  A general framework to describe a vast majority of biology-inspired systems
is to model them as stochastic processes in which multiple couplings are in
play at the same time. Molecular motors, chemical reaction networks, catalytic
enzymes, and particles exchanging heat with different baths, constitute some
interesting examples of such a modelization. Moreover, they usually operate out
of equilibrium, being characterized by a net production of entropy, which
entails a constrained efficiency. Hitherto, in order to investigate multiple
processes simultaneously driving a system, all theoretical approaches deal with
them independently, at a coarse-grained level, or employing a separation of
time-scales. Here, we explicitly take in consideration the interplay among
time-scales of different processes, and whether or not their own evolution
eventually relaxes toward an equilibrium state in a given sub-space. We propose
a general framework for multiple coupling, from which the well-known formulas
for the entropy production can be derived, depending on the available
information about each single process. Furthermore, when one of the processes
does not equilibrate in its sub-space, even if much faster than all the others,
it introduces a finite correction to the entropy production. We employ our
framework in various simple and pedagogical examples, for which such a
corrective term can be related to a typical scaling of physical quantities in
play.
