*7.3. Interpretation*

I have presented information about the eigenstates of *Hw* using both the subsystem and global perspectives. These perspectives can be brought together in the following way. Consider the state |*Eis*--*Eje*, a product of eigenstates of *Hs* and *He* (with particular values of *i* and *j*), and consider expanding that state in eigenstates of *Hw*. For the *EI* = 0.007 case, if a particular eigenstate |*Ekw* of *Hw* has a strong overlap with |*Eis*--*Eje*, then the (*k* + 1)th state is likely to have a much weaker overlap. This is expected given the way the energy distributions shift among the subsystems as the index is incremented, as illustrated in Figure 14. (A similar situation is considered for many body systems in [15].) Expanding in eigenstates of *Hw* with *EI* = 0.1 will work very differently. As illustrated in Figure 17, neighboring eigenstates will have energy distributions in the subsystems which are not radically different as the index is incremented. This suggests that the overlaps will vary much more smoothly with the index of *Ek*.

Similarly, the breadth of the distributions of |*Ew*-'s in the *s* and *e* energy distributions for *EI* = 0.1 suggests initial states with energy shared differently between *s* and *e* can still pick up similar overlaps with the |*Ew*-'s, accounting for the overall shape similarity among the different curves in the 2nd panel of Figure 18. Furthermore, since *Pw*(*E*), plus the phases, gives complete information about the global state |*ψw*, it is not surprising that, under equilibrium conditions (when the phases may be taken as random), states with similar *Pw*(*E*)'s also give similar *Ps*(*E*)'s and *Pe*(*E*)'s. (The |*Ew*- energy distributions in the *s* and *e* are not perfectly broad, so, not surprisingly, I have found examples of other initial states with particularly extreme energy distributions among *s* and *e* which have somewhat different shapes for *Pw*(*E*), and even the 2nd panel of Figure 18 shows noticeable variations.)

The initial states studied here are products of coherent states in *s* with energy eigenstates in *e*. That makes the simple illustration above less rigorous, but the coherent states are somewhat localized in energy, so the main thrust of the illustration should carry through. In addition, the energy distributions in *s* and *e* only contain some of the information relevant for calculating the overlap (*sEi*|*e Ej*--) · |*Ekw*, but again that information seems enough to capture some sense of what makes the *Pw*(*E*) curves so different for the two values of *EI*. Similar arguments can be used to relate my results to the Eigenstate Thermalization Hypothesis, which I do in Appendix C.

Finally, if one considers some process of extending this analysis to larger systems, one could imagine cases where the *Pw*(*E*) distributions become more narrow (perhaps einselected into sharp energies through weakly coupled environments as in the "quantum limit" discussed in [16]). The smooth qualities of *Pw*(*E*) we see for *EI* = 0.1 could correspond in such a limit to a relatively flat distribution within the allowed range. This could connect with ergodic ideas which count each state equally within allowed energies, making contact with conventional statistical mechanics.

On the other hand, looking at the *EI* = 0.007 case suggests another limit where *Pw*(*E*) could remain more jagged, preventing simple statistical arguments from taking hold. By envisioning limits in this way, it does seem like the primitive "thermalized" behaviors of the *EI* = 0.1 case discussed in Sections 5 and 6 are in some sense precursors to a full notion of thermalization for larger systems. Likewise, the alternative limit suggested by the *EI* = 0.007 case has parallels with Anderson and many body localization in large systems. The localized systems exhibit a lack of thermalization for the same sorts of reasons as the toy model considered here, namely the lack of full access to states that should be allowed based purely on energetic reasons. In addition, just as the localized case appears to reflect additional (approximately) conserved quantities [17], I have associated the special features of the *EI* = 0.007 case with the (partially broken) symmetry conserving *Hs*- and *He*- separately in the *EI* = 0 limit.

#### *7.4. The Effective Dimension as a Diagnostic*

One way to characterize the different qualities of the sets of curves in Figures 18 and 19 is using the "effective dimension"

$$d\_{eff}^w \equiv \frac{1}{\sum\_{i} (P^w(E\_i))^2}.\tag{9}$$

This quantity takes its minimum value of unity if *P*(*Ei*) is a delta function, and reaches its maximum possible value, *Nw*, if all *P*(*E*)'s are identical. Table 1 compiles information about the *dweff* values for the curves shown in Figure 18, as well as for the *Pw*(*E*)'s for *EI* = 1 and *EI* = 0.02 (which I have not displayed in graphical form and for which *deff* takes on intermediate values).

**Table 1.** The effective dimension (*dweff* , from Equation (9)), evaluated for and averaged over the five sets of initial states used in this paper. The effective dimension is larger when the function *P*(*E*) is broad and smooth. Comparing the curves in the top two panels of Figure 18 (as well as Figure 19) suggests it is not surprising that *deff* for *EI* = 0.1 is more than 60 times greater than the *EI* = 0.007 case. (The quantity Δ gives the variance of *deff* across the five solutions.)


The extremely different natures of the *EI* = 0.1 and *EI* = 0.007 curves are nicely captured by the large difference between their *deff* values. In addition, Δ, the variance of *deff* across the five different states gives one measure of "scatter" among the different *Pw*(*E*) curves for fixed *EI*. This scatter is smallest for the *EI* = 0.1 case, which is consistent with the observations made about energy distributions in Section 6 (although those were focused on energy distributions in the subsystems).

#### **8. Tuning of States and Parameters**

In the analysis presented here, the "thermalized"-like behavior seems to emerge as a special case for a particular value (presumably actually a small region of values) for *EI*. In much larger systems exhibiting localization discussed in the literature, it is typically the non-thermalized behavior that seems special, usually associated with specific parameter choices that lead to integrability. I simply note here that it is not surprising that such matters of tuning depend on measures implicit in the model being considered. I regard the ACL model as too simplistic to draw broad conclusions about tuning of parameters, except as an illustration of how measures can turn out differently. If the "emergent laws" perspective mentioned in the introduction is ever realized, that will come with its own perspective (and probably challenges) regarding measures.

I also note that tuning of the initial state is involved in the notion of equilibration. A special choice of initial state with a low entropy is required in order to see a system dynamically approach equilibrium. The fact that the actual Universe did indeed have such a special low entropy initial state is a source of grea<sup>t</sup> interest and curiosity to me, and although it is not often stated that way, it is related to the notorious "tuning problems" in cosmology (see [18] for pioneering work and Section 5 of [2] for a recent summary). That is certainly not (directly) the topic of this paper, although I can not help but note with interest the very different perspective I sometimes see in the statistical mechanics literature (for example, [12] which implies that the Universe should be taken to be in a typical state).

#### **9. Discussion and Conclusions**

This research originated with my curiosity about various behaviors of the ACL model that I encountered in earlier work [1,2]. On one hand, the equilibration process seemed so robust I wondered if there was a straightforward ergodicity picture to back it up. On the other hand, examination of the energy distributions that appeared in equilibrium made it clear that no conventional notion of temperature applied. Furthermore, standard arguments would interpret the part of the environment density of states *N*(*E*) that decreases with *E* (see Figure 9) as a "negative temperature". Would that introduce strange artifacts in our results?

In this work, I have examined the equilibration processes in the ACL model systematically. I have seen how the dephasing process is the solid foundation on which the equilibration takes place. Dephasing is able to drive equilibration under conditions where the notions of temperature and ergodicity do not apply. I have argued that this very basic form of equilibration is sufficient to support the use of the ACL model in studies of the equilibrium phenomena explored in [2].

Even though the notion of temperature does not apply, I have considered some primitive aspects of "thermalization." Specifically, I have considered the expectation that different initial states with the same global energy thermalize to the same subsystem energy distributions. The ACL model is only able to realize this expectation in equilibrium for certain values of the coupling strength. When this aspect of thermalization is realized, the energy distributions in the global space are smooth. In the other cases, the global energy distribution can be quite jagged. I have related these different behaviors to the intrinsic properties of the global energy eigenstates, and argued that the smooth behavior could be viewed as something of a precursor to ergodicity, which might take a more concrete form in the limit of larger system sizes. In addition, I have noted some rough parallels with discussions of the presence or absence of thermalization in large condensed matter systems.

Regarding negative temperature, even in the absence of a solid notion of temperature, the evolution of the energy distribution depicted in Figure 9 toward regions of lower energy but higher density of states might be seen as a more primitive version of the phenomena that can be associated with a negative temperature in other systems.

I have found it interesting to learn the degree to which the very simple ACL model is able to reflect certain familiar elements of equilibration, while still missing out on others due to its small size and other "unphysical" aspects. Understanding systems such as this one that are on the edge of familiar behaviors could prove useful in exploring selection effects in frameworks where the laws of physics themselves are emergent, one of the motivations for this research I discussed in the Introduction. Such work might ultimately help us understand the origin of the specific behaviors of the world around us that we call "physical".

### **10. Reflections**

It is a grea<sup>t</sup> pleasure to contribute to this volume honoring Wojciech Zurek's 70th birthday. I first met Wojciech at an Aspen Center for Physics workshop the summer after I completed my PhD in 1983. I have had the good fortune of having numerous connections with Wojciech since then, including as his postdoc later in the 1980s. Wojciech has been an inspiration to me in many ways. For one, his unbounded and energetic curiosity has led to some of the most joyful and adventurous conversations of my entire career. It is definitely in the spirit of this adventurous style that I have pursued the topics of this paper. I am also grateful to Wojciech for helping me develop a taste for natural hot springs. It is fitting that certain advances on this project were made while partaking of some of my local favorites (experiencing temperature, but fortunately not equilibrium).

**Funding:** This work was supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics QuantISED program under Contract No. KA2401032.

**Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** I am grateful to Rose Baunach, Zoe Holmes, Veronika Hubeny, Richard Scalettar, and Rajiv Singh for helpful conversations.

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Appendix A. Initial Conditions**

As discussed in [1], coherent states can be constructed for the truncated SHO using the standard formula

$$|a\rangle = \exp\left(a\mathbf{\hat{a}}^{\dagger} - a^\*\mathbf{\hat{a}}\right)|0\rangle. \tag{A1}$$

The five initial states used throughout this paper are constructed by selecting the *i*th eigenstate of *He*, with *i* drawn from {300, 400, 450, 500, 550} (ordered with increasing eigenvalue). Recall that *Ne* = 600.

I then adjust *α* so that the initial state |*αs*|*ie* has *Hw*- = 25. Note that *Hw*- includes the interaction term, so the value of *α* technically depends on *EI*. Except for the strongly coupled *EI* = 1 case, this dependence is very weak. The exact values of the different initial subsystem energies in each case can be read from Figures 4–7.

#### **Appendix B. Dephasing, Decoherence and Dissipation**

This work focuses on dissipative processes which cause energy to flow and then stabilize at equilibrium values. My focus on energy flow is natural for studying the topic of thermalization. In [1], we explored how decoherence and dissipation happen on different time scales in the ACL model, a difference which is much more pronounced for larger systems. In addition, I note here that the term dephasing can mean different things, for example, relating the phases of two different beams as discussed in [19], where the notion of depolarization is also explored and contrasted. I use dephasing to refer to the randomization of the *αiw* phases over time. The dephasing I observe is not absolute, since recurrences are to be expected eventually. However, the recurrence time lies far beyond the time ranges explored here (and in fact far outside the dynamic range of my computations, which are documented in [1]). This arrangemen<sup>t</sup> of the various relative time scales is well suited for the topics of this paper.

#### **Appendix C. Eigenstate Thermalization Hypothesis**

The Eigenstate Thermalization Hypothesis (ETH) [20–22] proposes that important statistical properties of thermalized systems can be expressed by single eigenstates of the global Hamiltonian. Even though the ETH is intended to apply to much larger systems, out of curiosity I have explored how the ACL behaviors might relate to the ETH. Focusing on the *Ps*(*E*)'s and *Pe*(*E*)'s, one can see from Figures 14 and 17 that the subsystem energy distributions look more similar for individual eigenstates of *Hw* for *EI* = 0.1 than for *EI* = 0.007. This similarity suggests that for the "thermalized" *EI* = 0.1 case individual eigenstates are starting to line up behind a common statement about the subsystem energy distributions, which would be a signature of the ETH. But the fact that significant differences remain among the *EI* = 0.1 curves suggests this signature is not expressed strongly.

Figure A1 presents a more systematic analysis of the *Hw* eigenstates relevant to the sets of initial conditions used here. (The selection of the eigenstates is illustrated in Figure A2.)

**Figure A1.** Exploring ETH: The energy distributions for the *EI* = 0.1 case from Figures 10 (system) and 11 (environment) are shown (blue and red curves, respectively). In addition, the energy distributions for individual eigenstates of *Hw* are plotted. As depicted (and color coded) in Figure A2, the particular *Hw* eigenstates are chosen to be representative of the global energy distributions that correspond to the original curves from Figures 10 and 11. No single eigenstate fully characterizes the equilibrium distributions (although they are not completely off the mark), and there is significant scatter among the distributions drawn from the individual eigenstates. These results sugges<sup>t</sup> only a very loose realization of the ETH.

The *Ps*(*E*) and *Pe*(*E*) curves for those eigenstates approach the energy distributions for our set of initial states reasonably well, but they still retain considerably greater scatter than seen for the non-*Hw* eigenstate states. Thus, the averaging over the entire distribution in *Pw*(*E*) space appears to have a significant role in realizing the sharpness of the convergence of the *Ps*(*E*) and *Pe*(*E*) curves. Although one could try to argue that there are elements of the ETH being expressed here, any such expression is only an approximate one.

**Figure A2.** Selected eigenstates: The eigenvalues of the *Hw* eigenstates chosen for Figure A1 are shown here (vertical lines), against the backdrop of the global energy distributions from the *EI* = 0.1 panel in Figure 18, which correspond to the original energy distributions depicted for the subsystems in Figure A1. There are three green and three purple lines, which are unresolved in the figure because they show adjacent eigenvalues (which only differ at the 0.01% level). The curves in Figure A1 are matched by color with the eigenvalues shown here.
