Path integral and transfer matrix approach for a 1D nonequilibrium system with disorder

A generating function for a nonequilibrium system is derived. The nonequilibrium system consists of a disordered 1D-direct-gap semiconductor excited by a coherent ultrashort laser pulse. The formulation with a generating function allows to perform exactly the ensemble average over the random potential. The ensemble average changes the time evolution of the 1D nonequilibrium system to nonlocal behaviour. However, a generator for infinitesimal space translations of the ensemble averaged system can still be obtained from the complete generating function so that a differential transfer matrix equation in the space coordinate x follows which already contains the ensemble average. The derivation of this transfer matrix equation is considered in detail.