Fluctuations in extended mass-action-law dynamics

Dynamics of chemical reactions, called mass-action-law dynamics, serves in this paper as a motivating example for investigating geometry of nonlinear non-equilibrium thermodynamics and for studying the ways to extend a mesoscopic dynamics to more microscopic levels. The geometry in which the physics involved is naturally expressed appears to be the contact geometry. Two extensions are discussed in detail. In one, the reaction fluxes or forces are adopted as independent state variables, the other takes into account fluctuations. All the time evolution equations arising in the paper are proven to be compatible among themselves and with equilibrium thermodynamics. A quantity closely related to the entropy production plays in the extended dynamics with fluxes and forces as well as in the corresponding fluctuating dynamics the same role that entropy plays in the original mass-action-law dynamics.