  We propose a housekeeping/excess decomposition of entropy production for
general nonlinear dynamics in a discrete space, including chemical reaction
networks and discrete stochastic systems. We exploit the geometric structure of
thermodynamic forces to define the decomposition; this does not rely on the
notion of a steady state, and even applies to systems that exhibit
multistability, limit cycles, and chaos. In the decomposition, distinct aspects
of the dynamics contribute separately to entropy production: the housekeeping
part stems from a cyclic mode that arises from external driving, generalizing
Schnakenberg's cyclic decomposition to non-steady states, while the excess part
stems from an instantaneous relaxation mode that arises from conservative
forces. Our decomposition refines previously known thermodynamic uncertainty
relations and speed limits. In particular, it not only improves an
optimal-transport-theoretic speed limit, but also extends the optimal transport
theory of discrete systems to nonlinear and nonconservative settings.
