# Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal

## Abstract

**:**

## 1. Introduction

#### The Models

## 2. Classical: $\mathit{N}$ Oscillators

#### 2.1. Ensemble Construction

#### 2.2. The Visiting Number $NV$

#### 2.3. $NV$ in the Ensemble Average

#### Partial Summary

- (1)
- $NV$ vanishes outside a central circular region in the $X,P$ plane, This is, of course, a trivial consequence of the upper bound of the energies in the finite model, alterable only by increase of N.
- (2)
- $NV$ is (approximately) circularly symmetric and its values are functions of the energies $\frac{1}{2}({X}^{2}+{P}^{2})$. (Recall that masses and bare frequencies are in the model unity.) With momenta containing also a time dimension, in contrast to the displacements, it seems that this result indicates that the visiting numbers and cumulative sojourn times are commensural, and the system is not subject to what is termed “weakly breaking ergodicity” [21,22,23,24,25,26], in which the two are distinguished.
- (3)
- $NV$ is neither monotonic nor uniform along a radial ray (this is not a two dimensionality effect), with a dip at low energies. 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 (as is usually deemed), then for a time integration to be equivalent to that, time durations with low energies must be compensated by an enhancing factor. (The next section argues this from a very different perspective).

#### 2.4. Energy Distribution of NV

## 3. Quantal: $\mathit{N}$ Spins

#### 3.1. Density of Reservoir-States

#### 3.2. A Time-Dependent Configuration Number Argument

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Ensemble Construction Rule in Equation (5)

## Appendix B. Parameter Dependencies of the Visiting Number NV

**Figure A2.**Visiting numbers with a stricter nearness criterion for visits. Chaotic entries in the small squares.

**Figure A3.**Increasing the number of interacting oscillators alone does not induce order in the basic square entries.

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**Figure 1.**The trajectory trace of the sub-system in phase- space plane at a run time ${t}_{r}=2000$ in units of inverse frequencies and number of oscillators $N=3$. Overall, a dense covering is apparent.

**Figure 2.**Visiting number $NV$ distributed over the phase-space, plotted by a matrix plot in which the color depths represent the numerical values in the small squares (deeper color meaning higher values). Plotted for a nearness parameter $np=0.02$ and run time ${t}_{r}=4000$ (in reciprocal units of the frequencies). $NV$ is seen to be distributed regularly. Detailed discussion is in the text.

**Figure 3.**A contour plot for the visiting number $NV$ distribution in the phase space plane, using the same parameters and data as in the previous figure.

**Figure 4.**A three dimensional plot for the visiting number $NV$ distribution in the phase space plane, using the same parameters and data as in the previous figure.

**Figure 5.**MatrixPlot for the visiting numbers of a large ensemble (for a solutions of the same Hamiltonian, but 100,000 different initial conditions for the dynamic variables [$X1(t),P1(t)$]), with nearness parameter $np=0.02$. A fairly regular radial dependence is shown, with nearly circular shape (discussed in the text). The number of visits at a typical phase - space point is 400 with a peak of 1152 (in the course of ${10}^{5}$ runs!).

**Figure 6.**Distribution of visiting numbers $NV$ at a given time averaged over ${10}^{5}$ randomly chosen members in the ensemble. $np=0.02$.

**Figure 8.**Distribution of visiting numbers $NV$ as function of the energy $\frac{1}{2}({X}_{1}{(t)}^{2}+{P}_{1}{(t)}^{2})$.

**Figure 9.**Dots: Eigenenergies (in arbitrary units, numbering ${2}^{N}$ values) plotted against the serial number r for an N- spin system ($N=5,6,7$). There are two regions: the first sharply linearly increasing, followed by a more moderate tendency to rise. Curve to fit the dot values (after their arbitrary shifts vertically) described in the text.

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Englman, R.
Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal. *Symmetry* **2020**, *12*, 1642.
https://doi.org/10.3390/sym12101642

**AMA Style**

Englman R.
Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal. *Symmetry*. 2020; 12(10):1642.
https://doi.org/10.3390/sym12101642

**Chicago/Turabian Style**

Englman, Robert.
2020. "Ergodic Tendencies in Sub-Systems Coupled to Finite Reservoirs—Classical and Quantal" *Symmetry* 12, no. 10: 1642.
https://doi.org/10.3390/sym12101642