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We study theoretically the equilibrium fluctuations of steps on vicinal surfaces. From an analytical continuum description of the step edge, we find a single Langevin equation governing the motion of an isolated step around its equilibrium position that includes attachment/detachment of atoms, diffusion over the terrace, diffusion along the edge, and evaporation. We then extend this approach to treat an array of steps, i.e., a vicinal surface. We also present, in an appendix, an alternative formalism in which detachment to terrace and to step-edge diffusion can take place independently. In established as well as some new limits, and for numerous special cases, we study the wave-vector dependence---both exponent and prefactor---of the relaxation time of fluctuations. From this we recover scaling relations for early-time dependence of the mean-square fluctuations. We discuss how to extract the mesoscopic transport coefficients associated with different atomistic mechanisms of surface mass transport and how to distinguish between mechanisms having the same power-law dependence on wavelength in the capillary-wave analysis. To examine the crossovers between limiting regimes, we compute and explore an effective exponent for this power law and show that the crossover occurs over a narrow region of phase space. Furthermore, we find that single-sided approximations are valid only in the limit of extreme Schwoebel barriers.
Khare et al. (Sun,) studied this question.