  We discuss non-equilibrium thermodynamics of the mean field Ising model from
a geometric perspective, focusing on the thermodynamic limit. When the number
of spins is finite, the Gibbs equilibria form a smooth Legendrian submanifold
in the thermodynamic phase space whose points describe the stable macroscopic
states of the system. We describe the convergence of these smooth Legendrian
submanifolds, as the number of spins goes to infinity, to a singular Legendrian
submanifold, admitting an analytic continuation that contains both the stable
and metastable states. We also discuss the relaxation to a Gibbs equilibrium
when the physical parameters are changed abruptly. The relaxation is defined
via the gradient flow of the free energy with respect to the Wasserstein metric
on microscopic states, that is, in the geometric language, via the gradient
flow of the generating function of the equilibrium Legendrian with respect to
the ghost variables. This leads to a discrete Fokker-Planck equation when the
number of spins is finite. We show that in the thermodynamic limit this
description is closely related to the seminal model of relaxation proposed by
Glauber. Finally, we find a special range of parameters where such relaxation
happens instantaneously, along the Reeb chords connecting the initial and the
terminal Legendrian submanifolds.
