  The second law of thermodynamics can be formulated as a restriction on the
evolution of the entropy of any system undergoing Markovian dynamics. Here I
show that this form of the second law is strengthened for multi-dimensional,
complex systems, coupled to multiple thermodynamic reservoirs, if we have a set
of \textit{a priori} constraints restricting how the dynamics of each
coordinate can depend on the other coordinates. As an example, this
strengthened second law (SSL) applies to complex systems composed of multiple
physically separated, co-evolving subsystems, each identified as a coordinate
of the overall system. In this example, the constraints concern how the
dynamics of some subsystems are allowed to depend on the states of the other
subsystems. Importantly the SSL applies to such complex systems even if some of
its subsystems can change state simultaneously, which is prohibited in a
multipartite process. The SSL also strengthens previously derived bounds on how
much work can be extracted from a system using feedback control, if the system
is multi-dimensional. Importantly, the SSL does not require local detailed
balance. So it potentially applies to complex systems ranging from interacting
economic agents to co-evolving biological species.
