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Asymptotics of a solution of an \(N\)-partial Liouville equation for large \(N\) and refutation of the chaos hypothesis for density functions. (English. Russian original) Zbl 0860.35106

Math. Notes 56, No. 2, 872-874 (1994); translation from Mat. Zametki 56, No. 2, 153-155 (1994).
The authors consider the question whether the \(N\)-particle density function \(\rho_N\), which solves the Liouville equation \(\partial_t \rho_N= \{H_N, \rho_N\}\), decomposes into a product of 1-particle distribution functions in the limit \(N\to \infty\), at any time \(t\), provided we assume this asymptotic decomposition holds for \(t=0\). This would generalize the so-called “chaos hypothesis”, which deals with the decomposition of the \(k\)-particle marginal distribution function, with fixed \(k\). The authors answer the question in the negative, by announcing the following results: (1) the decomposition of \(\rho_N\) may fail; (2) there is a formula for the \(L^1\)-limit of \(\rho_N (t)\) for Hamiltonians with smooth interactions with slow growth.

MSC:

35Q40 PDEs in connection with quantum mechanics
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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