Some Special Cases of Khintchine's Conjectures in Statistical Mechanics: Approximate Ergodicity of the Auto-Correlation Functions of an Assembly of Linearly Coupled Oscillators

TL;DR

This paper rigorously defines 'normal state' in Hamiltonian models, proves the existence of such states in large oscillator assemblies, and validates Khintchine's conjectures on approximate ergodicity in non-ergodic systems.

Contribution

It provides a mathematical proof of normal states in oscillator assemblies, supporting Khintchine's conjectures on weak ergodicity without requiring full ergodicity.

Findings

Existence of normal cells of trajectories in the heat-bath model.

Verification of Khintchine's conjectures in specific oscillator cases.

Approximate validity of autocorrelation estimates without ergodicity.

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.