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Approximate Inertial Manifold Approach to Non-Equilibrium Thermodynamics

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.