Lorentz Gas Research Articles

Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres

Abstract We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$ -horospheres in the space of affine lattices.

Optimizing the performance of the thermal transistor based on negative differential thermal resistance

Thermal transistors have significant potential in thermal management due to their ability to precisely control the heat fluxes. However, the current design has a limited working temperature range and cannot meet the demand for heat control. In this work, we optimize the performance of a thermal transistor based on negative differential thermal resistance using a modified Lorentz gas model, focusing on the effects of gate control and thermal conductance. We find that the control of the gate has an asymmetrical property, and the sufficient control can expand the working region by more than 1.5 times. In addition, the switching and amplifying functions of the thermal transistor are inversely dependent on thermal conductance, and an increase in thermal conductance can lead to a 27 times amplification in heat flux. The synergistic regulation of the two factors results in the optimal performance of a 44% working region and a 33 times amplification in heat flux. These findings refresh the understanding of the performance limits of thermal transistors, promoting efficient thermal management and addressing heat dissipation issues.

Kinetic Theory for the Low-Density Lorentz Gas

The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including semiconductor physics and radiative transfer. A rigorous derivation of the linear Boltzmann equation from the underlying particle dynamics was given, for random scatterer configurations, in three seminal papers by Gallavotti, Spohn and Boldrighini-Bunimovich-Sinai. The objective of the present study is to develop an approach for a large class of deterministic scatterer configurations, including various types of quasicrystals. We prove the convergence of the particle dynamics to transport processes that are in general (depending on the scatterer configuration) not described by the linear Boltzmann equation. This was previously understood only in the case of the periodic Lorentz gas through work of Caglioti-Golse and Marklof-Strömbergsson. Our results extend beyond the classical Lorentz gas with hard sphere scatterers, and in particular hold for general classes of spherically symmetric finite-range potentials. We employ a rescaling technique that randomises the point configuration given by the scatterers’ centers. The limiting transport process is then expressed in terms of a point process that arises as the limit of the randomised point configuration under a certain volume-preserving one-parameter linear group action.

Introduction of Asymmetry to Enhance Thermal Transport in Porous Metamaterials at Low Temperature

Introducing porosity with different degrees of disorder has been widely used to regulate thermal properties of materials, which generally results in decrease of thermal conductivity. We investigate the thermal conductivity of porous metamaterials in the ballistic transport region by using the Lorentz gas model. It is found that the introduction of asymmetry and Gaussian disorder into porous metamaterials can lead to a strong enhancement of thermal conductivity. By dividing the transport process into ballistic transport, non-ballistic transport, and unsuccessful transport processes, we find that the enhancement of thermal conductivity originates from the significant increase ballistic transport ratio. The findings enhance the understanding of ballistic thermal transport in porous materials and may facilitate designs of high-performance porous thermal metamaterials.

Jamming, relaxation, and memory in a minimally structured glass former.

Structural glasses form through various out-of-equilibrium processes, including temperature quenches, rapid compression (crunches), and shear. Although each of these processes should be formally understandable within the recently formulated dynamical mean-field theory (DMFT) of glasses, the numerical tools needed to solve the DMFT equationsup to the relevant physical regime do not yet exist. In this context, numerical simulations of minimally structured (and therefore mean-field-like) model glass formers can aid the search for and understanding of such solutions, thanks to their ability to disentangle structural from dimensional effects. We study here the infinite-range Mari-Kurchan model under simple out-of-equilibrium processes, and we compare results with the random Lorentz gas [J. Phys. A 55, 334001 (2022)10.1088/1751-8121/ac7f06]. Because both models are mean-field-like and formally equivalent in the limit of infinite spatial dimensions, robust features are expected to appear in the DMFT as well. The comparison provides insight into temperature and density onsets, memory, as well as anomalous relaxation. This work also further enriches the algorithmic understanding of the jamming density.

Local large deviations for periodic infinite horizon Lorentz gases

We prove optimal local large deviations for the periodic infinite horizon Lorentz gas viewed as a ℤd-cover (d = 1,2) of a dispersing billiard. In addition to this specific example, we prove a general result for a class of nonuni-formly hyperbolic dynamical systems and observables associated with central limit theorems with nonstandard normalisation.

Nonmonotonic dependence of thermal conductivity on surface roughness: A multiparticle Lorentz gas model.

Utilizing surface roughness to manipulate thermal transport has aided important developments in thermoelectrics and heat dissipation in microelectronics. In this paper, through a multiparticle Lorentz gas model, it is found that thermal conductivity oscillates with the increase of surface roughness, and the oscillating thermal conductivity gradually disappears with the increase of nonlinearity. The transmittance analyses reveal that the oscillating thermal conductivity is caused by localized particles due to boundary effects. Nonlinearity will gradually break the localization. Thus, localization still exists in the weak nonlinear system, where there exists an interplay between nonlinear interaction and localization. Furthermore, it is also found that boundary shapes have a great influence on the oscillating thermal conductivity. Finally, we have also studied the oscillating thermal rectification effects caused by rough boundaries. This study gains insight into the boundary effect on thermal transport and provides a mechanism to manipulate thermal conductivity.

The Lorentz Gas with a Nearly Periodic Distribution of Scatterers

We consider the Lorentz gas in a distribution of scatterers which microscopically converges to a periodic distribution, and prove that the Lorentz gas in the low density limit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz gas, which does not satisfy the Boltzmann equation in the limit.

Hall effect in a two-dimensional disordered Lorentz gas

Hall effect in a two-dimensional disordered Lorentz gas

Large fluctuations and transport properties of the Lévy–Lorentz gas

Le gaz de Lévy–Lorentz modélise le déplacement d’une particule sur l’axe des nombres réels en présence d’obstacles distribués de telle façon que les distances entre les obstables sont des variables aléatoires i.i.d. à queue lourde et de moyenne finie. La dynamique est donnée par l’interpolation, en temps continu et vitesse fixe, de la marche aléatoire symétrique sur les obstacles. Cet article étudie les grandes fluctuations du processus en temps continu et les propriétés de transport du modèle sous-jacent. Les résultats obtenus dans les cas de désordre “annealed” et “quenched” confirment et généralisent des résultats précédents issus de la physique dans le cas “annealed”. En particulier, sous l’hypothèse que la queue de la loi des distances inter-obstacles est à variation régulière à l’infini, nous prouvons dans le cas “annealed” un principe de grandes déviations et obtenons l’expression asymptotique des moments, qui montrent l’existence d’un regime de sur-diffusion. Dans le cas quenched, nous obtenons l’expression asymptotique des moments et une borne supérieure sur les grandes deviations des fluctuations. Cela nous permet de montrer, pour des configurations d’obstacles typiques, que le régime asymptotique de diffusion est la diffusion normale, et non la sur-diffusion. Bien que le gaz de Lévy–Lorentz soit en général utilisé pour modéliser la diffusion anormale, notre résultat suggère que le regime sur-diffusif est seulement transitoire. Cela soulève également la question de la nature de la transition entre diffusion normale “quenched” et sur-diffusion “annealed”.

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) .

Projective Cones for Sequential Dispersing Billiards

We construct Birkhoff cones for dispersing billiards, which are contracted by the action of the transfer operator. This construction permits the study of statistical properties not only of regular dispersing billiards but also of sequential billiards (the billiard changes at each collision in a prescribed manner), open billiards (the dynamics exits some region or dies when hitting some obstacle) and many other examples. In particular, we include applications to chaotic scattering and the random Lorentz gas.

Aperiodic and linearly repetitive Lorentz gases of finite horizon are not exponentially mixing

We prove that aperiodic and linearly repetitive Lorentz gases with finite horizon are not mixing with exponential or stretched exponential speed in any dimension for any class of H\"older observables. We also bound the polynomial speed of mixing for observables in the H\"older space $H_{\alpha}$ depending on $\alpha$.

Extensions of Exact and K-Mixing Dynamical Systems

We consider extensions of non-singular maps which are exact, respectively K-mixing, or at least have a decomposition into positive-measure exact, respectively K-mixing, components. The fibers of the extension spaces have countable (finite or infinite) cardinality and the action on them is assumed surjective or bijective. We call these systems, respectively, fiber-surjective and fiber-bijective extensions. Technically, they are skew products, though the point of view we take here is not the one generally associated with skew products. Our main results are an Exact and a K-mixing Decomposition Theorem. The latter can be used to show that a large number of periodic Lorentz gases (the term denoting here general group extensions of Sinai billiards, including Lorentz tubes and slabs, in any dimension) are K-mixing.

Infinite measure mixing for some mechanical systems

We show that if an infinite measure preserving system is well approximated on most of the phase space by a system satisfying the local limit theorem, then the original system enjoys mixing with respect to global observables, that is, the observables which admit an infinite volume average. The systems satisfying our conditions include the Lorentz gas with Coulomb potential, the Galton board, and piecewise smooth Fermi-Ulam pingpongs.

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.

Derivation of the Linear Boltzmann Equation from the Damped Quantum Lorentz Gas with a General Scatterer Configuration

It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper, we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence—in the absence of damping the limiting behaviour depends on the exact configuration under consideration, and indeed, the linear Boltzmann equation is not expected to appear for periodic and other highly ordered configurations.

Statistical Properties of Lorentz Gases on Aperiodic Tilings, Part 1

We consider the Lorentz gas model of category A (that is, with no corners and of finite horizon) on aperiodic repetitive tilings of $\mathbb{R}^2$ of finite local complexity. We show that the compact factor of the collision map has the K property, from which we derive mixing for pattern-equivariant functions as well as the planar ergodicity of the Lorentz gas flow.

Gradient descent dynamics and the jamming transition in infinite dimensions

Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the simplest examples, namely that of soft repulsive particles in the limit of infinite spatial dimension d. The gradient descent dynamics then displays a jamming transition: at low density, it reaches zero-energy states in which particles’ overlaps are fully eliminated, while at high density the energy remains finite and overlaps persist. At the transition, the dynamics becomes critical. In the d → ∞ limit, a set of self-consistent dynamical equations can be derived via mean field theory. We analyze these equations and we present some partial progress towards their solution. We also study the random Lorentz gas in a range of d = 2…22, and obtain a robust estimate for the jamming transition in d → ∞. The jamming transition is analogous to the capacity transition in supervised learning, and in the appendix we discuss this analogy in the case of a simple one-layer fully-connected perceptron.

Equilibrium fluctuations in mean-field disordered models.

Mean-field models of glasses that present a random first order transition exhibit highly nontrivial fluctuations. Building on previous studies that focused on the critical scaling regime, we here obtain a fully quantitative framework for all equilibrium conditions. By means of the replica method we evaluate Gaussian fluctuations of the overlaps around the thermodynamic limit, decomposing them in thermal fluctuations inside each state and heterogeneous fluctuations between different states. We first test and compare our analytical results with numerical simulation results for the p-spin spherical model and the random orthogonal model, and then analyze the random Lorentz gas. In all cases, a strong quantitative agreement is obtained. Our analysis thus provides a robust scheme for identifying the key finite-size (or finite-dimensional) corrections to the mean-field treatment of these paradigmatic glass models.