The statistical distribution of levels of an integrable system is claimed to be a Poisson distribution. In this paper, we numerically generate an ensemble of N dimensional random diagonal matrices as a model for regular systems. We evaluate the corresponding nearest-neighbor spacing (NNS) distribution, which characterizes the short range correlation between levels. To characterize the long term correlations, we evaluate the level number variance. We show that, by increasing the size of matrices, the level spacing distribution evolves from the Gaussian shape that characterizes ensembles of 2\times2 matrices tending to the Poissonian as N \rightarrow \infty. The transition occurs at N \approx 20. The number variance also shows a gradual transition towards the straight line behavior predicted by the Poisson statistics.
Random matrix theory [1,2] provides a framework for describing the statistical properties of spectra for quantum systems, whose classical counterpart is chaotic. It models the Hamiltonian of the system by an ensemble of N-dimensional random matrices, subject to some general symmetry constraints. For example, time-reversal-invariant quantum systems are represented by a Gaussian orthogonal ensemble (GOE) of random matrices when the system has rotational. A complete discussion of the level correlations for a GOE is a difficult task. Most of the interesting results are obtained for the limit of N→∞. Analytical results have long ago been obtained for the case of N=2 [3].It yields simple analytical expressions for the nearest-neighbor-spacing (NNS) P(s), renormalized to make the mean spacing equal one. The spacing distribution for a two-dimensional GOE given by -² 4 ( ) , 2 s p s se
is known as Wigner’s surmise.The two-dimensional GOE obviously ignores the long range correlations within the spectra of chaotic systems. In spite of this limitation, the Wigner surmiseprovides a surprisingly accurate representation for NNS distributions of large matrices.
Berry and Tabor [4] conjectured that the fluctuations of quantum systems whose classical counterpart is completely integrable are the same as
those of an uncorrelated sequence of levels. An infinitely large independentlevel sequence can be regarded as a Poisson random process. The NNS distribution is given by ( ) exp(-). p s s (1.2) An integrable system in quantum mechanics has, in principle, a known complete set of eigenvectors. The Hamiltonian matrix will naturally be diagonal in the basis that consists with this set. It is thus reasonable to model the integrable systems by an ensemble of diagonal random matrices. Interestingly, the NNS distribution derived from a 2×2 random matrix model is Gaussian and not Poissonian. This suggests that the limit of large N is reached in integrable systemsmuch later that in the chaotic systems. The purpose of this paper is to estimate the minimal size of the random matrix ensemble that may be used to model large quantum systems.
In the present paper, weconsider both the short and long term correlations between levels characterized by the nearest-neighbor spacing (NNS) distribution P(s) and the variance Σ 2 , respectively, to discuss the eigenvalues statistics of N dimensional diagonal random matrices. First, we show that the formula of NNS distribution of ensembles of the 2×2 random matrix ensemble has a Gaussian shape that characterize. An analogous derivation was given before by Chau [5] and Berry [6]. Thenwe numerically discuss the statisticspreviously mentioned above and compare the result with Poisson distribution and the distribution of Gaussian shape by 2×2 matrices.
By consider a (2×2) real symmetric matrix ( ,
exp . 42
In integrable systems, where the dynamical motion is integrable, the state functions are known in principle. They can be used as a basis for the matrix elements of the Hamiltonian. In this case, the Hamiltonian can be represented as a diagonal matrix. We shall therefore consider. the case when the diagonal elements have equal variances 2 .In this case, Eq.
(2.2) reads
Introducing the new variables
and imposing the condition of a unit mean spacing, the NNS distribution becomes (2.4) This distribution is not exactly the Poisson distribution P(s) =exp (-s).The 2×2 random-matrix model cannot describe the NNS distributions for large regular systems, which are known to be described by the Poisson distribution.
We numerically generate an ensemble of N-dimensional diagonal random matrices whose entries are pseudorandom values drawn from the standard normal distribution. In other words, we shall use a random number generator to generate values of the matrix elements so that they have a Gaussian probability density function.
For a physical system, the level density depends on the properties of the system under consideration. It varies from one system to another. One of the achievements of quantum chaology is that the fluctuation properties of energy spectra are universal when the spectra are “unfolded”. The same is assumed for systems having regular classical dynamics. Unfolding consists in separating the secular variation from the oscillation terms. For this reason, unfolding is used to generate a spectrum whose mean level density is 1. The sequence of eigenvalues of each matrix of the ensemble generated according to the above procedure, {E
1 , E 2 … E N }, after ordering does not have uniform average level density. To analyze the fluctuation properties, this spectrum has to be unfolded, i.e. specific mean level density must be removed from the data [7].
Every sequence is taken from the eigenvalues when unfolded, is transformed into a new sequence with unit mean level spacing. This is done by fitting a theoretical expression to the number N (E) of levels by use cubic polynomial or a Gauss
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