  The physical picture of interacting magnetic islands provides a useful
paradigm for certain plasma dynamics in a variety of physical environments,
such as the solar corona, the heliosheath, and the Earth's magnetosphere. In
this work, we derive an island kinetic equation to describe the evolution of
the island distribution function (in area and in flux of islands) subject to a
collisional integral designed to account for the role of magnetic reconnection
during island mergers. This equation is used to study the inverse transfer of
magnetic energy through the coalescence of magnetic islands in 2D. We solve our
island kinetic equation numerically for three different types of initial
distribution: delta-distribution, Gaussian and power-law distribution. The time
evolution of several key quantities is found to agree well with our analytical
predictions: magnetic energy decays as $\tilde t^{-1}$, the number of islands
decreases as $\tilde t^{-1}$, and the averaged area of islands grows as $\tilde
t$, where $\tilde t$ is the time normalized to the characteristic reconnection
time scale of islands. General properties of the distribution function and the
magnetic energy spectrum are also studied.
