  The rank-size regularity known as Zipf's law is one of scaling laws and
frequently observed within the natural living world and in social institutions.
Many scientists tried to derive the rank-size scaling relation by
entropy-maximizing methods, but the problem failed to be resolved thoroughly.
By introducing a pivotal constraint condition, I present here a set of new
derivations based on the self-similar hierarchy of cities. First, I derive a
pair of exponent laws by postulating local entropy maximizing. From the two
exponential laws follows a general hierarchical scaling law, which implies
general Zipf's law. Second, I derive a special hierarchical scaling law with
exponent equal to 1 by postulating global entropy maximizing, and this implies
the strong form of Zipf's law. The rank-size scaling law proved to be one of
the special cases of the hierarchical law, and the derivation suggests a
certain scaling range with the first or last data point as an outlier. The
entropy maximization of social systems differs from the notion of entropy
increase in thermodynamics. For urban systems, entropy maximizing suggests the
best equilibrium state of equity for parts/individuals and efficiency for the
whole.
