Multiscale thermodynamics: Energy, entropy, and symmetry from atoms to bulk behavior

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8 Scopus citations


Here, we investigate how the local properties of particles in a thermal bath may influence the thermodynamics of the bath, and consequently alter the statistical mechanics of subsystems that comprise the bath. We are guided by the theory of small‐system thermodynamics, which is based on two primary postulates: that small systems can be treated self‐consistently by coupling them to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt this “nanothermodynamics” to investigate 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 an ensemble of “regions” of variable size. The mechanism of region formation could come from quantum exchange symmetry that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by their distinct locations. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by a theorem in quantum mechanics for sub‐additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for particles that may be meters apart. Other models presented here are based on Ising‐like spins, which are solved analytically in one dimension. Focusing on the bonds between the spins, we find similarity in the equilibrium 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. Another result using Ising‐like spins involves simulations that include a nonlinear correction to Boltzmann’s factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin exchange on intermediate lengths. These simulations exhibit 1/f‐like noise at low frequencies (f), and white noise at higher f, similar to the equilibrium thermal fluctuations found in many materials.

Original languageEnglish (US)
Article number721
Issue number4
StatePublished - Apr 2021


  • Corrections to Boltzmann’s factor
  • Finite thermal baths
  • Fluctuations
  • Gibbs’ paradox
  • Ideal gas
  • Ising model
  • Maximum entropy
  • Nanothermodynamics
  • Statistics of indistinguishable particles

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)


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