  Stochastic and dynamical processes lie at the heart of all physical,
chemical, and biological systems. However, kinetic and thermodynamic properties
which characterize these processes have largely been treated separately as they
can be obtained independently for many systems at thermodynamic equilibrium. In
this work we demonstrate the existence of a class of relations between kinetic
and thermodynamic factors which holds even in the hydrodynamic limit, and which
must be satisfied for all systems that satisfy detailed balance and Boltzmann
distribution at equilibrium. We achieve this by proving that for systems with
inhomogeneous equilibrium states governed by dynamics such as the Cahn-Hilliard
(CH) dynamics, the chemical potential and self-diffusivity must mutually
constrain each other. We discuss common issues in the literature which result
in inconsistent formulations, construct the consistency requirement
mathematically, develop a class of self-diffusivities that guarantee
consistency, and discuss how the requirement originates from detailed balance
and Boltzmann distribution, and is therefore applicable to both conserved and
non-conserved dynamics.
