S S symmetry Article Multiscale Thermodynamics: Energy, Entropy, and Symmetry from Atoms to Bulk Behavior Ralph V. Chamberlin 1,*, Michael R. Clark 1, Vladimiro Mujica 2 and George H. Wolf 2 1 Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA; [email protected] 2 School of Molecular Science, Arizona State University, Tempe, AZ 85287-1604, USA; [email protected] (V.M.); [email protected] (G.H.W.) * Correspondence: [email protected] 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 Citation: Chamberlin, R.V.; Clark, particles become indistinguishable, and as required by a theorem in quantum mechanics for sub- M.R.; Mujica, V.; Wolf, G.H. additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a Multiscale Thermodynamics: Energy, semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for particles Entropy, and Symmetry from Atoms that may be meters apart. Other models presented here are based on Ising-like spins, which are to Bulk Behavior. Symmetry 2021, 13, solved analytically in one dimension. Focusing on the bonds between the spins, we find similarity 721. https://doi.org/10.3390/ in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state sym13040721 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 Academic Editors: Sergei D. Odintsov result using Ising-like spins involves simulations that include a nonlinear correction to Boltzmann’s and Enrique Maciá Barber factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin Received: 1 February 2021 exchange on intermediate lengths. These simulations exhibit 1/f -like noise at low frequencies (f ), Accepted: 15 April 2021 and white noise at higher f, similar to the equilibrium thermal fluctuations found in many materials. Published: 19 April 2021 Keywords: nanothermodynamics; fluctuations; maximum entropy; finite thermal baths; corrections Publisher’s Note: MDPI stays neutral to Boltzmann’s factor; ideal gas; Ising model; Gibbs’ paradox; statistics of indistinguishable particles with regard to jurisdictional claims in published maps and institutional affil- iations. 1. Introduction Thermodynamics and statistical mechanics provide two theoretical approaches for interpreting the thermal behavior shown by nature [1–7]. In statistical mechanics, the local Copyright: © 2021 by the authors. symmetry of particles in the system is well known to influence their behavior, yielding Licensee MDPI, Basel, Switzerland. Maxwell–Boltzmann, Bose–Einstein, or Fermi–Dirac statistics. Lesser known is the im- This article is an open access article pact of local symmetry on thermodynamics. Difficulty in understanding local behavior in distributed under the terms and standard thermodynamics is due to its two basic postulates: that all systems must be macro- conditions of the Creative Commons scopic, and homogeneous. Hence, accurate theories of local thermal properties require Attribution (CC BY) license (https:// that finite-size effects are added to thermodynamics, such as in fluctuation theorems [8,9] creativecommons.org/licenses/by/ and stochastic thermodynamics [10,11]. These approaches are remarkably successful at 4.0/). Symmetry 2021, 13, 721. https://doi.org/10.3390/sym13040721 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 721 2 of 26 describing thermal dynamics, including far-from-equilibrium processes. However, in gen- eral, these theories require that, at least at some time during the dynamics, the system must have ideal thermal contact to an ideal heat bath; specifically, the system must be weakly, rapidly, and homogeneously coupled to an effectively infinite heat reservoir. Here, we focus on the theory of small-system thermodynamics [12–14], which is based on two novel postulates: that small systems can be treated self-consistently by coupling them to similarly 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 infi- nite heat reservoir. Thus, this “nanothermodynamics” provides a fundamental foundation for connecting thermal properties across multiple size scales: from microscopic particles, through mesoscopic subsystems, to macroscopic behavior. Although small-system thermodynamics was originally developed to treat individual molecules and isolated clusters, we use nanothermodynamics as a guide to study how large systems are influenced by internal heterogeneity, especially on the scale of nanome- ters [15–17]. Here, we focus on how thermal equilibrium often involves subdividing a large system into an ensemble of subsystems, usually requiring that the number of particles and volume of each subsystem fluctuate freely, without external constraint. We call these freely fluctuating subsystems “regions,” and the self-consistent system of independently fluctuating nanometer-sized regions the “nanocanonical” ensemble. A common assumption in standard thermodynamics is additivity. For example, it is assumed that if the size of a system is doubled, then all of its extensive variables (e.g., in- ternal energy E, entropy S, and number of particles N) will precisely double, and all of its intensive variables (e.g., temperature T, pressure P, and chemical potential m) will remain unchanged. In fact, the Gibbs–Duhem relation (found from assuming that all extensive variables increase linearly with N) is often used as a test of consistency in thermodynamics. Thus, Gibbs’ paradox [18–21] comes from the apparent discrepancy between the predictions of thermodynamics and classical statistical mechanics. Specifically, standard thermody- namics states that when a partition is reversibly removed between two identical systems the total entropy should be exactly twice the entropy of each initial system, while Maxwell– Boltzmann statistics of a classical ideal gas predicts an additional term proportional to Nln(2) due to the entropy of mixing. Here, we obtain several results where various thermo- dynamic quantities (including E, S, and m) contain terms that depend nonlinearly on N, and we focus on how these finite-size effects are necessary to conserve total energy and maximize total entropy. The broad generality of thermodynamics can be a distraction to many scientists who prefer the concrete models and microscopic details that constitute statistical mechanics. For similar reasons, statistical mechanics is often said to be a foundation for thermody- namics. However, the fundamental physical laws are in the thermodynamics, and strict adherence to these laws is necessary before statistical mechanics can fully represent the real world. Here, 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. 2. Background 2.1. Standard Thermodynamics Thermodynamics establishes equations and inequalities between thermodynamic variables which must be obeyed by nature. Thermodynamic variables usually come in conjugate pairs, whose product yields a contribution to the internal energy of a system. The fundamental equation for reversible changes in thermodynamics (aka the Gibbs equa- tion, which combines the first and second laws) states that the total internal energy of the system can be changed by changing one (or more) of the thermodynamic variables. For example, the fundamental equation of N particles in volume V is Symmetry 2021, 13, 721 3 of 26 dE = TdS − PdV + mdN (1) Equation (1) gives three ways to reversibly change the internal energy of a system (dE): heat can be transferred (TdS), work can be done (PdV), or the number of particles can be changed (mdN). The primary conjugate variables in Equation (1) used to define thermal equilibrium [1] are T and S. Temperature is the familiar quantity that is “hotness measured on some definite scale” [4], but entropy is a more subtle concept [22–25]. Although entropy is often associ- ated with randomness, any system with sufficient information about the randomness may also have low entropy. Thus, a more general definition of entropy comes from quantifying the amount of missing information. Entropy is a main focus of our current