Application of the Wigner Distribution to
Nonequilibrium Problems at Surfaces:
Relaxation, Growth, and Scaling of Capture Zones
CSIC Building (#406),
Seminar Room 4122.
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Application of the Wigner Distribution to
Nonequilibrium Problems at Surfaces: Relaxation,
Growth, and Scaling of Capture Zones
Professor
Theodore L. Einstein
University of Maryland

Abstract:
The generalized
Wigner distribution (GWD), arising from randommatrix theory, has proved
amazingly successful at describing the distribution of spacings in a long list
of fluctuating systems, including nuclear levels, fermions in 1D, universal conductance
fluctuations in disordered wires, intervals between parked cars, and terrace
widths on vicinal surfaces. The GWD has
the form P_{β}= a_{β} s^{β} exp(–b_{β} s^{2}) [with a_{β} and b_{β} being constants assuring normalization
and unit mean, and s being the fluctuating quantity divided by its
mean. After a review of these
equilibrium situations, we describe three new applications to nonequilibrium
problems in surface/interface physics.
1) We develop a FokkerPlank formalism
to describe how steps on a vicinal surface relax to equilibrium (described by
the GWD) from an arbitrary initial configuration.
2) We sketch how growth affects the terrace
width distribution of a vicinal surface, leading to a narrower distribution, i.e.
a GWD with higher β.
3) We discuss in detail applications to the
problem of island growth, longstudied numerically but elusive
analytically. Using published kinematic
Monte Carlo data, we find the GWD describes well the
distribution of the areas of Voronoi polygons (proximity cells) around
nucleation centers, i.e. the capture zones (CZ). In this case, β = i + d/2, where i is the size
of the critical nucleus and d the spatial dimensionality. We demonstrate
excellent fits of numerical data for both d = 1 and d = 2. To
clarify the underlying physics, we present a phenomenological derivation by
constructing a Langevin equation similar to that used in deriving the
FokkerPlank equation for relaxation of TWDs. We discuss implications for
processing and analysis of experimental data.Work at UMd supported by the MRSEC, NSF Grant DMR 0520471,
and partially by DOE CMSN grant DEFG0205ER46227. AP's visits to UMd by supported
by a CNRS Travel Grant.
[slides] 
