On the evolution of the empirical measure for the Hard-Sphere dynamics

Propagation processes of correlations of hard spheres

The paper develops an approach to the description of the evolution of correlations for many hard spheres based on a hierarchy of evolution equations for the cumulants of the probability distribution function governed by the Liouville equation. It is established that the constructed dynamics of correlations underlies the description of the evolution of the states of many hard spheres described by the BBGKY hierarchy for reduced distribution functions or the hierarchy of nonlinear evolution equations for reduced correlation functions. As an application of the developed approach to describing the…

Spectral Gaps for Reversible Markov Processes with Chaotic Invariant Measures: The Kac Process with Hard Sphere Collisions in Three Dimensions

We develop a method for producing estimates on the spectral gaps of reversible Markov jump processes with chaotic invariant measures, and we apply it to prove the Kac conjecture for hard sphere collision in three dimensions.

The analyticity region of the hard sphere gas. Improved bounds

We find an improved estimate of the radius of analyticity of the pressure of the hard-sphere gas in $d$ dimensions. The estimates are determined by the volume of multidimensional regions that can be numerically computed. For $d=2$, for instance, our estimate is about 40% larger than the classical one.

On $p$-adic Hard Sphere model with three states on a Cayley tree

We study $p$-adic model of hard spheres with three states on the Cayley tree. We show that there exist three translation-invariant $p$-adic Gibbs measures and two periodic measures on a Cayley tree of oreder two.

Derivation of the Boltzmann equation: hard spheres, short-range potentials and beyond

We review some results concerning the derivation of the Boltzmann equation starting from the many-body classical Hamiltonian dynamics. In particular, the celebrated paper by O. E. Lanford III and the more recent papers [13, 23] are discussed.

Densest columnar structures of hard spheres from sequential deposition

The rich variety of densest columnar structures of identical hard spheres inside a cylinder can surprisingly be constructed from a simple and computationally fast sequential deposition of cylinder-touching spheres, if the cylinder-to-sphere diameter ratio D is within [1,2.7013]. This provides a direction for theoretically deriving all these densest structures and for constructing such densest packings with nano-, micro-, colloidal or charged particles, which all self-assemble like hard spheres.

Long time validity of the linearized Boltzmann equation for hard spheres: a proof without billiard theory

We study space-time fluctuations of a hard sphere system at thermal equilibrium, and prove that the covariance converges to the solution of a linearized Boltzmann equation in the low density limit, globally in time. This result has been obtained previously in [7], by using uniform bounds on the number of recollisions of dispersing systems of hard spheres (as provided for instance in [9]). We present a self-contained proof with substantial differences, which does not use this geometric result. This can be regarded as the first step of a program aiming to derive the fluctuation theory of the rar…

Correlation decay for hard spheres via Markov chains

We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of $2$ and $1.7$, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results.

Book review: From Newton to Boltzmann: Hard Spheres and Short-range Potentials, by Isabelle Gallagher, Laure Saint-Raymond and Benjamin Texier

Review of From Newton to Boltzmann: Hard Spheres and Short-range Potentials, by Isabelle Gallagher, Laure Saint-Raymond and Benjamin Texier, European Mathematical Society, Zürich, 2014, xi+135 pp., ISBN 978-3- 03719-129-3.

On the cardinality of collisional clusters for hard spheres at low density

We resume the investigation, started in arXiv:1408.6571, of the statistics of backward clusters in a gas of $N$ hard spheres of small diameter $\varepsilon$. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We obtain an estimate of the average cardinality of clusters with respect to the equilibrium measure, global in time, uniform in $\varepsilon, N$ for $\varepsilon^2 N=1$ (Boltzmann-Grad regime).

一维硬球动力学中台球轨迹的外尔群表示

Weyl Group Representation of Billiard Trajectories for One-dimensional Hard Sphere Dynamics

本文针对一维 Tonks 气体模型，给出了硬球动力学基于外尔群求和的精确显式公式，并利用该公式简洁证明了相应台球流以刘维尔测度为不变测度。

On the hard sphere model and sphere packings in high dimensions

We prove a lower bound on the entropy of sphere packings of $\mathbb R^d$ of density $Θ(d \cdot 2^{-d})$. The entropy measures how plentiful such packings are, and our result is significantly stronger than the trivial lower bound that can be obtained from the mere existence of a dense packing. Our method also provides a new, statistical-physics-based proof of the $Ω(d \cdot 2^{-d})$ lower bound on the maximum sphere packing density by showing that the expected packing density of a random configuration from the hard sphere model is at least $(1+o_d(1)) \log(2/\sqrt{3}) d \cdot 2^{-d}$ when the …

Min-type Morse theory for configuration spaces of hard spheres

We study configuration spaces of hard spheres in a bounded region. We develop a general Morse-theoretic framework, and show that mechanically balanced configurations play the role of critical points. As an application, we find the precise threshold radius for a configuration space to be homotopy equivalent to the configuration space of points.

硬球模型的强空间混合完美采样

Perfect Sampling for Hard Spheres from Strong Spatial Mixing

提出一种基于强空间混合的硬球模型完美采样算法，期望运行时间与体积线性，优于已知近似采样器，并推广至有限范围排斥势的吉布斯点过程。