  Here we investigate how local properties of particles in a thermal bath
influence the thermodynamics of the bath. We utilize nanothermodynamics, based
on two postulates: that small systems can be treated self-consistently by
coupling to an ensemble of similarly small systems, and that a large ensemble
of small systems forms its own thermodynamic bath. We adapt these ideas to
study how a large system may subdivide into an ensemble of smaller subsystems,
causing internal heterogeneity across multiple size scales. For the
semi-classical ideal gas, maximum entropy favors subdividing a large system of
atoms into regions of variable size. The mechanism of region formation could
come from quantum exchange that makes atoms in each region indistinguishable,
while decoherence between regions allows atoms in separate regions to be
distinguishable by location. Combining regions reduces the total entropy, as
expected when distinguishable particles become indistinguishable, and as
required by theorems for sub-additive entropy. Combining large volumes of small
regions gives the entropy of mixing for a semi-classical ideal gas, resolving
Gibbs paradox without invoking quantum symmetry for distant atoms. Other models
we study are based on Ising-like spins in 1-D. We find similarity in the
properties of a two-state model in the nanocanonical ensemble and a three-state
model in the canonical ensemble. Thus, emergent phenomena may alter the thermal
behavior of microscopic models, and the correct ensemble is necessary for
fully-accurate predictions. We add a nonlinear correction to Boltzmann's factor
in simulations of the Ising-like spins to imitate the dynamics of spin exchange
on intermediate lengths, yielding the statistics of indistinguishable states.
These simulations exhibit 1/f-like noise at low frequencies (f), and white
noise at higher f, similar to the thermal fluctuations found in many materials.
