical description to the most probable evolution. More interestingly, it also quantifies the probability of any dynamical evolution departing from solutions of the Boltzmann equation. This picture is fully compatible with the heuristic classical view of irreversibility, but makes it much more precise in various ways. We also explain that this large deviation action provides a natural gradient structure for the Boltzmann equation.

Boltzmann–Grad asymptotic behavior of collisional dynamics

This paper discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.

On the Boltzmann–Grad Limit for Smooth Hard-Sphere Systems

The problem is posed of the prescription of the so-called Boltzmann–Grad limit operator ( $$\mathcal {L}_{BG}$$ ) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator $$\mathcal {L}_{BG}$$ , the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente (Found Phys 45:404, 2015) that there is “no time-asymmetric ingredient” in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The proof is couched on the “ab initio” axiomatic approach to the classical statistical mechanics recently developed (Tessarotto et al. in Eur Phys J Plus 128:32, 2013). Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out.

Low-Density Asymptotic Behavior of Observables of Hard Sphere Fluids

The paper deals with a rigorous description of the kinetic evolution of a hard sphere system in the low-density (Boltzmann–Grad) scaling limit within the framework of marginal observables governed by the dual BBGKY (Bogolyubov–Born–Green–Kirkwood–Yvon) hierarchy. For initial states specified by means of a one-particle distribution function, the link between the Boltzmann–Grad asymptotic behavior of a nonperturbative solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables and a solution of the Boltzmann kinetic equation for hard sphere fluids is established. One of the advantages of such an approach to the derivation of the Boltzmann equation is an opportunity to describe the process of the propagation of initial correlations in scaling limits.

One-sided convergence in the Boltzmann–Grad limit

We review various contributions on the fundamental work of Lanford [21] deriving the Boltzmann equation from (reversible) hard-sphere dynamics in the low density limit.

Irreversibility in the Derivation of the Boltzmann Equation

Uffink and Valente (Found Phys 45:404–438, 2015) claim that there is no time-asymmetric ingredient that, added to the Hamiltonian equations of motion, allows to obtain the Boltzmann equation within the Lanford’s derivation. This paper is a discussion and a reply to that analysis. More specifically, I focus on two mathematical tools used in this derivation, viz. the Boltzmann–Grad (B–G) limit and the incoming configurations. Although none of them are time-asymmetric ingredients, by themselves, I claim that the use of incoming configurations, as taken within the B–G limit, is such a time-asymmetric ingredient. Accordingly, this leads to reconsider a kind of Stoszahlansatz within Lanford’s derivation.

From Hard Sphere Dynamics to the Stokes–Fourier Equations: An Analysis of the Boltzmann–Grad Limit

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of $N$ hard spheres of diameter $\varepsilon$ in two space dimensions, when~$N\to \infty$, $\varepsilon \to 0$, $N\varepsilon =\alpha \to \infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy \cite{lanford}, and on the pruning procedure developed in \cite{BGSR1} to improve the convergence time. The main novelty here is that uniform $L^2$ a priori estimates combined with a subtle symmetry argument provide a useful cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.

From Newton’s Law to the Linear Boltzmann Equation Without Cut-Off

We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann–Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford’s strategy. Our proof relies then on a combination of Lanford’s strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.

The Boltzmann–Grad limit of a hard sphere system: analysis of the correlation error

We present a quantitative analysis of the Boltzmann-Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order $k$ measures the event where $k$ particles are connected by a chain of interactions preventing the factorization. We show that, provided $k < \varepsilon^{-\alpha}$, such an error flows to zero with the average density $\varepsilon$, for short times, as $\varepsilon^{\gamma k}$, for some positive $\alpha,\gamma \in (0,1)$. This provides an information on the size of chaos, namely, $j$ different particles behave as dictated by the Boltzmann equation even when $j$ diverges as a negative power of $\varepsilon$. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.

Superdiffusion in the Periodic Lorentz Gas

We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times t and low scatterer densities (Boltzmann–Grad limit). The normalization factor is $${\sqrt{t {\rm log} t}}$$ , where t is measured in units of the mean collision time. This result holds in any dimension and for a general class of finite-range scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.

Efficient algorithms for general periodic Lorentz gases in two and three dimensions

We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly efficient as the radius of the obstacles tends to 0, the so-called Boltzmann–Grad limit. The 2D algorithm applies continued fractions to obtain the exact disc with which a particle will collide at each step, instead of using periodic boundary conditions as in the classical algorithm. The 3D version incorporates the 2D algorithm by projecting to the three coordinate planes. As an application, we calculate distributions of free path lengths close to the Boltzmann–Grad limit for certain Lorentz gases. We also show how the algorithms may be applied to deal with general crystal lattices.

From hard spheres dynamics to the Stokes–Fourier equations: An [formula omitted] analysis of the Boltzmann–Grad limit

We derive the Stokes–Fourier equations in dimension 2 as the limiting dynamics of a system of N hard spheres of diameter ε when N→∞, ε→0, Nε=α→∞, using the linearized Boltzmann equation as an intermediate step. Our proof is based on the strategy of Lanford [6], and on the pruning procedure developed in [3] to improve the convergence time. The main novelty here is that uniform a priori estimates come from a L2 bound on the initial data, the time propagation of which involves a fine symmetry argument and a systematic study of recollisions.

Theory of collisional invariants for the Master kinetic equation

Abstract The paper investigates the integral conservation properties of the Master kinetic equation, which provides an exact kinetic statistical description for the Boltzmann–Sinai classical dynamical system. It is proved that, besides the customary Boltzmann collisional invariants, this equation admits also a class of generalized collisional invariants (GCI). The result applies only when the number N and the diameter σ of hard-spheres are finite. This includes the case of dilute gases for which suitable asymptotic ordering conditions hold. However, when the Boltzmann–Grad limit is performed on the Master kinetic equation, it is shown that the existence of GCI is not permitted anymore.

Diffusion in the Lorentz Gas

The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann—Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.