A thermodynamic analysis of energy eigenvalues

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Abstract

The use of the thermodynamic “internal energy” U as a tool for characterizing eigenvalue spectra is examined via Monte Carlo studies. We show that for pure and complete spectra, internal energy provides significant separation between individual Gaussian orthogonal ensemble (GOE) and Poisson spectra for spectra with as few as 30–40 levels. We also demonstrate that internal energy is sensitive to the presence of missing or spurious levels in GOE spectra. Comparison of the behavior of U and Δ3 for imperfect spectra shows that the relative variance of U is smaller; therefore, U is less sensitive to experimental limitations affecting the quality of data than is Δ3.

Introduction

Random-matrix theory (RMT) was first proposed to describe statistical properties of nuclear systems by Wigner in the 1950s [1]. Early developments, both theoretical and experimental, in this area can be followed through the collection of reprints in the book by Porter [2]; of special note for our current results is Wigner's result [3] that the level density for the Gaussian orthogonal ensemble (GOE) version of RMT is semicircular. A series of papers by Dyson and Mehta in the early 1960s [4], [5], [6], [7], [8] further developed the statistical description of eigenvalue distributions. Dyson and Mehta utilized circular ensembles, rather than the Gaussian ensembles normally used today, with the argument that the asymptotic behavior was the same but some of the mathematics was more tractable with the circular ensembles. Dyson showed [4] that there is a direct relation between the eigenvalue distributions of a random matrix and the distribution of the positions of unit charges confined to a unit circle at a certain temperature; therefore, the two-dimensional Coulomb gas serves as a physical analog to the ensembles.

Work on RMT continued through the 1960s and into the 1970s as summarized in the review by Brody et al. [9]. A revival of interest in RMT followed the conjecture by Bohigas et al. [10] of a connection between quantum chaos and RMT. A more recent review by Guhr et al. [11] discusses applications of RMT throughout quantum physics and includes over 800 references. A discussion of many of the mathematical details of RMT has been given by Mehta [12].

One of the issues that one faces when comparing RMT to nuclear energy levels is that the various statistics that have been used are all rather sensitive to missing or spurious levels in the data. In fact, for a significant time after the introduction of RMT, it was difficult to ascertain whether nuclear spectra disagreed with RMT or whether there were simply quality problems with the data. It was only with the combining of different data sets to form the Nuclear Data Ensemble [13], [14] in the early 1980s that this question seemed to be firmly resolved with the conclusion that neutron and proton resonance data are well described by the GOE version of RMT. This observation allows one to assume that the GOE describes resonances and to utilize deviations from the GOE to estimate the fraction of levels missing from an experimental sequence [15], [16]; such an estimate, in turn, allows better estimates of level densities [17]. However, nuclear resonances span only a limited range of energies and angular momentum; the general applicability of RMT to nuclei when one considers variations in energy, angular momentum, and mass is still an open question. For that reason alone, additional statistical measures should be useful. In addition, the great sensitivity of the available statistics to imperfect data can make it quite difficult to determine whether observed deviations from GOE are real or are due to data of insufficient quality. Different statistics that are less sensitive to imperfections in the data but can still serve as indicators of GOE behavior would address this issue as well as provide an important complement to current methods.

One aspect of RMT that was introduced over 40 years ago by Dyson [4] but has received significantly less attention than some other aspects since then is that of the thermodynamics of the ensembles. Quantities such as free energy, internal energy, entropy, and specific heat were studied, and expected values were obtained for each of the various ensembles (orthogonal, unitary, and symplectic) commonly discussed in this context. In their evaluation of early neutron resonance data, Dyson and Mehta used several statistics but concluded that the data quality was not sufficient to decide whether the data agreed with GOE or not. One of the statistics they introduced at that time—the Dyson–Mehta Δ3 statistic [7]—has become the standard measure of long-range order in spectra. In that early paper Dyson and Mehta also used one of the thermodynamic variables (the internal energy) as a test. However, this test has apparently not been utilized to evaluate experimental data since. In this paper, we investigate the internal energy U and study its behavior for GOE and Poisson spectra. In particular, we examine how this statistic behaves as a function of spectrum size and how it is affected by the presence of missing or spurious levels in the spectrum. We then compare the behavior of U with that of the Δ3 statistic.

In Section 2, we define the internal energy U for a collection of eigenvalues and look at its expected behavior for both GOE and Poisson spectra. In Section 3, we describe its use to characterize pure and complete GOE spectra. The effects of missing and spurious levels are presented in Section 4, and a comparison between U and Δ3 is presented in Section 5. We summarize and draw conclusions in Section 6.

Section snippets

The internal energy U

Dyson's use of a 2D Coulomb gas within the context of a circular orthogonal ensemble (COE) provides a starting point for the discussion of internal energy for random matrix ensembles. If we have N unit charges free to move along a circle of radius 1 and if the positions of those charges are specified by angular variables θ1,θ2θN, then the potential energy of the system is given byW=-j>iln|eiθi-eiθj|.The minimum value of W, denoted by W0, occurs when the charges are equally spaced around the

Application to pure and complete spectra

The first step in determining the usefulness of U is to apply it to pure and complete spectra. Ensembles of 2500 GOE and Poisson spectra ranging in size from N=10 to 1000 levels have been generated. Because GOE spectra have an energy-dependent level density and because U depends on energy differences, it is necessary to “unfold” the level density of the GOE spectra before further analysis in order to ensure that all levels are treated on an equal footing. To perform this unfolding, we actually

Effects of missing and spurious levels

As discussed earlier, one of the challenges for any spectral statistic is how it is affected by imperfect data. True levels may be missed, often because of limited resolution, or spurious levels may be included in a data set, either because of a mistake in identifying the level itself or more often because of difficulties in identifying the appropriate quantum numbers for the level. Discussions of the effects of missing levels on spectral statistics have been given recently by Bohigas and Pato

Comparison to Δ3

We have shown that the internal energy U does a good job of identifying a GOE spectrum as long as the spectrum exceeds a certain minimum size (50 levels), even if the data are imperfect to some degree. We now examine how U compares to the Dyson–Mehta Δ3 statistic that is commonly used in this context. For an energy interval [Emin,Emax], Δ3 is defined in terms of the “staircase” function N(E)—the number of levels with energy less than or equal to E—as follows:Δ3(L)=minA,B1Emax-EminEminEmax[N(E)

Conclusions

We have utilized Monte Carlo methods to examine the use of internal energy U as a statistic to characterize energy eigenvalue spectra. We have shown that U shows very different behavior for GOE and Poisson spectra and, therefore, provides an additional measure that can be used to evaluate this aspect of such spectra.

We have also simulated the effects of missing and spurious levels on U and compared its behavior to that of the commonly used statistic Δ3. The effect of either missing or spurious

Acknowledgments

This work was supported in part by the US Department of Energy under Grants DE-FG02-96ER40990 and DE-FG02-97ER41042, by the US National Science Foundation under Grant no. INT-0112421, and by the Brazilian agencies CNPq and FAPESP.

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