Physical insight into superdiffusive dynamics of Sinai billiard through collision statistics

In 1981 Bunimovich and Sinai established the statistical properties of the planar periodic Lorentz gas with finite horizon. Our aim is to extend their theory to the multidimensional Lorentz gas. In that case the Markov partitions of the Bunimovich-Sinai type, the main tool of their theory, are not available. We use a crude approximation to such partitions, which we call Markov sieves. Their construction in many dimensions is essentially different from that in two dimensions; it requires more routine calculations and intricate arguments. We try to avoid technical details and outline the construction of the Markov sieves in mostly qualitative, heuristic terms, hoping to carry out our plan in full detail elsewhere. Modulo that construction, our proofs are conclusive. In the end, we obtain a stretched-exponential bound for the decay of correlations, the central limit theorem, and Donsker's Invariance Principle for multidimensional periodic Lorentz gases with finite horizon.

The Lorentz gas (LG) acts in the theory of dynamical systems as a paradigm allowing us to address fundamental issues of statistical mechanics, for instance, transport processes, such as diffusion in the configuration space [1, 2]. The static periodic LG comprises a regular lattice of circular fixed scatterers and an ensemble of non-interacting particles travelling freely between collisions and scattering elastically off the circular obstacles. The transport properties of such a system are determined by the billiard’s geometry, that is the specific lattice symmetry and the lattice constant. If the maximum free path length is not bounded from above, then the setup possesses a so-called infinite horizon (IH) and the diffusion in configuration space is anomalous [4, 5]. For a more compact packing of the scatterers, i.e. finite horizon (FH), arbitrarily long flights are not possible and the system exhibits normal diffusion.

We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term $$c=2-3\ln 2+\frac{27\zeta(3)}{2\pi^2}$$ in the asymptotic formula $$h(T)=-2 \ln \epsilon +c+o(1)$$ of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.

Heat transport coefficients are calculated for various random walks with internal states (the Markov partition of the Sinai billiard connects these walks with the Lorentz gas among a periodic configuration of scatterers). Models with reflecting or absorbing barriers and also those without or with local thermal equilibrium are investigated. The method is unified and is based on the Keldysh expansion of the resolvent of a matrix polynomial.

We consider a quantum particle interacting with $N$ obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii potential). We show convergence of the $N$ dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for $N$ large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.

The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor \(\sqrt{t}\) are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by \(\sqrt {t\ln t}\), with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.

Certain classical models, such as dispersing billiards, exhibit strong chaotic behavior but are highly nonlinear and contain singularities. It was a long standing conjecture that, due to singularities, the rate of the decay of correlations in such models is subexponential. Recently, L.-S. Young disproved this conjecture — she established an exponential decay of correlations for a periodic Lorentz gas with finite horizon. We extend this result to all the major classes of dispersing billiards. We also discuss many-particle interacting systems with exponential decay of correlations.

Chaotic electron motion in macroscopic and mesoscopic antidot lattices

Fluctuations of the diffusion tensor in a lattice Lorentz gas model are calculated numerically from a sample of mesoscopic-size systems, and used as input in a mode-coupling expression that predicts long-time tail amplitudes. The results are in good agreement with previously published direct measurements of the tail amplitude over a wide range of scatterer concentrations. We argue that the previous, long-standing disagreement between theory and simulations comes from concentration-independent configurational fluctuations in the diffusion tensor, which we have measured separately and which are not taken into account by existing theories.

Future–Mass Projection (FMP) gravity replaces local dark matter sources by a nonlocal bilocal kernel acting on the baryonic energy–momentum tensor. In covariant formulations this kernel is defined on a closed time path (CTP) with a finite horizon ΔT, but its Newtonian limit in realistic, nonlinear galactic configurations remains opaque. In this exploratory work we organise the Newtonian FMP kernel in terms of an “entropic” linear–response ansatz on the space of coarse–grained surface density profiles Σ(R) of axisymmetric stellar discs. Starting from a coarse–grained functional S[Σ] = Sloc + Spair we define a background–dependent linear–response kernel KS(R, R′; ¯Σ ) as the Hessian of S around a chosen background disc ¯Σ(R). Restricting to Gaussian statistics in the fluctuations and to a simple, radially modulated covariance kernel, we obtain an entropic contribution to the FMP source, ΣF(R)=∫0∞\ddR′Kent(R,R′;Σ¯)δΣ(R′), which is linear in the fluctuations δΣ = Σ − ¯Σ but parametrically dependent on the background disc. We model Kent as a superposition of a local term and a finite–width nonlocal term, controlled by a radial weight function and a Gaussian covariance kRernel. For an exponential Milky Way–like disc we explicitly enforce the radial zero–DC condition $\int \Sigma_F(R)\,2\pi R\,\dd R=0$ and show how it fixes the ratio of local to nonlocal amplitudes. With a small fluctuation parameter ε ≪ 1 we find entropic boost factors D(R) of order unity, with D(R) ≃ 1.0 at R ≲ 3 kpc, D(R) ≃ 2.3 at R ≃ 8 kpc, and D(R) ≃ 1.2 at R ≃ 20 kpc, consistent with the range required by previous FMP fits to Milky Way rotation curves. The construction is deliberately phenomenological and does not claim to derive the FMP kernel from microphysics; instead it provides a state–dependent surrogate that organises the Newtonian kernel in terms of coarse–grained disc properties and highlights where a future CTP–based derivation would need to reproduce or replace these ingredients.