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Fluctuations of Gibbs ensembles. (English. Russian original) Zbl 1317.82006
Dokl. Math. 90, No. 2, 622-625 (2014); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 458, No. 1, 22-26 (2014).
The author considers the Gibbs ensembles of a continuum, identical, and homogeneous Hamiltonian systems. Investigations of the quasi-Hamiltonian systems are based on the phase space \((g_j,p_j)\), where the phase density \(\rho(t)\) is a nonnegative integrable function that satisfies the Liouville equation. The phase density \(\rho(t)\) is a function of time oscillation and will not be limited as \(t\to 0\), but for all time the following holds \[ \int_\Gamma \rho(t)\,d\mu= 1. \] Here \(d\mu= \sum^m_{j= 1}{^d}q^d_j d_j\) is the Liouville measure, and \(\dim\Gamma= 2m\). Considering that \(t\) weakly converges to \(\overline\rho\) if \[ \varliminf_{t\to\infty} \int_\Gamma \varphi\rho(t)\,d\mu= \int_\Gamma\varphi \overline\rho\,d\mu, \] the author investigates fluctuations of the Gibbs ensembles.
The main result of these investigations is the binomial law of probability that \(k\) particles are in the subdomain \(c\) of the parallelepiped \(\Pi: C^k_N \alpha^k\beta^{N-k}\). Here \(\alpha= {\text{mes\,}\Phi\over \text{mes\,}\Pi}\), \(\beta= 1-\alpha\).
The obtained probability does not depend on the initial distribution of colliding particles.
MSC:
82B05 Classical equilibrium statistical mechanics (general)
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
70F10 \(n\)-body problems
82C40 Kinetic theory of gases in time-dependent statistical mechanics
60G20 Generalized stochastic processes
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References:
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