  We construct the exact partition function of the Potts model on a complete
graph subject to external fields with linear and nematic type couplings. The
partition function is obtained as a solution to a linear diffusion equation and
the free energy, in the thermodynamic limit, follows from its semiclassical
limit. Analysis of singularities of the equations of state reveals the
occurrence of phase transitions of nematic type at not zero external fields and
allows for an interpretation of the phase transitions in terms of shock
dynamics in the space of thermodynamics variables. The approach is shown at
work in the case of a q-state model for q=3 but the method generalises to
arbitrary q.
