Spectral modes from the expansion in Lorentzian lines: the case of spectra of current autocorrelations

The general theory of exponential expansion of time correlation functions in many-body statistical mechanics [1] has a very simple but remarkable consequence for the second derivative of a time correlation function a(t). The frequency spectrum of a(t) is made of central Lorentzians and/or distorted inelastic Lorentzians, characterized by parameters zj (where j numbers the spectral modes) which are either real widths or complex quantities representing damping and frequencies of oscillatory modes, respectively. Then, d2a/dt2 has a spectrum composed by lines with the same set of zj's. Any spectral mode appears in both spectra with different amplitudes only, and a simple and explicit relation holds between corresponding amplitudes. This result (i) proves that the time autocorrelation of a dynamical variable and that of its derivative of any order contain the same information on the system considered, and (ii) has an immediate application to the case of current autocorrelations. The dynamic structure factor and the longitudinal-current correlation spectrum form the best-known pair of frequency distributions corresponding to time correlations related to each other by a double-derivative operation. Both are commonly used for the description of the collective dynamics of a system. We illustrate the above mentioned property with examples in the case of simple dynamical models and draw.