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.