  The classical Langevin dynamics of a particle in a periodic potential energy
landscape are studied via the intermediate scattering function (ISF). By
construction, the particle performs coupled vibrational and activated jump
motion with a wide separation of the vibrational period and the mean residence
time between jumps. The long time limit of the ISF is a decaying tail
proportional to the function that describes ideal jump motion in the absence of
vibrations. The amplitude of the tail is unity in idealized jump dynamics
models, but is reduced from unity by the intra-well motion. Analytical
estimates of the amplitude of the jump motion signature are provided by
assuming a \textcolor{blue}{factorization of the conditional probability
density of the particle position at long times, motivated by the separation of
time scales associated with inter-cell and intra-cell motion}. The assumption
leads to a factorization of the ISF at long correlation times, where one factor
is an ideal jump motion signature, and the other component is the amplitude of
the signature. The amplitude takes the form of a single-particle anharmonic
Debye-Waller factor. The factorization approximation is exact at the
diffraction conditions associated with the periodic potential. Numerical
simulations of the Langevin equation in one and two spatial dimensions confirm
that for a strongly corrugated potential the analytical approximation provides
a good qualitative description of the trend in the jump signature amplitude,
between the points where the factorization is exact.
