Localized states versus band states in a model for small polarons

A general formal theory of irreversible processes is applied to Holstein's one-dimensional model for small polarons in order to find the appropriate representation of states for the system as a function of temperature. We find conditions under which the exact equations for the occupation probabilities of a band state or a localized state become markoffian. The temperature region in which a representation is adequate is determined by the markofficity conditions and by the requirement that it is possible to prepare the system in one of the states of that representation. We find that the band picture is appropriate at low temperatures while the localized-states picture is adequate at high temperatures. Both representations become inadequate approximately at the same point.