We consider systems that start from and/or end in thermodynamic equilibrium
while experiencing a finite rate of change of their energy density or of other
intensive quantities $q$ at intermediate times. We demonstrate that at these
times, during which the global intensive quantities $q$ vary at a finite rate,
the size of 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 ({\it arbitrarily far separated}) sites $i$ and $j$ may, on
average, become finite. Such non-vanishing connected correlations between
asymptotically distant sites are general and may also appear in theories that
only have local interactions. In simple models, these correlations may be
traced to the generic macroscopic entanglement of finite temperature states.
Once the global mean $q$ no longer changes, the average of $|G_{ij}|$ over all
spatial separations $|i-j|$ 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 derive new
uncertainty relation based inequalities that connect the heat capacity to the
dynamics in general open thermal systems. We further briefly comment on
parallels between quantum measurements and unitary quantum evolution and
thermalization.