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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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