Hard Ball Systems Research Articles

Long time behavior of stochastic hard ball systems

We study the long time behavior of a system of $n=2,3$ Brownian hard balls, living in $\mathbb{R}^d$ for $d\ge2$, submitted to a mutual attraction and to elastic collisions.

Polarization and Finite Size Effects in Correlation Functions of Dusty Plasmas

AbstractAn approach is presented to derive a pseudopotential model of interaction between dust particles that simultaneously takes into account the polarization, finite size and screening effects. The consideration starts from the assumption that the dust particles are hard balls made of a conductive material such that their mutual interaction and interaction with the electrons and ions of the buffer plasma can analytically be interpreted within the method of image charges. Then, the renormalization theory of plasma particles interaction, leading to the so‐called generalized Poisson‐Boltzmann equation, is applied to obtain the interaction potential of two isolated dust grains immersed into the buffer plasma of electrons and ions. After that the Ornstein‐Zernike relation in the hyper‐netted chain approximation (HNC) is numerically solved to study the radial distribution function and the static structure factor of the dust grains. In doing so the system of hard balls is actually replaced by a system of point‐like charges with properly adjusted number density in the form of van der Waals correction. A straightforward comparison is made with the Monte‐Carlo simulation to find a fairly good agreement for the radial distribution function at relatively high dust couplings. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hard-sphere fluids with chemical self-potentials

Existence, uniqueness and stability of solutions is studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at is the sum of the matter reservoir's contribution and a self contribution which is computed by convoluting the fluid density distribution with a van der Waals, a Yukawa, or a Newton kernel. We prove the existence of a grand canonical phase transition, and a petit canonical phase transition which is embedded in the former.

Algebro-geometric methods for hard ball systems

For the study of hard ball systems, the algebro-geometric approach appeared in 1999 --- in a sense surprisingly but quite efficiently --- for proving the hyperbolicity of typical systems (see [26]). An improvement by Simányi [22] also provided the ergodicity of typical systems, thus an almost complete proof of the Boltzmann--Sinai ergodic hypothesis. More than that, at present, the best form of the local ergodicity theorem for semi-dispersing billiards, [6] also uses algebraic methods (and the algebraicity condition on the scatterers). The goal of the present paper is to discuss the essential steps of the algebro-geometric approach by assuming and using possibly minimum information about hard ball systems. In particular, we also minimize the intersection of the material with the earlier surveys [29] and [20].

Infinite system of Brownian balls with interaction: the non-reversible case

We consider an infinite system of hard balls in undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.

Finite-range interactions and phases

We consider a one-dimensional system of hard balls on the positive half-line with a general finite-range interaction in thermodynamical equilibrium. We show that, under the action of a weak gravitational field, one can distinguish two spatially separated subsystems: a dense one, which we call the condensate, and a dilute one which we call the gas; between these two, there is a narrow interface where both subsystems mix.

Dynamics and statistics in few-body hard-ball systems

The dynamic and statistical properties of several hard balls restricted in a box are studied.The collision distribution in configuration space is main object of the research. Numerical simulations indicate that the distribution function is uniform (horizontal line) when the radius of balls is very small. This uniform p art decreases with increasing size of the balls, and a double-peak distribution is found when the radius exceeds a critical value. Brownian motion is simulated by using hard-ball systems. It can be found that the mean square displacement is proportional to the square of time, i.e., the transport process is dominated by ballistic transport, rather than a diffusion behavior.

Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

We consider the system of $N$ ($\ge2$) hard balls with masses $m_1,...,m_N$ and radius $r$ in the flat torus $\Bbb T_L^\nu=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ of size $L$, $\nu\ge3$. We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection $(m_1,...,m_N; L)$ of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case $\nu=2$. The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.

Viscosity in molecular dynamics with periodic boundary conditions.

We report a study of viscosity by the method of Helfand moment in systems with periodic boundary conditions. We propose a new definition of Helfand moment which takes into account the minimum image convention used in molecular dynamics with periodic boundary conditions. Our Helfand-moment method is equivalent to the method based on the Green-Kubo formula and is not affected by ambiguities due to the periodic boundary conditions. Moreover, in hard-ball systems, our method is equivalent to that developed by Alder, Gass, and Wainwright [J. Chem. Phys. 53, 3813 (1970)]. We apply and verify our method in a fluid composed of N> or =2 hard disks in elastic collisions. We show that the viscosity coefficients already take values in good agreement with Enskog's theory for N=2 hard disks in a hexagonal geometry.

Multi-Dimensional Semi-Dispersing Billiards: Singularities and the Fundamental Theorem

The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987, see [S-Ch (1987)] and an improved version in [K-S-Sz (1990)]. It provides sufficient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently ([B-Ch-Sz-T (2000)]) that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are defined by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models - among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Rehacek stadia.

Ball-avoiding theorems

Consider a nice hyperbolic dynamical system (singularities not excluded). Statements about the topological smallness of the subset of orbits, which avoid an open subset of the phase space (for every moment of time, or just for a not too small subset of times), play a key role in showing hyperbolicity or ergodicity of semi-dispersive billiards, especially, of hard-ball systems. As well as surveying the characteristic results, called ball-avoiding theorems, and giving an idea of the methods of their proofs, their applications are also illustrated. Furthermore, we also discuss analogous questions (which had arisen, for instance, in number theory), when the Hausdorff dimension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball-avoiding subsets are naturally related to repellers extensively studied by physicists. For the interested reader we also sketch some analytical and rigorous results about repellers and escape times.

An infinite system of Brownian balls with infinite range interaction

We study an infinite system of Brownian hard balls, moving in R d and submitted to a smooth infinite range pair potential. It is represented by a diffusion process, which is constructed as the unique strong solution of an infinite-dimensional Skorohod equation. We also prove that canonical Gibbs states associated to the sum of the hard core potential and the pair potential are reversible measures for the dynamics.

Non-integrability of cylindric billiards and transitive Lie group actions

A conjecture is formulated and discussed which provides a necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup ${\cal G}$ of the orthogonal group $SO(d)$ ($d$ being the dimension of the billiard in question) be transitive on the unit sphere $S^{d-1}$. If $C_1, \dots, C_k$ are the cylindric scatterers of the billiard, then ${\cal G}$ is generated by the embedded Lie subgroups ${\cal G}_i$ of $SO(d)$, where ${\cal G}_i$ consists of all transformations $g\in SO(d)$ of ${\Bbb R}^d$ that leave the points of the generator subspace of $C_i$ fixed ($1 \le i \le k$). In this paper we can prove the necessity of our conjecture and we also formulate some notions related to transitivity. For hard ball systems, we can also show that the transitivity holds in general: for an arbitrary number $N\ge 2$ of balls, arbitrary masses $m_1, \dots, m_N$ and in arbitrary dimension $\nu \ge 2$. This result implies that our conjecture is stronger than the Boltzmann–Sinai ergodic hypothesis for hard ball systems. We also note a somewhat surprising characterization of the positive subspace of the second fundamental form for the evolution of a special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of sufficiency of an orbit segment.

Chaotic and ergodic properties of cylindric billiards

Chaotic and ergodic properties are discussed for various subclasses of cylindric billiards. A common feature of the studied systems is that they satisfy a natural necessary condition for ergodicity and hyperbolicity, the so-called transitivity condition. The relation of our discussion to former results on hard ball systems is twofold. On the one hand, by slight adaptation of the proofs we may discuss hyperbolic and ergodic properties of 3 or 4 particles with (possibly restricted) hard ball interactions in any dimensions. On the other hand, a key tool in our investigations is a kind of connected path formula for cylindric billiards, which together with the conservation of momenta gives back, when applied to the special case of hard ball systems, the classical connected path formula.

Hard Ball Systems are Completely Hyperbolic

We consider the system of N (\ge 2) elastically colliding hard balls with masses m_1,..., m_N, radius r, moving uniformly in the flat torus T_L^{\nu}= R^\nu/L \cdot Z^\nu, \nu \ge 2. It is proved here that the relevant Lyapunov exponents of the flow do not vanish for almost every (N+1)-tuple (m_1,...,m_N;L) of the outer geometric parameters.

Ergodicity in hard-ball systems and Boltzmann's entropy.

The ergodic property of the hard-ball systems is investigated numerically by comparing the microcanonical single-particle distributions and the corresponding numerical results in two- and three-dimensional cases. A Boltzmann-entropy-like quantity H(${\mathit{n}}_{\mathit{c}}$) is defined and its evolution is discussed. The deviation of this entropy from the equilibrium value obeys a power decay law in the long time limit, and the origin of this tail is explored heuristically. \textcopyright{} 1996 The American Physical Society.

The K-property of 4D billiards with nonorthogonal cylindric scatterers

The K-property of cylindric billiards given on the 4-torus is established. These billiards are neither “orthogonal,” where general necessary and sufficient conditions were obtained by D. Szasz, nor isomorphic to hard-ball systems, where the connecting path formula of N. Simanyi is at hand.

The K-property ofN billiard balls II. Computation of neutral linear spaces

The present second part of the article “The K-property ofN billiard balls” finishes the proof of the Kolmogorov mixing property for hard ball systems in tori, provided that the number of balls dos not exceed the dimension of the torus. The missing computation of the linear spaces of non-expanding and non-contracting tangent vectors at phase points with combinatorially rich symbolic collision sequences is presented hereby.

Classical equilibrium generalized hydrodynamic correlation Green's functions for a system of hard balls with weak interaction

Classical equilibrium generalized hydrodynamic correlation Green's functions for a system of hard balls with weak interaction