A Continued-Fraction Representation of the Time-Correlation Functions

TLDR

The paper discusses applying a continued‑fraction representation to anomalous relaxation and transport near second‑order phase transitions. They derive a continued‑fraction expansion of the Laplace transform of time‑correlation functions that expresses generalized susceptibilities and transport coefficients in terms of static correlations, provides convenient long‑ and short‑time approximations, and yields an expansion for dynamical evolution, linking the formalism to irreversible processes and offering physical insight into dissipation.

Abstract

A continued-fraction expansion of the Laplace transform of the time-correlation functions is obtained, which enables us to express the generalized susceptibilities and the transport coefficients in terms of the static correlation functions of a set of quantities. This expansion has a different feature from the moment and cumulant expansions, and has a convenient form to introduce the long-time approximation as well as the short-time approximation. Its application to the anomalous relaxation and transport phenomena near the second-order phase transition points is discussed. An expansion formula is also obtained for the time evolution of dynamical quantities in order to describe the various modes of motion involved according to their characteristic time constants. These two expansions are closely related to the time-correlation function formalism of irreversible processes, and allow us to have physical intutition in calculating dissipative properties.