S S symmetry Article Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal Robert Englman Physics Department, Ariel University, Ariel 40700, Israel; [email protected] Received: 2 September 2020; Accepted: 24 September 2020; Published: 6 October 2020 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. Keywords: ergodic theorems; spin chain; coupled oscillators; ensembles 1. Introduction In a broadly phrased description, ergodic theorems equate time and ensemble averages [1–6], 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. As regards the former, physical applications of ergodicity (as in References [9,10]) typically steer clear of the niceties (e.g., the qualification of “almost all”) of set-theoretical precision [11,12], as is being done here. The varieties of theorems are features of both developments in the course of times and at a particular time of the subject matters which are at interest. Historically, Poincaré [13] is regarded, through his recurrence theorem, as the progenitor of Ergodic Theory, which had led to Boltzmann’s Statistical Mechanics and Entropy concept for the dynamical theory of gases [14]. At the basis of ergodic theory as applied to dynamical systems is what is termed “the invariant measure”, whose meaning is roughly the uniformity of the statistical weight of the system over states that it potentially partakes of. (It is this weight that enters logarithmically the entropy concept of Boltzmann, taught in every elementary course on Statistical Mechanics. Related to this is the “Metrical Decomposability” of von Neumann [15]). Microcanonical ensembles, that is, those whose manifold sub-systems have a fixed total energy, possess this property and ergodicity in its multiplicity of set-theoretical senses referred in its primary applicability to these. (In truth, the potential applicabilities of ergodicity are extremely broad, encompassing beyond the exact sciences also biology, sociology, medicine and humanities, for example, Reference [16]). Constituent parts, “sub-systems” of systems forming a microcanonical ensemble do not possess “the invariant measure” property (as elucidated in further detail later on in this article), and general theories do not appear to hold for these. Thence, there is some value in studying some particular models (and, at it, some simple ones), with a view of gaining some knowledge of ergodic behaviors or otherwise, in these systems, where they are not mandated. Previous work [17,18] went in this direction. The present article amplifies the knowledge by including classical, as well as quantum systems (in Sections2 and3, respectively). The net conclusion arising from our (mainly numerical) results is that ergodic behavior is a property also of systems lacking “invariant measures”, but only approximately so and with a critical amendment in the process of time averaging, namely by the Symmetry 2020, 12, 1642; doi:10.3390/sym12101642 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1642 2 of 16 appropriate and non-uniform weighting of the time (t, here termed the “sojourn duration”) in the time integration. A further deviation of the present work from the routine are the “finitenesses (!)” of the objects, this in contrast to the mainstream of ergodicity studies, which deal with systems of infinite size and duration ("the thermodynamic limit") and are capable to yield formal results. Moreover, in the following investigation of the onset of ergodicity in finite and transient systems, of size (N < ¥) and time durations (tr < ¥), consideration will be given to another, somewhat imprecise term that enters the following, accepted formulation of ergodicity: “In the ergodic limit, the path trajectory approaches arbitrarily near to every point in phase space” [1]. This nearness concept had been given crucial importance early on by the Ehrenfests [19,20]. In what follows the term “near” will enter in a numerically precise and important way through the nearness parameter “np”, which is the measure of the distance between a chosen phase-space point (X, P for a single classical particle) and the corresponding value on the trajectory. “Nearness” enters our computations in the following way: In the sequel we establish the number of times the trajectory approaches closely to a given point X, P in phase space. If this approach takes place to within values of “np” for both the coordinate and the momenta, then this is counted as a visit; if it fails to do so for either, then this approach is not counted as a visit. Typical values to achieve ergodic tendencies in our finite systems are found with np not falling below a 0.02 fraction of the characteristic values in X and P (which are of the same order, in our system of units). Thus, we find that below this value of “np” the number of approaches, or “visits” (here designated NV, for visiting numbers) take effectively random values for different points in phase space; only above this tolerance (np > 0.02) are the visiting numbers distributed in a regular way. The regularities are discussed in the text. 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 also not ergodic) and were intensively studied by Barkai and collaborators, for example, for anomalous diffusers temporarily residing in traps [22–25] and more recently in Reference [26]. In contrast to these, the dynamic, continuously moving system studied here in the classical section shows no signs of weakly breaking of ergodicity. The Models The objects of our treatment in the classical domain are coupled oscillators, Section2 a subject which lies at the basis of lattice dynamics and has been extensively studied for one-to-three dimensional systems. Finite numbers of interacting oscillators belong in reality to the realm of infra-red molecular spectra. In the quantum regime we treat N interacting 1/2-spins, Section3. (Notably, the quantum description in this section centering in the interacting spins is conceptually simpler than the classical one, in so much as the phase space is one dimensional, labelled with r by the series of discrete eigen-energy values, regarded as non-degenerate. A reduction in complexity, in terms of information contents, of a quantum description of a random process relative to a classical one was noted in Reference [27]. A further distinction is on the minimal grid size in coordinate-momentum phase space of quantal systems required by the uncertainty principle. Owing to the finite number of spins involved, cooperative effects like ferromagnetism are not applicable). Multiple interacting spin systems (chains)are currently at interest for their potentialapplications with a view to quantum metrology, as in Reference [28], and to information transfer, as in Reference [29]. Interacting systems having finite Hilbert state dimensions have also been theoretically researched for entanglement and other properties [30–38]. The more general topic of open systems was comprehensively treated in Reference [39]. Symmetry 2020, 12, 1642 3 of 16 On the experimental side, a quantum processor consisting of N = 8 coupled qubits and subject to time dependent interactions was studied with the view of achieving feasible quantum computing [40]. Finite number of interacting spins, (namely 15 spins forming a V15 cluster) constituting a molecular magnet were the subject of Reference [41]: the nature of couplings in this system differs from the one considered in the present work, in that PI includes also Dzyaloshinsky-Moriya and hyperfine couplings. In contrast; our finite spin systems reach only up to only 9 spins, and for these numerical results are presented, they extrapolate nicely to the formal results for infinite, thermal reservoirs of Rechtman and Penrose [42]. To round up the picture, there remains the study of effects arising from the mutual interaction between the two entities,spins and oscillators. These have in the first instance entered the field as the Jaynes-Cummings model, subsequently they have later formed the subject of the broadly scoped review of Reference [43]. One also notes a further work framed in the adiabatic approximation in Reference [44]. From the present viewpoint of ergodicity in finite systems, a study of their properties is under consideration by us. In a nutshell, 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. 2. Classical: N Oscillators Here the time dependent displacement variable X1(t) of the sub-system is coupled to the displacement variables [X2(t), ...., XN(t)] of the reservoir. A model is designed with the aim of tracing the (long time, but not infinitely long duration) 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 (classical) phase space, represented by the position and momentum labels X, P.