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

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