  We study conserved one-dimensional models of particle diffusion, attachment
and detachment from clusters, where the detachment rates decrease with
increasing cluster size as gamma(m) ~ m^{-k}, k>0. Heuristic scaling arguments
based on random walk properties show that the typical cluster size scales as
(t/ln(t))^z, with z=1/(k+2). The initial symmetric flux of particles between
neighboring clusters is followed by an effectively assymmetric flux due to the
unbalanced detachement rates, which leads to the above logarithmic correction.
Small clusters have densities of order t^{-mz(1)}, with z(1) = k/(k+2). Thus,
for k<1, the small clusters (mass of order unity) are statistically dominant
and the average cluster size does not scale as the size of typically large
clusters does. We also solve the Master equation of the model under an
independent interval approximation, which yields cluster distributions and
exponent relations and gives the correct dominant coarsening exponent after
accounting for the effects of correlations. The coarsening of large clusters is
described by the distribution P_t(m) ~ 1/t^y f(m/t^z), with y=2z. All results
are confirmed by simulation, which also illustrates the unusual features of
cluster size distributions, with a power law decay for small masses and a
negatively skewed peak in the scaling region. The detachment rates considered
here can apply in the presence of strong attractive interactions, and recent
applications suggest that even more rapid rate decays are also physically
realistic.
