  We propose a method to derive the stationary size distributions of a system,
and the degree distributions of networks, using maximisation of the
Gibbs-Shannon entropy. We apply this to a preferential attachment-type
algorithm for systems of constant size, which contains exit of balls and urns
(or nodes and edges for the network case). Knowing mean size (degree) and
turnover rate, the power law exponent and exponential cutoff can be derived.
Our results are confirmed by simulations and by computation of exact
probabilities. We also apply this entropy method to reproduce existing results
like the Maxwell-Boltzmann distribution for the velocity of gas particles, the
Barabasi-Albert model and multiplicative noise systems.
