A Morse index invariant reduction of non-equilibrium thermodynamics

We consider the finite dimensional reduction of the well known non-equilibrium thermodynamics theory developed through the last two decades by a team of researchers lead by Giovanni Jona-Lasinio, realized by considering a simplified version – a reaction-di¤usion-like dynamics – of that theory. We begin with a clear axiomatic format of that framework, showing that the reaction-di¤usion dynamics emerge in a direct way after a few assumptions. Our goal is to put focus on the relations between the reduced and the full theory and to underline some topological features of this theory, more precisely, by first showing that the Morse index distribution of the equilibria of the finite dimensional reduced system is exactly the same of the full original system, thus giving us eventually a good measure of the robustness of the reduction, and secondly, moving our framework to a Morse–Smale setting, by proposing an alternative way to compute the Morse index of the equilibria. In order to realize this last program, we propose a weak infinite-dimensional Maslov–Ho¨rmander theorem.