The irreversible thermalization of energy lost by an electron in amorphous condensed matter makes successive losses incoherent. This allows us to extend a stochastic treatment of electron degradation down to the thermal range. The probability of a given momentum and energy transfer is viewed as a macroscopic property of each material, which reduces to the structure factor $S(\mathbf{q},\ensuremath{\omega})$ in the Born approximation. The identification of such bulk properties presents novel challenges for experimentalists.