Step Evolution Toward Equilibrium: Fokker-Planck Equation and the Wigner Surmise

The generalized Wigner surmise $P_w(s)=as^\varrho\exp(-bs^2)$, drawn from random-matrix theory, has been shown to provide arguably the best (both conceptually and quantitatively) description of the equilibrium terrace-width distribution (TWD) of steps on a vicinal surface,\footnote{Hailu Gebremariam, S.D. Cohen, H.L. Richards, \& T.L. Einstein, Phys. Rev. B {\bf 69}, 125404 (2004) and references therein.} but with limited formal justification for non-special $\varrho$.\footnote{H.L. Richards \& T.L. Einstein, cond-mat/0008089.} Using a mean-field approach to Dyson's Brownian motion model,\footnote{F.J. Dyson, J. Math. Phys. {\bf 3}, 1191 (1962).} we show that $P_w(s)$ can be derived from a Fokker-Planck equation, analogously to the derivation of the Heston model of econophysics.\footnote{A.A. Dragulescu and V.M. Yakovenko, Quantitative Finance {\bf 2}, 443 (2002) [cond-mat/0203046].} From this formulation we can find how the system evolves from some arbitrary initial distribution toward $P_w(s)$. For a simple initial TWD such as uniformly-spaced straight steps, we can find the solution analytically. In parallel we carry out Monte Carlo studies within the terrace-step-kink model$^2$ with such initial distributions and confirm that the variance of the TWD evolves as predicted.