Molecular time scale generalized Langevin equation theory and polynomial maximum entropy imaging of spectral densities

Abstract

Molecular time scale generalized Langevin equation (MTGLE) theory is discussed as an approach to condensed phase dynamics. A polynomial maximum entropy (MaxEnt) process for imaging required MTGLE spectral densities based on knowledge of the moments of the spectral density is introduced. The process is based on the use of interpolation polynomials which serve both to image the spectral density as well as provide a numerical procedure to compute the inverse Hessian matrix in a Newton-type minimization. A default model is added to allow for the inclusion of additional information in forming the image. The polynomial MaxEnt imaging process is found to be a fast, numerically stable, computational procedure which produces images comparable in quality to images obtained by other imaging processes. The polynomial MaxEnt imaging process is examined in the context of imaging MTGLE bath spectral densities with special emphasis on a coupled linear chain model. Standard harmonic oscillator, Hamiltonian bath models such as Ohmic-exponential and Ohmic-Gaussian are shown to possess regions of parameter space for which the MTGLE adiabatic frequency is imaginary. When the adiabatic frequency is zero, it is shown that imaging of the friction kernel is the best approach.