The Boltzmann-Sinai Ergodic Hypothesis for Hard Ball Systems
英文摘要:
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Nandor Simanyi, Domokos Szasz
This paper has been withdrawn by the authors, due a crucial error.
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In 1963 Ya. G. Sinai formulated a modern version of Boltzmann's ergodic hypothesis, what we now call the ``Boltzmann-Sinai Ergodic Hypothesis'': The billiard system of $N$ ($N\ge 2$) hard balls of unit mass moving on the flat torus $\mathbb{T}^ν=\mathbb{R}^ν/\mathbb{Z}^ν$ ($ν\ge 2$) is ergodic after we make the standard reductions by fixing the values of trivial invariant quantities. It took fifty years and the efforts of several people, including Sinai himself, until this conjecture was finally proved. In this short survey we provide a quick review of the closing part of this process, by show…
In the ergodic theory of semi-dispersing billiards the Local Ergodic Theorem, proved by Chernov and Sinai in 1987, plays a central role. So far, all existing proofs of this theorem had to use an annoying global hypothesis, namely the almost sure hyperbolicity of singular orbits. (This is the so called Chernov--Sinai Ansatz.) Here we introduce some new geometric ideas to overcome this difficulty and liberate the proof from the tyranny of the Ansatz. The presented proof is a substantial generalization of my previous joint result with N. Chernov (which is a $2D$ result) to arbitrary dimensions. A…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.
We consider the system of $N$ ($\ge2$) hard disks of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^2$. We prove the ergodicity (actually, the B-mixing property) of such systems for almost every selection $(m_1,...,m_N;r)$ of the outer geometric parameters.
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, provided that almost every singular orbit is geometrically hyperbolic (sufficient), i. e. the so called Chernov-Sinai Ansatz is true. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs excl…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, without exceptional values. The present proof does not use at all the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools of geometric analysis.
The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. Since almost three decades, the question of ergodicity is still open. The subject of this work is to contribute to the solution of the ergodicity conjecture for three falling balls with a specific mass ratio. The latter is executed in the following three points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of least expanding points and, third, we explain that the proper alignment condition can still be verifi…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^ν$, $ν\ge2$. In the case $ν=2$ we prove (the full hyperbolicity and) the ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, without exceptional values. In higher dimensions, for hard ball systems in $\Bbb T^ν$ ($ν\ge3$), we prove that every such system (is fully hyperbolic and) has open ergodic components.
The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. For almost three decades new, the question of its ergodicity remains open. We contribute to the solution of the erogodicity conjecture for three falling balls with a specific mass ratio in the following two points: First, we prove the Chernov-Sinai ansatz. Second, we prove that there is an abundance of sufficiently expanding points. It is of special interest that for the aforementioned specific mass ratio, the configuration space can be unfolded to …
This work results from our attempts to solve Boltzmann-Sinai's hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface $J$ in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of $J$ grows at least linearly in time. Our proof is independent of the solution of the Boltzmann-Sinai hypothesis, …
Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.
Boltzmann's Billiard Systems: Computation of the Billiard Mapping and Some Numerical Results
本文针对 Boltzmann 提出的台球系统,通过几何与数值分析研究其动力学,发现多数系统呈现混沌,其中一个系统可能遍历,从而检验其遍历性断言。
Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences, equipped with an $\mathcal O_K$-invariant probability measure associated to $\mathcal F_k$. We prove that this action is ergodic, has pure point spectrum and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the paper by the first author and Si…
We consider the billiard flow of elastically colliding hard balls on the flat $ν$-torus ($ν\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-Sinai Ergodic Hypothesis.
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