  Conceptual analogies among statistical mechanics and classical (or quantum)
mechanics often appeared in the literature. For classical two-body mean field
models, an analogy develops into a proper identification between the free
energy of Curie-Weiss type magnetic models and the Hamilton-Jacobi action for a
one dimensional mechanical system. Similarly, the partition function plays the
role of the wave function in quantum mechanics and satisfies the heat equation
that plays, in this context, the role of the Schrodinger equation in quantum
mechanics. We show that this identification can be remarkably extended to
include a wide family of magnetic models classified by normal forms of suitable
real algebraic dispersion curves. In all these cases, the model turns out to be
completely solvable as the free energy as well as the order parameter are
obtained as solutions of an integrable nonlinear PDE of Hamilton-Jacobi type.
We observe that the mechanical analog of these models can be viewed as the
relativistic analog of the Curie-Weiss model and this helps to clarify the
connection between generalised self-averaging and in statistical thermodynamics
and the semi-classical dynamics of viscous conservation laws.
