The Characteristic Function As a Unifying Framework for Linear Response in Surface Diffusion

In this short perspective, we analyze the different linear response functions relevant to surface diffusion studied by helium atom scattering, organizing them around a single object: the intermediate scattering function (ISF), which is also a characteristic function (CF) in the sense of probability theory. This organizing role of the CF is, to our knowledge, not made explicit elsewhere in the surface-diffusion literature, even though the time exponential function it predicts in the diffusive regime is a special case of the classical continuous-time-random-walk (CTRW) theory. Special emphasis is placed on this diffusive regime, established at times much greater than the inverse of the friction coefficient, where quantum features of the diffusion process are washed out. We show how the entire hierarchy of response functions—the after-effect function, the generalized susceptibility, the relaxation function, and the Green function—can be written directly in terms of the time moments of the ISF at $t=0$, and how the Pauli master equation and the Chudley-Elliott (CE) jump model follow as particular lattice realizations of a general compound-Poisson process. The extension to finite surface coverage is discussed within the interacting single adsorbate (ISA) model.