  Disordered systems show deviations from the standard Debye theory of specific
heat at low temperatures. These deviations are often attributed to two-level
systems of uncertain origin. We find that a source of excess specific heat
comes from correlations between quanta of energy if phonon-like excitations are
localized on an intermediate length scale. We use simulations of a simplified
Creutz model for a system of Ising-like spins coupled to a thermal bath of
Einstein-like oscillators. One feature of this model is that energy is
quantized in both the system and its bath, ensuring conservation of energy at
every step. Another feature is that the exact entropies of both the system and
its bath are known at every step, so that their temperatures can be determined
independently. We find that there is a mismatch in canonical temperature
between the system and its bath. In addition to the usual finite-size effects
in the Bose-Einstein and Fermi-Dirac distributions, if excitations in the heat
bath are localized on an intermediate length scale, this mismatch is
independent of system size up to at least 10^6 particles. We use a model for
correlations between quanta of energy to adjust the statistical distributions
and yield a thermodynamically consistent temperature. The model includes a
chemical potential for units of energy, as is often used for other types of
particles that are quantized and conserved. Experimental evidence for this
model comes from its ability to characterize the excess specific heat of
imperfect crystals at low temperatures.
