**1. Introduction**

In [1], my collaborators and I introduced a toy model which adapted the Caldeira– Leggett model for numerical analysis. This "Adapted Caldeira–Leggett" (ACL) model was designed to optimize decoherence and einselection between a simple harmonic oscillator (SHO) and an environment. The SHO and environment are treated together as a closed, unitarily evolving quantum system. The ACL model naturally equilibrates when evolved for a sufficient length of time. We used equilibrium states thus obtained in [2] to study the extent to which equilibrium states can exhibit einselection [3] despite the absence of an arrow of time. That work was motivated especially by cosmological considerations.

In this paper, I dig deeper into the equilibrium behavior exhibited by the ACL model. I show that a range of equilibration behaviors is possible, some of which show attributes that might be seen as a primitive form of "thermalization," and others that definitely do not. I show how the different behaviors are controlled by parameters in the ACL Hamiltonian and relate these behaviors to properties of the global energy eigenstates.

One motivation for this work is to make sure the equilibrium states of the ACL model form a suitable foundation for the studies in [2] (they do). However, by design, the ACL model eschews a number of physically realistic features (such as locality in the environment) to allow decoherence to function efficiently with limited computational resources. One might ask, what is the point of studying equilibration in a system with significant unphysical features? While in [1], and also in [4], we do make some connection to results from Nuclear Magnetic Resonance experiments, for the purposes of this paper, I regard the physically unrealistic aspects of the ACL model as a strength.

I have long been fascinated by the question of the emergence of physical laws through the identification of (possibly multiple) semiclassical domains in large quantum systems (as

**Citation:** Albrecht, A. Equilibration and "Thermalization" in the Adapted Caldeira–Leggett Model. *Entropy* **2022**, *24*, 316. https://doi.org/ 10.3390/e24030316

Academic Editors: Sebastian Deffner, Raymond Laflamme, Juan Pablo Paz and Michael Zwolak

Received: 2 February 2022 Accepted: 17 February 2022 Published: 23 February 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

discussed, for example, in [5,6]). One feature that seems to be important is the capability of large numbers of degrees of freedom to behave in very simple ways that have a semiclassical description. In physically realistic situations, this is often achieved by the processes of equilibration and thermalization. I am interested in turning the question around and learning in general terms what sorts of physical systems can achieve these behaviors. I am curious if the need to have equilibration and thermalization can help choose—in some selection process associated with their emergence—key features of the laws of physics as we know them. This paper takes a very small step in that direction by exploring the various behaviors of the ACL model.

I introduce the ACL model in Section 2 and in Section 3 demonstrate how the process of dephasing lies at the root of the wide range of equilibration processes that the model exhibits. I point out that, quite generically, the dephasing processes provide all that is needed to produce suitable equilibrium states for the work in [2]. In Section 4, I show that the processes depicted in Section 3 are missing a key feature associated with thermalization. Sections 5 scan a range of results produced by varying a parameter in the Hamiltonian. I show how an appropriate choice of this parameter allows the ACL model to include the thermalizing features, and argue that additional approximately conserved quantities are present in cases in which such features are absent. The discussion up to Section 5 has been shaped by tracking the energies of the SHO and the environment, specifically the first moments of the energy distributions for each of these subsystems. Section 6 expands the discussion to scrutinize the full energy distributions for these subsystems, starting with their initial forms and tracking them as they settle into equilibrium. I explore how these also depend on the parameters of the system and study the presence or absence of the thermalizing behavior in this context. Section 7 examines the properties of the global energy eigenstates, first establishing their general properties and then relating those to various behaviors reported earlier in the paper. The question of the tuning of parameters and the initial states is examined briefly in Section 8, and Section 9 presents some further discussion and conclusions. This is an invited paper for a special volume honoring the 70th birthday of Wojciech Zurek, and I offer some appropriate reflections in Section 10.

#### **2. ACL Model**

In [1], my collaborators and I introduced a toy model which adapted the Caldeira– Leggett model for numerical analysis. This "Adapted Caldeira–Leggett" (ACL) model has a "world" Hamiltonian *Hw* of the form

$$H\_{\rm w} = H\_{\rm s} \otimes \mathbf{1}^{\varepsilon} + H^{I} + \mathbf{1}^{s} \otimes H\_{\rm c} \tag{1}$$

where the "system" Hamiltonian *Hs* represents a truncated simple harmonic oscillator (SHO).

The interaction term is given by *H<sup>I</sup>* = *qs* ⊗ *H<sup>I</sup> e*, where *qs* is the SHO position operator and *H<sup>I</sup> e* has the form

$$H\_c^I = E\_I R\_I^c + E\_I^0. \tag{2}$$

The matrix *Re I* is a random matrix constructed by drawing each of the real and imaginary parts of each independent matrix element of this *Ne* × *Ne* Hermitian matrix from a distribution that is uniform over the interval [−0.5, 0.5]. Throughout this paper, I will use *Ne* = 600 (the size of the environment Hilbert space) and *Ns* = 30 for the truncated SHO (as used in [1,2]).

The environment self-Hamiltonian is given by

$$H\_{\mathfrak{k}} = E\_{\mathfrak{k}} \mathbb{R}^{\mathfrak{k}} + E\_{\mathfrak{c}}^{\mathfrak{U}} \tag{3}$$

where *Re* is constructed in the same manner as *Re I*, but as a separate realization. In Equations (2) and (3), *EI* and *Ee* are c-numbers which parameterize the overall energy scales. Both *Re I*and *Re* are fixed initially and are not changed during the time evolution. The full Hamiltonian of the ACL model is time independent. All the results in this paper use *E*<sup>0</sup> *I* = *E*<sup>0</sup> *e* = 0, but nonzero values for these offset parameters have been useful in other contexts such as [2].

In [1], we demonstrated how the ACL model is able to numerically reproduce decoherence phenomena typically studied with the original Caldeira–Leggett model, and argued that the specific form of *Hw* enables the numerical studies to reproduce these phenomena in an efficient manner. In [2], we used ACL model calculations to address the relationship between the arrow of time and the emergence of classicality (a topic we motivated with cosmological considerations), and, in [4], we explored new phenomena at the early stages of decoherence.

We made a point in [1] of demonstrating the capability of the ACL model to equilibrate, and these equilibrated states played a key role in [2]. The focus of this paper is to more fully understand the equilibration processes in the ACL model, and place them in the context of modern ideas from quantum statistical mechanics.

#### **3. Basic Equilibration and Dephasing**

The basic equilibration process of the ACL model is demonstrated in Figure 1. In this example, energy flows from the SHO to the environment for a period of time, and then the energies in both systems stabilize, up to small fluctuations.

**Figure 1.** Equilibration in the ACL model: Entropy increases and energy flows from the environment to the SHO for a period of time, and then stabilizes up to small fluctuations. (**a**) entanglement entropy between the SHO and the environment; (**b**) subsystem energies *Hs*- (blue) and *He*- (red), and interaction energy *H<sup>I</sup>* (grey).

For these curves, I used *Ee* = 1 and *EI* = 0.02. Throughout this paper, I use units where *h*¯ *ωSHO* = 1. The initial state is a product of a coherent state for the SHO and an eigenstate of *He*, each with energies as shown in the plot.

As inferred in Appendix A of [2], the basic mechanism for this equilibration is "dephasing." When expanding the global state as

$$\left|\psi\_{w}\right\rangle = \sum\_{i} a\_{w}^{i}(t) \left|E\_{w}^{i}\right\rangle \tag{4}$$

where --*Eiw* are the eigenstates of *Hw*, special relationships are required among the *αiw*(*<sup>t</sup>* = 0) to realize the initial product form of |*ψw*-. The dimension of the global space (*Nw* = *Ns* × *Ne* = 18,000) is sufficiently large, and the eigenvalues of *Hw* are sufficiently incommensurate that these special relationships come undone over time. The equilibrium state corresponds to the state where the phases of the *αiw*(*<sup>t</sup>* = 0) are fully randomized. This is demonstrated explicitly in Figure 2 where additional curves are included from an initial state where the phases of *αiw* were randomized "by hand" at *t* = 0.

**Figure 2.** Dephasing: In addition to the curves shown in Figure 1, I have added dotted curves from the calculations with randomized *αiw* phases discussed in the text. The convergence of the dotted and solid curves at later times reflects the dephasing nature of the equilibration process.

This dephasing process is well-known in the quantum statistical mechanics literature (as reviewed, for example, in [7]. Some nice historical reflections can be found in [8]). Papers such as [9,10] also demonstrate its general relevance to decoherence. In Appendix B I elaborate a bit on how how I used the term dephasing here, and its connection to other topics such as decoherence.

Figure 3 shows the effect of choosing different random number seeds on curves from Figure 2.

In this figure, the curves from Figure 2 are reproduced along with four additional curves. The additional curves were generated the same way, except with different seeds for the random number generator used for randomizing phases and for the random entries in *He* and *<sup>H</sup>Ie*. The similarity of these sets of curves reflects the fact that artifacts of individual random number seeds show up only in the small scale fluctuations.

I note that the equilibration process driven by dephasing presented here is more than sufficient as a basis for the equilibrium states studied in [2]. For that work, the crucial piece is that detailed balance should be respected, so that fluctuations and their time reverse are equally likely to appear. Next, I turn to more nuanced aspects of equilibration in this model which I find interesting, although not specifically in the context of [2].

**Figure 3.** Random artifacts: The curves from Figure 2 are shown with additional curves (in brighter shades) giving the same calculations using different random number seeds. The small panels zoom in on the *H*-'s during equilibrium, and show the same state in each panel, with and without the randomized phases. The matched pairs converge tightly (supporting the dephasing picture), while subtle changes to the Hamiltonian from changing the random number seed generate a slightly larger scatter from one set to another. The small panels have black horizontal lines that are the same for both sets as a reference to aid in tracking the vertical scatter between the sets.

#### **4. Equilibration without "Thermalization"**

Figure 4 plots the entanglement entropy and subsystem energies from Figure 1 along with equivalent curves produced with different initial conditions.

In each case, the total energy was set at *Hw*- = 25, but the initial energy was distributed differently between the environment and the SHO. As with Figure 1, all SHO initial states were coherent states with initial energies as shown in the plot, and the environment initial states were eigenstates of *He*. More details about how the initial conditions are constructed appear in Appendix A.

One expects isolated physical systems with the same global energy to thermalize to the same distribution of energies among subsystems, regardless of initial conditions. To the extent that that has not happened in the examples shown in Figure 4, it appears that this equilibration process does not exhibit that aspect of thermalization. There appear to be interesting parallels between the behaviors of the ACL model and the behaviors associated with localization phenomena in condensed matter systems [11]. I will touch on this a couple of times in this paper but warn the reader that I use the term "thermalize" with a grain of salt since no example given here has all the features one associates with full thermalization.

**Figure 4.** Varying the initial conditions: In addition to the curves from Figure 1, I have added results from different initial conditions (all with the same total energy *Hw*-). The new curves have a different shade of the same color and each set (*S*, *Hs*- and *He*-) is matched by line type. Note that, while the total energy is the same, at the end of the equilibration process, the energy distribution between the environment and the SHO is different in each case. Here, *EI* = 0.02.

#### **5. Varying the Coupling Strength** *EI*

It is instructive to consider the special case of *EI* = 0. In that case, no energy will flow between the environment and the SHO, and the energies *He*- and *Hs*- are separately conserved. When *EI* = 0, formally, *any* initial state is already "equilibrated" in terms of the values *He*- and *Hs*-, insofar as after an extended period of evolution these will be unchanged. In this section, we examine the different behaviors that emerge as *EI* is varied. We will see that the *EI* = 0 case offers a useful reference point for this exploration and helps us interpret the lack of "thermalization" discussed with Figure 4.

Figure 5 uses *EI* = 0.007, a factor of 0.35 down from the case shown in Figure 4. The sets of initial conditions and the other parameters in *Hw* are identical for the two figures.

Here, the energy in each subsystem changes very little, and the equilibrium energy values achieved from different initial conditions are further apart than in Figure 4. This is as expected since one is closer to the *EI* = 0 case. One might have the intuition that any nonzero value of *EI* should allow equilibration, with smaller *EI*'s leading to longer equilibration times. I believe that intuition is only valid for much larger systems, and, in any case, it is certainly not valid for the results reported here.

Figure 6 corresponds to *EI*= 0.1, considerably larger than values used for Figures 4 and 5.

 The energy for each subsystem converges to the same value after equilibration, realizing the sense of "thermalization" considered in Section 4. I will further explore the ways this equilibration process is different from what is seen for other values of *EI* in later sections of this article. In addition, note that the interaction energy *HI*- (black curves) shows larger fluctuations than for the previous plots, but that these curves still settle down to the same value, which is considerably smaller than the equilibrium energies in either subsystem. This allows one to still consider this a "weakly coupled" case.

Figure 7 shows a strongly coupled case, with *EI* 10 times larger than the value used for Figure 6.

With such a large coupling, each subsystem has much less of an individual identity in terms of its evolution and the interpretation of *Hs*- and *He*-. However, the dephasing process still leads to stable values for *Hs*-, *He*- and *S* at later times, with only small fluctuations.

**Figure 5.** *EI* = 0.007: These curves correspond to the curves in Figure 4, but evolved with a lower value of *EI*, closer to the limit of complete decoupling. These curves exhibit less energy flow, and greater differences among the equilibrium values for each subsystem, despite the fixed value of the global energy. This is what one expects as one approaches the *EI* = 0 limit.

**Figure 6.** *EI* = 0.1: Similar to Figures 4 and 5, but evolved with a larger value of *EI* (5 times larger than for Figure 4). All the initial conditions converge to the same energy value for each subsystem, realizing the primitive notion of "thermalization" discussed in Section 4.

**Figure 7.** *EI* = 1: This case is strongly coupled. Under such conditions, each subsystem has less of an individual identity, but the dephasing process still creates some notion of equilibration.

#### **6. Energy Distributions**

Throughout this paper, I take the global system to be in a pure quantum state |*ψw*-. In general, each subsystem will be described by a density matrix according to

$$\rho\_{\mathbb{s}} \equiv \operatorname{Tr}\_{\mathfrak{c}}(|\psi\rangle\_{\mathfrak{w}^{\mathfrak{w}}} \langle \psi|) \tag{5}$$

and

$$\rho\_{\varepsilon} \equiv \operatorname{Tr}\_{s}(|\psi\rangle\_{\mathcal{W}^{\operatorname{uv}}} \langle \psi|). \tag{6}$$

When written on the basis of eigenstates of *Hs* and *He*, respectively, the diagonal elements of *ρs* and *ρe* give the probabilities assigned to different values of the subsystem energies. I define

$$P\_s(E) \equiv \operatorname{diag}(\rho\_s^E) \tag{7}$$

and

$$P\_{\mathcal{C}}(E) \equiv \operatorname{diag}(\rho\_{\mathcal{C}}^E) \tag{8}$$

where the superscript *E* indicates that the energy eigenbasis is used. Note that, for the ACL model, the argumen<sup>t</sup> *E* on the left side of these expressions is drawn from the discrete set of energy eigenvalues.

These figures show the case for which the energies (namely, the first moments of these distributions) are shown in Figure 1. One can see that, as we have already seen in the case of the first moments, the whole distribution stabilizes at late times, up to small fluctuations.

Figures 10 and 11 show the late time distributions for the full range of initial conditions and choices of *EI* considered above. It is certainly not surprising that the cases where the subsystem energies equilibrated to different values show significantly different late time forms for the overall distributions. It is interesting though that, in the case (*EI* = 0.1) where the subsystem energies equilibrated to the same values for different initial states, the entire distribution appears to equilibrate to the same form, encompassing many more moments than just the first.

Figure 8 shows the time evolution of *Ps*(*E*) and Figure 9 shows the corresponding *Pe*(*E*) evolution.

**Figure 8.** Time evolution of SHO energy distribution *Ps*(*E*) for the case shown in Figure 1. The histogram in the lower panel is the density of energy eigenstates for the SHO (which is uniform). I have connected the discrete set of points given by *Ps*(*E*) here for ease of viewing.

**Figure 9.** Time evolution of environment energy distribution *Pe*(*E*) for the case shown in Figure 1. The histogram in the lower panel shows the density of energy eigenstates for the environment (which reflects the Wigner semicircle form expected for a random Hamiltonian). The *Ne* = 600 different eigenvalues have been binned as shown in the histogram and I plot *Pbe* (*E*), which is the total probability in the corresponding bin. This discrete set of points is connected for visualization purposes.

**Figure 10.** Late time energy distributions for the SHO. Each panel shows a set corresponding to the five different initial conditions, evolved using the value of *EI* as marked. For *EI* = 0.1, the SHO energies (first moments of *Ps*(*E*)) converged at late times for the different initial conditions (see Figure 6). These results indicate that many more than just the first moment converges for this value of *EI*. The *EI* = 0.1 panel includes the phase randomized version of each curve as well.

In Section 4, I discussed a primitive notion of "thermalization" based on the expectation that a thermalized system should share energies in the same proportions among different subsystems, regardless of the initial state, as long as each initial state had the same total energy. Among the cases considered in Section 5, we saw that only the *EI* = 0.1 case met that criterion. Figures 10 and 11 show that, for *EI* = 0.1, the system meets a stronger criterion, namely that many moments of the final energy distributions are independent of the initial state. This is certainly what one gets in the case of true thermalization of realistic physical systems, although it is worth emphasizing that none of the distributions shown are truly thermal in the sense of having the Gibbs form, as a function of an actual temperature. Still, having a parameter to dial which can turn on or off the rudimentary features of thermalization discussed here suggests that further explorations might reveal some insights into the notion of thermalization in general. I undertake such explorations in what follows.

It is tempting to make contact with the notion of "generalized canonical state" as discussed, for example, in [12]. However, in the places I have seen the generalized Gibbs distribution discussed it has taken a more idealized form. For example, a thermodynamic limit is taken or an idealized notion of "passivity" [13,14] is utilized. Those idealizations would preclude the sort of small fluctuations that appear in the my results.

**Figure 11.** Late time energy distributions for the environment. Each panel shows a set corresponding to the five different initial conditions, evolved using the value of *EI* as marked. For *EI* = 0.1, the environment energies (first moments of *Pe*(*E*)) converged at late times for the different initial conditions (see Figure 6). These results indicate that many more than just the first moment converges for this value of *EI*. The *EI*= 0.1 panel includes the phase randomized version of each curve as well.

Before concluding this section, I want to comment about the strongly coupled *EI* = 1 case. I have included it in this paper for completeness, but it should be emphasized that, due to the strong coupling, there is little meaning to the *s* and *e* subsystems. The quantities plotted in that case (*Hs*-, *Hs*-, *Ps*(*E*) and *Pe*(*E*)), are mathematically well-defined, but they do not have natural physical interpretations. It seems unlikely that the *EI* = 1 case admits a physically useful interpretation of the full space as a tensor product of any subspaces— certainly not the specific *e* and *s* ones considered here. One point one can make about this case is that it demonstrates that a rudimentary process of equilibration, driven by dephasing, is possible without any reference to energy flow, or any sense in which one subsystem is acting as a "bath" to another. Comments along these lines appear in [12].

#### **7. The** *Hw* **Eigenstates**

#### *7.1. Energy Distributions in the Subspaces*

Here, I explore the relationship between the phenomena discussed above and the form of the eigenstates of the global Hamiltonian *Hw*. I will focus here on the *EI* = 0.007 and *EI* = 0.1 cases. The *EI* = 0.02 case exhibits behavior intermediate between those two, and, as discussed above, the *EI* = 1 is not amenable to deeper analysis due to the strong coupling.

Recall that, for the *EI* = 0 case, the eigenstates of *Hw* are products of eigenstates of *Hs* and *He*. Once the interaction is turned on, in general, *Hw* eigenstates will appear as density matrices in the *s* and *e* subspaces. I will utilize the techniques from Section 6 to focus on the energy distributions *Ps*(*E*) and *Pe*(*E*), given by the diagonal elements of the density matrices according to Equations (7) and (8). While these give incomplete information (only certain matrix elements will be shown, and *Pe*(*E*) will appear binned as above), that information is sufficient to ge<sup>t</sup> a sense of what is going on. I have studied the properties of these states with more complete information than presented here and have confirmed that the information I do present gives a reasonable characterization for the points I want to make.

Figures 12 and 13 show information about a broad range of *Hw* eigenstates in terms of *Ps*(*E*) and *Pe*(*E*), respectively, for the *EI* = 0.007 case.

**Figure 12.** A selection of eigenstates of *Hw* (running from minimum to maximum eigenvalues) represented in terms of distributions in SHO energy. Here, *EI* = 0.007. Note that, while these are not perfect delta functions, they are reasonably sharply peaked, as one would expect for a situation close to the *EI*= 0 limit.

For this case, one can see that, while the eigenstates of *Hw* may not be perfect product states of *Hs* and *He* eigenstates, the energy distributions are still quite localized, as one would expect for very weak coupling.

Figure 14 shows energy distributions in both *s* and *e* subspaces for three adjacent energy eigenstates of *Hw*.

While the three pairs of curves represent eigenvalues of *Hw* which differ by at most 0.04%, the energy distribution between *s* and *e* is very different for each of the three cases. This is simply a reflection of the fact that there *are* many ways of distributing a fixed total energy among the two subsystems, and the peaked nature of the energy distributions for this very weakly coupled case allows that fact to play out in a simple and vivid way in the eigenstates of *Hw*.

**Figure 13.** The same eigenstates shown in Figure 12 are shown here in terms of the (binned) distributions in environment energy. These too are reasonably sharply peaked, as expected.

**Figure 14.** Energy distributions shown in both *s* (blue) and *e* (red) for three adjacent eigenstates of *Hw*, with *EI* = 0.007. Although the associated values of *Ew* are essentially identical the energy is distributed in very different ways between the two subsystems.

Figures 15 and 16 show information about a broad range of *Hw* eigenstates in terms of *Ps*(*E*) and *Pe*(*E*), respectively, this time for the *EI* = 0.1 case.

**Figure 15.** A selection of eigenstates of *Hw* (running from minimum to maximum eigenvalues) represented in terms of distributions in SHO energy. Here, = *EI* = 0.1. While this case is weakly coupled by some measures, the interaction is strong enough to mix many of the energy eigenstates of the SHO, creating much broader distributions than seen in Figure 12 for the *EI* = 0.007 case, at least away from the extreme ends of the spectrum.

**Figure 16.** The same eigenstates shown in Figure 15 are shown here in terms of the distributions in environment energy. These distributions are also much more broad than those shown in Figure 13 for the *EI* = 0.07 case.

While in many respects the *EI* = 0.1 case might be thought of as "weakly coupled"— for example, note the small relative values of the interaction energy shown in black in

Figure 6—the interaction term is strong enough to mix many eigenstates of *Hs* and of *He*, leading to much broader distributions, except at the extremes of the spectrum. Note that the relevant measure for understanding the breadth of these distributions is the size of the interaction term relative to the *spacing* of the energy eigenvalues of the respective subsystems.

Figure 17 shows energy distributions in both *s* and *e* subspaces for three adjacent energy eigenstates of *Hw*, here with *EI* = 0.1.

**Figure 17.** Energy distributions shown in both *s* (blue) and *e* (red) for three adjacent eigenstates of *Hw* with *EI* = 0.1. For these broad distributions, the differences between neighboring eigenstates are more subtly compared with the more weakly coupled case shown in Figure 14.

In contrast with what was seen for the more weakly coupled case in Figure 14, the changes in the subsystem energy distributions as one steps between adjacent eigenstates of *Hw* are more subtle, although the differences can be discerned upon inspection.

#### *7.2. Energy Distributions in the Global Space w*

Having established significant differences between eigenstates of *Hw* as they appear in the subsystems, depending on the strength of the coupling *EI*, I will now consider how the initial conditions are represented in these different sets of eigenstates. Figure 18 shows the distributions *Pw*(*E*), defined in the same manner as *Ps* and *Pe* (Equations (7) and (8)) but in the global "world" space *w* using *ρw* = |*ψw<sup>w</sup>ψ*|. The *Pw*'s are time independent, and the different curves correspond to the set of five different initial conditions used to create each of Figures 4–7. (Note that I have only been using the same set of five initial conditions throughout this paper. I have been evolving and analyzing each one using *Hw*'s with different values of *EI*.)

**Figure 18.** The (binned) global energy distributions *P<sup>b</sup> w*(*E*) corresponding to the five initial states used throughout this paper. One can see that these curves have very different behaviors depending on the coupling strength *EI*. For example, the *EI* = 0.007 curves frequently approach zero, and the *EI* = 0.1 curves do not. The bottom panel shows a histogram representing the density of energy eigenstates *<sup>N</sup>*(*Ew*). Technically, *N*(*Ew*) will be different for the two values of *EI*, but both the values shownherearesmallenoughnottochangetheformof *N*(*Ew*) significantly.

 Figure 19 shows zoomed-in portions of the top two panels of Figure 18.

**Figure 19.** A zoomed-in look at the first two panels of Figure 18. One can see that not only do the *EI* = 0.007 curves approach zero frequently, in contrast to the *EI* = 0.1 curves, but the curves for each state exhibit very different patterns of large and small values.

The large oscillations of the *EI* = 0.007 curves are especially clear in Figure 19, and one can see that the individual curves have very different locations of their peaks and minima.
