  Using the mass balance equations for chemical reactions, we show how the
system relaxes towards a steady state in and out of the Onsager region. In the
chemical affinities space, after fast transients, the relaxation process is a
straight line when operating in the Onsager region, while out of this regime,
the evolution of the system is such that the projections of the evolution
equations for the forces and the shortest path on the flows coincide. For
spatially-extended systems, similar results are valid for the evolution of the
thermodynamic mode (i.e., the mode with wave-number k = 0). These results allow
us to obtain the expression for the affine connection of the space covered by
the thermodynamic forces, close to the steady states. Through the affine
connection, the nonlinear closure equations are derived.
