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

physical review. e, statistical, nonlinear and soft matter physics

Modified Bose-Einstein and Fermi-Dirac statistics if excitations are  localized on an intermediate length scale: Application to non-Debye specific  heat