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Time dependent signals in experimental techniques such as Nuclear MagneticResonance (NMR) and Muon Spin Relaxation (muSR) are often the result of anensemble average over many microscopical dynamical processes. While there are anumber of functions used to fit these signals, they are often valid only inspecific regimes, and almost never properly describe the "spectral diffusion"regime, in which the dynamics happen on time scales comparable to thecharacteristic frequencies of the system. Full treatment of these problemswould require one to carry out a path integral over all possible realizationsof the dynamics of the time dependent Hamiltonian.  In this paper we present a numerical approach that can potentially be used tosolve such time evolution problems, and we benchmark it against a Monte Carlosimulations of the same problems. The approach can be used for any sort ofdynamics, but is especially powerful for any dynamics that can be approximatedas Markov processes, in which the dynamics at each step only depend on theprevious state of the system. The approach is used to average both classicaland quantum observables; in the latter case, a formalism making use ofLiouvillians and density matrices is used.

A Dyson equation approach for averaging of classical and quantum observables on multiple realizations of Markov processes