Abstract

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.

Highlights

  • Thermodynamics and statistical mechanics provide two theoretical approaches for interpreting the thermal behavior shown by nature [1,2,3,4,5,6,7]

  • We stress how nanoscale thermal properties must govern large systems that subdivide into a heterogeneous distribution of subsystems, and how nanothermodynamics impacts the statistical mechanics of specific models

  • We argue that this inability to explain heterogeneity can be traced to sources of energy and entropy that arise from finite-size effects, especially sources that occur on the scale of nanometers, which require nanothermodynamics to be treated in a self-consistent and complete manner

Read more

Summary

Introduction

Thermodynamics and statistical mechanics provide two theoretical approaches for interpreting the thermal behavior shown by nature [1,2,3,4,5,6,7]. We focus on the theory of small-system thermodynamics [12,13,14], which is based on two novel postulates: that small systems can be treated self-consistently by coupling them to small systems (without correlations between the small systems so that they form an ensemble), and that a large ensemble of small systems becomes its own effectively infinite heat reservoir. This “nanothermodynamics” provides a fundamental foundation for connecting thermal properties across multiple size scales: from microscopic particles, through mesoscopic subsystems, to macroscopic behavior. We briefly explain how the laws of thermodynamics can be extended to length scales of nanometers, and why applying the resulting nanothermodynamics to statistical models may improve their accuracy and relevance to real systems

Standard Thermodynamics
Standard Statistical Mechanics
An Introduction to Nanothermodynamics
Extending Statistical Mechanics to Treat Multiscale Heterogeneity
The Subdivided Ising Model
Finite Chains of Effectively Indistinguishable Ising-Like Spins
Entropy and Heat in an Ideal 1-D Polymer
Findings
Conclusions
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call