Recent work employed an influence functional approach to analyze and quantify the validity of linear response theory for processes in complex environments. A cumulant expansion of the influence functional was obtained, in which the nth term is given by an n-dimensional time-integral involving the n-time correlation function of the environment. The lowest order (quadratic) term in this expansion maps the environment onto an effective harmonic medium and corresponds to the linear response approximation. It was shown that this effective harmonic bath model is exact for arbitrary values of the overall coupling strength if the system-bath coupling is diluted over an infinite number of degrees of freedom.
Further, the cumulant expansion of the influence functional, where all terms are related to multi-time correlation functions of the environment (which are subject to decoherence that truncates the extent of nonlocality), suggests a decomposition of the path integral that leads to a general iterative methodology similar in spirit to the i-QuAPI method for a harmonic bath. In the general case of an anharmonic environment, the influence functional is not known analytically but must be evaluated numerically for each given pair of forward and backward system path segments. The forward-backward semiclassical dynamics (FBSD) approximation to the influence functional provides an efficient tool for this purpose.