Further Developments of Sinai’s Ideas: the Boltzmann-sinai Hypothesis
نویسنده
چکیده
0.1. Preface. In 1963 Ya. G. Sinai [Sin(1963)] formulated a modern version of Boltzmann’s ergodic hypothesis, what we now call the “Boltzmann-Sinai Ergodic Hypothesis”: The billiard system of N (N ≥ 2) hard balls of unit mass moving on the flat torus T = R/Z (ν ≥ 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 showing how Sinai’s original ideas developed further between 2000 and 2013, eventually leading the proof of the conjecture.
منابع مشابه
Flow-Invariant Hypersurfaces in Semi-Dispersing Billiards
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-disper...
متن کاملThe Boltzmann - Sinai Ergodic Hypothesis in Full Generality ( Without Exceptional Models )
We consider the system of N (≥ 2) elastically colliding hard balls of masses m1, . . . , mN and radius r on the flat unit torus T , ν ≥ 2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection (m1, . . . , mN ; r) of the external geometric parameters, without exceptional values. Primary subject classification: ...
متن کاملConditional Proof of the Boltzmann-sinai Ergodic Hypothesis (assuming the Hyperbolicity of Typical Singular Orbits)
We consider the system of N (≥ 2) elastically colliding hard balls of masses m1, . . . , mN and radius r on the flat unit torus T , ν ≥ 2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection (m1, . . . , mN ; r) of the external geometric parameters, under the assumption that almost every singular trajectory i...
متن کاملUnconditional Proof of the Boltzmann-sinai Ergodic Hypothesis
We consider the system of N (≥ 2) elastically colliding hard balls of masses m1, . . . , mN and radius r on the flat unit torus T , ν ≥ 2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection (m1, . . . , mN ; r) of the external geometric parameters. The present proof does not use the formerly developed, rathe...
متن کاملProof of the Boltzmann - Sinai Ergodic Hypothesis
We consider the system of N (≥ 2) elastically colliding hard balls of masses m1, . . . , mN and radius r on the flat unit torus T , ν ≥ 2. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection (m1, . . . , mN ; r) of the external geometric parameters. The present proof does not use the formerly developed, rathe...
متن کامل