The equilibrium theory for a train of steps is revisited. The analysis
is based on a nonlocal Langevin equation derived from the Burton-Carb
rera-Frank model. The Ehrlich-Schwoebel effect, diffusion along the st
ep edge, as well as elastic interactions between steps have been incor
porated. We discuss several static correlation functions and give an i
mproved estimate for the terrace width distribution. By exploiting the
dispersion relation, the time dependence of the step fluctuations has
been calculated. In the limit of well separated length scales there a
re several time intervals where the temporal step correlation function
follows a power law with one of the exponents 1/2, 1/3, or 1/4. In th
e opposite situation, neither power laws nor simple scaling behaviors
are obtained. We provide precise conditions on which regime must be ex
pected in a given real situation. Moreover, it is shown that different
physical mechanisms can give rise to the same exponent. This study is
thus crucial for the discrimination between various physical regimes
in a real experiment. The range of validity of the approximation and t
he crossover times are discussed for steps on Si(111).