Abstract
Whereas ergodic theories relate to limiting cases of infinite thermal reservoirs and infinitely long times, some ergodicity tendencies may appear also for finite reservoirs and time durations. These tendencies are here explored and found to exist, but only for extremely long times and very soft ergodic criteria. “Weak ergodicity breaking” is obviated by a judicious time-weighting, as found in a previous work [Found. Phys. (2015) 45: 673–690]. The treatment is based on an N-oscillator (classical) and an N-spin (quantal) model. The showing of ergodicity is facilitated by pictorial presentations.
Highlights
In a broadly phrased description, ergodic theorems equate time and ensemble averages [1,2,3,4,5,6], and take various forms both in the precise definition of the term “equate” and of the variety of items whose averages are considered
A model is designed with the aim of tracing the movement of a privileged particle, this being part of a set of similar entities and calculating the frequency of its visiting each point in its phase space, represented by the position and momentum labels X, P
Whereas the drop at high energies is due to the upper-boundedness of our finite system, the sharp initial rise is remarkable and we wish to claim on the basis of this, that, if in any ensemble averaging the points along a radial ray possess identical measures, for a time integration to be equivalent to that, time durations with low energies must be compensated by an enhancing factor. (The section argues this from a very different perspective)
Summary
In a broadly phrased description, ergodic theorems equate time and ensemble averages [1,2,3,4,5,6], and take various forms both in the precise definition of the term “equate” (which is usually done in the senses of Set Theory, for example, References [7,8]) and of the variety of items whose averages are considered. The term “visiting number” leads us to consideration for classical dynamics of an allied quantity, the “sojourn time” already mentioned, being the time that the trajectory spends in the vicinity (defined here through np) of the phase-space point Cases when this quantity is in violation of ergodicity were named by Bouchaud [21] “weak ergodic breaking” (distinct from “strong ergodicity breaking”, for which NV is not ergodic) and were intensively studied by Barkai and collaborators, for example, for anomalous diffusers temporarily residing in traps [22,23,24,25] and more recently in Reference [26]. The main conspicuous outcomes arising from the treatments in the present work, for either the classical or the quantum systems, are the extended establishment of ergodicity (beyond its basic theoretically warranted domain), but in a manner that depends quantitatively on some parameter values, and the requirement of unequal weighting of the time in taking the time averages
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