  Dynamic heterogeneity has often been modeled by assuming that a
single-particle observable, fluctuating at a molecular scale, is influenced by
its coupling to environmental variables fluctuating on a second, perhaps
slower, time scale. Starting from the most simple Gaussian Markov process we
model the exchange between 'slow' and 'fast' environments by treating the
fluctuating single-particle variable as a projection from a higher-dimensional
Markov process. The moments of the resulting stochastic process are calculated
from the corresponding Master equations or Langevin equations, depending on the
model. The calculations show the importance of the way to treat exchange
processes. The resulting stochastic process is non-Markovian for all models.
However, the deviations from a Gaussian behavior depend on the details of the
models. A comparison of our results with other model treatments and experiments
should provide further insight into the concept of dynamic heterogeneity.
