in eqs \[eq:g1qgeneral\] and \[eq:g1sandr\] may formally be phrased as averages over displacement distribution functions such as $P(\Delta x, t)$, which gives the time-dependent probability that a scattering particle will displace through $\Delta x$ during time $t$. Two previous papers[@phillies2005a; @phillies2012a] examined how $g^{(1s)}(q,t)$ and $g^{(1)}(q,t)$ are actually related to the displacement distribution functions. The two prior papers were primarily concerned with establishing formal relationships between dynamic structure factors and probabilities for scatterer displacement. The significance of these relationships for the interpretation of experimental measurements was at most a secondary consideration. This paper focuses on interpreting experimental measurements.

Section II of this paper presents the correct general relationship between $g^{(1s)}(q,t)$ and $P(\Delta x, t)$. Section III discusses the special case of probe particles in a purely Newtonian fluid. Section IV notes experimental findings bearing on the relative significance of Sections II and III. Section V considers paths for interpreting probe diffusion spectra. Section VI treats the determination of $P(\Delta x, t)$, relationships between $g^{(1s)}(q,t)$ and trapping/hopping behavior, and, closing on a positive note, cases in which quasielastic scattering from diffusing probes, correctly interpreted, has given valuable information about complex fluids and the objects diffusing in them.

General Case\[sectiongeneralcase\]
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This 