  Urban transformations within large and growing metropolitan areas often
generate critical dynamics affecting social interactions, transport
connectivity and income flow distribution. We develop a statistical-mechanical
model of urban transformations, exemplified for Greater Sydney, and derive a
thermodynamic description highlighting critical regimes. We consider urban
dynamics at two time scales: fast dynamics for the distribution of population
and income, modelled via the maximum entropy principle, and slower dynamics
evolving the urban structure under spatially distributed competition. We
identify phase transitions between dispersed and polycentric phases, induced by
varying the social disposition---a factor balancing the suburbs'
attractiveness---in contrast with the travel impedance. Using the Fisher
information we identify critical thresholds and quantify the thermodynamic cost
of urban transformation, as the minimal work required to vary the underlying
parameter. Finally, we introduce the notion of thermodynamic efficiency of
urban transformation, as the ratio of the order gained during a change to the
amount of required work, showing that this measure is maximised at criticality.
