The \(K\)-property of four billiard balls
The dynamics of a system of \(N(\geq 2)\) hard balls of radii \(r>0\) on the \(\nu\)-dimensional unit torus \(T^ \nu\), \(\nu\geq 2\), determined by their uniform motion with elastic collisions is exprimed, in an isomorphic manner, by the billiard flow \(\{S^ t: t\in\mathbb{R}\}\) in \(\mathbb{Q}\) with specular reflections at \(\partial\mathbb{Q}\) and they conserve the Liouville measure \(d\mu=\text{const}\cdot dq\cdot dv\). Here \(\mathbb{Q}=\{Q\in\tilde\mathbb{Q},\sum^ N_{i=1}q_ i=0\}\), where by \((q_ i,v_ i)\in T^ \nu\times \mathbb{R}^ \nu\) is denoted the phase point of the \(i\)th ball and the configuration space \(\tilde\mathbb{Q}\) of the \(N\) balls is a subset of \(T^{N\cdot\nu}\) which is obtained from \(T^{N\cdot\nu}\) by cutting out \({N\choose 2}\) cylindric scatters: \(\tilde C_{i,j}=\{Q=(q_ 1,\dots,q_ N)T^{N\cdot\nu}:\;| q_ i-q_ j| < 2r\}\), \(1\leq i< j\leq N\). (Without loss of generality, the authors assume that \(H={1\over 2}\sum^ N_{i=1}v^ 2_ i={1\over 2}\), the total momentum \(P=\sum^ N_{i=1}v_ i=0\) and, moreover, that the sum of spatial components \(B=\sum^ N_{i=1}q_ i=0\).) For these values of \(H\), \(P\) and \(B\), the phase of the system reduces to \(M=\mathbb{Q}\times{\mathcal S}_{N\cdot\nu-\nu-1}\) where \({\mathcal S}_ k\) denotes the \(k\)-dimensional unit sphere. The main result proved in this paper is the further step toward establishing the celebrated Boltzmann-Sinai ergodic hypothesis: For \(N=4\), \(\nu\geq 3\) and \(r<{1\over 8}\) the dynamical system \((M,\{S^ t\},\mu)\) is a \(K\)-flow. In the abstract and in the introduction the authors explain the main novelties of the proof: ``all part of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened''.
scientific article; zbMATH DE number 54738