Space Covering by Growing Rays
P. L. Krapivsky and E. Ben-Naim
We study kinetic and jamming properties of a space covering process in
one dimension. The stochastic process is defined as follows: Seeds are
nucleated randomly in space and produce rays which grow with a
constant velocity. The growth stops upon collision with another
ray. For arbitrary distributions of the growth velocity, the exact
coverage, velocity and size distributions are evaluated for both
simultaneous and continuous nucleation. In general, simultaneous
nucleation exhibits a stronger dependence on the details of the growth
velocity distribution in the asymptotic time regime. The coverage in
the continuous case exhibits a universal 1/t approach to the
jammed state, while an inhomogeneous version of the process leads
to nonuniversal algebraic decays.
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