  In this paper a reaction-diffusion type equation is the starting point for
setting up a genuine thermodynamic reduction, i.e. involving a finite number of
parameters or collective variables, of the initial system. This program is
carried over by firstly operating a finite Lyapunov-Schmidt reduction of the
cited reaction-diffusion equation when reformulated as a variational problem.
In this way we gain an approximate finite-dimensional o.d.e. description of the
initial system which preserves the gradient structure of the original one and
that is similar to the approximate inertial manifold description of a p.d.e.
introduced by Temam and coworkers. Secondly, we resort to the stochastic
version of the o.d.e., taking into account in this way the uncertainty (loss of
information) introduced with the above mentioned reduction. We study this
reduced stochastic system using classical tools from large deviations,
viscosity solutions and weak KAM Hamilton-Jacobi theory. In the last part we
highlight some essential similarities existing between our approach and the
comprehensive treatment non equilibrium thermodynamics given by Jona-Lasinio
and coworkers. The starting point of their axiomatic theory --motivated by
large deviations description of lattice gas models-- of systems in a stationary
non equilibrium state is precisely a conservation/balance law which is akin to
our simple model of reaction-diffusion equation.
