  We consider general systems that start from and/or end in thermodynamic
equilibrium while experiencing a finite rate of change of their energy density
or other intensive quantities $q$ at intermediate times. We demonstrate that at
these times, during which $q$ varies at a finite rate, the associated
covariance, the connected pair correlator $G_{ij} = \langle q_{i} q_{j} \rangle
- \langle q_{i} \rangle \langle q_{j} \rangle$, between any two (far separated)
sites $i$ and $j$ in a macroscopic system may, on average, become finite. Once
the global mean $q$ no longer changes, the average of $G_{ij}$ over all site
pairs $i$ and $j$ may tend to zero. However, when the equilibration times are
significant (e.g., as in a glass that is not in true thermodynamic equilibrium
yet in which the energy density (or temperature) reaches a final steady state
value), these long range correlations may persist also long after $q$ ceases to
change. We explore viable experimental implications of our findings and
speculate on their potential realization in glasses (where a prediction of a
theory based on the effect that we describe here suggests a universal collapse
of the viscosity that agrees with all published viscosity measurements over
sixteen decades) and non-Fermi liquids. We discuss effective equilibrium in
driven systems and derive uncertainty relation based inequalities that connect
the heat capacity to the dynamics in general open thermal systems. These
rigorous thermalization inequalities suggest the shortest possible fluctuation
times scales in open equilibrated systems at a temperature $T$ are typically
"Planckian" (i.e., ${\cal{O}}(\hbar/(k_{B} T))$). We briefly comment on
parallels between quantum measurements, unitary quantum evolution, and
thermalization and on how Gaussian distributions may generically emerge.
