  The global steady state of a system in thermal equilibrium exponentially
favors configurations with lesser energy. This principle is a powerful
explanation of self-organization because energy is a local property of a
configuration. For nonequilibrium systems, there is no such property for which
an analogous principle holds, hence no common explanation of the diverse forms
of self-organization they exhibit. However, a flurry of recent work
demonstrates that a local property of configurations called "rattling" predicts
the global steady states of a broad class of nonequilibrium systems. We
interpret this emerging physical theory in terms of Markov processes to
generalize and prove its main claims. Surprisingly, we find that the idea at
the core of rattling theory is so general as to apply to equilibrium and
nonequilibrium systems alike. Its predictions hold to an extent determined by
the relative variance of, and correlation between, the local and global "parts"
of an arbitrary steady state. We show how these key quantities characterize the
local-global relationships of random walks on random graphs, various spin-glass
dynamics, and models of animal collective behavior.
