  We study the collective phenomena and constraints associated with the
aggregation of individual cooling units from a statistical mechanics
perspective. These units are modelled as Thermostatically Controlled Loads
(TCLs) and represent zones in a large commercial or residential building. Their
energy input is centralized and controlled by a collective unit -- the Air
Handling Unit (AHU) -- delivering cool air to all TCLs, thereby coupling them
together. Aiming to identify representative qualitative features of the
AHU-to-TCL coupling, we build a realistic but also sufficiently simple model
and analyze it in two distinct regimes: the Constant Supply Temperature (CST)
and the Constant Power Input (CPI) regimes. In both cases, we center our
analysis on the relaxation dynamics of individual TCL temperatures to a
statistically steady state. We observe that while the dynamics are relatively
fast in the CST regime, resulting in all TCLs evolving around the control
setpoint, the CPI regime reveals emergence of a \emph{bi-modal probability
distribution and two, possibly strongly separated, time scales}. We observe
that the two modes in the CPI regime are associated with all TCLs being in the
same low and high-temperature states, respectively, with occasional (and
therefore possibly rare) collective transition between the modes akin in the
Kramer's phenomenon of statistical physics. To the best of our knowledge, this
phenomenon was overlooked in the context of the multi-zone energy building
engineering, even thought it has direct implications on the operations of
centralized cooling systems in buildings. It teaches us that a balance needs to
be struck between occupational comfort -- related to variations in the
individual temperatures -- and power output predictability -- the main focus of
the DR schemes.
