  Inspired by a continuously increasing interest in modeling and framing
complex systems in a thermody- namic rationale, in this paper we continue our
investigation in adapting well known techniques (originally stemmed in fields
of physics and mathematics far from the present) for solving for the free
energy of mean field spin models in a statistical mechanics scenario. Focusing
on the test cases of bipartite spin systems embedded with all the possible
interactions (self and reciprocal), we show that both the fully interacting
bipartite ferromagnet as well as the spin glass counterpart, at least at the
replica symmetric level, can be solved via the fundamental theorem of calculus,
trough an analogy with the Hamilton-Jacobi theory and lastly with a mapping to
a Fourier diffusion problem. All these technologies are shown symmetrically for
ferromagnets and spin-glasses in full details and contribute as powerful tools
in the investigation of complex systems.
