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Some Special Cases of Khintchine's Conjectures in Statistical Mechanics: Approximate Ergodicity of the Auto-Correlation Functions of an Assembly of Linearly Coupled Oscillators | | Joseph F. Johnson
; | | Date: |
16 Aug 2011 | | Abstract: | We give Sir James Jeans’s notion of ’normal state’ a mathematically precise
definition. We prove that normal cells of trajectories exist in the Hamiltonian
heat-bath model of an assembly of linearly coupled oscillators that generates
the Ornstein--Uhlenbeck process in the limit of an infinite number of degrees
of freedom. This, in some special cases, verifies some far-reaching conjectures
of Khintchine on the weak ergodicity of a dynamical system with a large number
of degrees of freedom. In order to estimate the theoretical auto-correlation
function of a time series from the sample auto-correlation function of one of
its realisations, it is usually assumed without justification that the time
series is ergodic. Khintchine’s conjectures about dynamical systems with large
numbers of degrees of freedom justifies, even in the absence of ergodicity,
approximately the same conclusions.
Para emplear el correlograma de los valores muestrales de un proceso
estoc’astico para estimar su funci’on te’orica de autocorrelaci’on, por
regla general se asume, sin justificaci’on, que el proceso es erg’odico. Pero
en 1943, Khintchine conjetur’o proposiciones de gran importancia en este
asunto, que justificar’i an una aproximaci’on a las mismas estimaciones a’un
sin la ergodicidad del sistema. Mostraremos casos particulares de las
conjeturas de Khintchine para asambleas de osciladores lineales. | | Source: | arXiv, 1108.3151 | | Services: | Forum | Review | PDF | Favorites | |
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