A Stochastic Differential Equation Approach to Spectral Fluctuations in systems with Mixed Dynamics

The distribution of energy level spacing in mixed ballistic cavities (where regular and chaotic dynamics may coexist), can be understood in terms of a Wigner-Dyson distribution with fluctuating variance $\sigma^2$. In a recent work [1-4], a detailed analysis of the spacing distribution in mixed cavities was made using the method of statistic superposition (or superstatistics). It was found that $\sigma^2$ is well described by a chi square distribution, which in turn is used to obtain the corresponding spacing distribution via Bayes theorem. Another interesting result is the estimation of the time scales in the problem via the time correlation function for the spacing S, yielding two time scales $\tau_1$ and $\tau_2$, and the kurtosis of the spacing distribution, yielding the time scale $T$. The estimated $T$ turned out to be one order of magnitude larger than $\tau _i$, which was used to justify the superstatistical method for deriving the spacing distribution. However, no model description was given for the time correlation function of S [3]. Furthermore, the time correlation function of $\sigma^2$, which could yield additional large time scales [4], was not studied. In this work, we describe the fluctuations of energy spacing levels in ballistic quantum cavities with mixed dynamics using a coupled system of stochastic differential equations (SDE). The main advantage of our method is the possibility to account for all time scales of the problem, thus describing in the same dynamical model both the spacing distribution and its time correlation function. The results of the superstatistical method are recovered in the limit where all time scales ratios go to infinity. Our SDE model is an extension of recent generalization of statistical ensembles to multiscale systems, in which an Ornstein-Uhlenbeck inverse gamma (OUIG) process with M time scales is coupled to a single variable Langevin equation with one time-scale [5]. In our model we couple Dyson?s random matrix process with N time-scales to an OUIG process with M time-scales. A detailed analysis of some particular cases ($N= 2$, $M = 1$) and ($N = 2$, $M= 2$) is provided and both the spacing distribution and the time correlation functions are calculated from the same model.