SOME SPECIAL CASES OF KHINTCHINE’S CONJECTURES IN STATISTICAL MECHANICS: APPROXIMATE ERGODICITY OF THE AUTO-CORRELATION FUNCTIONS OF AN ASSEMBLY OF LINEARLY COUPLED OSCILLATORS

Resumen

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. In
1943, Khintchine made some visionary conjectures about dynamical systems with large numbers of degrees of freedom which would justify, even in the absence of ergodicity, approximately the same conclusions. We prove Khintchine’s conjectures in some special cases of a linearly coupled assembly of harmonic oscillators.

KEY WORDS: Time series, ergodic, auto-correlation function, statistical mechanics

RESUMEN
Para emplear el correlograma de los valores muestrales de un proceso estoc´astico para estimar su funcion teorica 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´ıan una aproximacion a las mismas estimaciones a un sin la ergodicidad del sistema. Mostraremos casos particulares de las conjeturas de Khintchine para asambleas de osciladores lineales