  Nanothermodynamics extends standard thermodynamics to facilitate finite-size
effects on the scale of nanometers. A key ingredient is Hill's subdivision
potential that accommodates the non-extensive energy of independent small
systems, similar to how Gibbs' chemical potential accommodates distinct
particles. Nanothermodynamics is essential for characterizing the thermal
equilibrium distribution of independently relaxing regions inside bulk samples,
as is found for the primary response of most materials using various
experimental techniques. The subdivision potential ensures strict adherence to
the laws of thermodynamics: total energy is conserved by including an
instantaneous contribution from the entropy of local configurations, and total
entropy remains maximized by coupling to a thermal bath. A unique feature of
nanothermodynamics is the completely-open nanocanonical ensemble. Another
feature is that particles within each region become statistically
indistinguishable, which avoids non-extensive entropy, and mimics
quantum-mechanical behavior. Applied to mean-field theory, nanothermodynamics
gives a heterogeneous distribution of regions that yields stretched-exponential
relaxation and super-Arrhenius activation. Applied to Monte Carlo simulations,
there is a nonlinear correction to Boltzmann's factor that improves agreement
between the Ising model and measured non-classical critical scaling in magnetic
materials. Nanothermodynamics also provides a fundamental mechanism for the 1/f
noise found in many materials.
