Maximal entropy spectral fluctuations and the sampling of phase space

Abstract
An analytical functional form for the distribution of intensities in an absorption spectrum is derived. Deviations from the purely statistical ‘‘Porter–Thomas’’ distribution are shown to be directly related to finite time information on the dynamics in phase space. The predicted distribution is wider than the purely statistical one with a higher proportion of very low intensity transitions. The derivation is based on a maximum entropy form of the spectrum. The constraints used are the values of the survival amplitude at finite number of times. The amplitude is obtainable as the Fourier transform of an observed spectrum or as the result of a dynamical computation. The optimal choice of the time points which characterize the spectrum, is discussed and a numerical algorithm is provided. Extensive spectral fluctuations occur when more than one time scale is needed to characterize the dynamics. This separation of time scales is also manifested as a clump structure in the spectrum of maximal entropy. The formalism also provides the distribution of line spacings and the ‘‘correlation hole’’ in the time autocorrelation function is discussed as an illustration.