# Multiscale Thermodynamics: Energy, Entropy, and Symmetry from Atoms to Bulk Behavior

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Standard Thermodynamics

^{3}) possible ensembles [26]. In practice, however, only seven of these ensembles are well defined in standard thermodynamics. Examples include the fully-closed microcanonical ensemble, as well as the partially-open ensembles: canonical, Gibbs’, and grand-canonical. The fully-open generalized ensemble, involving three intensive environmental variables (e.g., µ, P, T), is ill defined in standard thermodynamics because at least one extensive environmental variable is needed to control the size of the system. Thus, nanothermodynamics is the only way to treat fully-open systems in a consistent manner, allowing the system to find its equilibrium distribution of subsystems without external constraints. In fact, the nanocanonical ensemble is necessary for the true thermal equilibrium of any system having independent internal regions, especially from localized internal fluctuations. Because large and homogeneous systems all yield equivalent behavior for all ensembles, it is sometimes said that the choice of ensemble can be made merely for convenience [14]. However, for small systems, and for bulk samples that subdivide into an ensemble of small regions, the choice of ensemble is crucial, so that the correct ensemble must be used for realistic behavior. Indeed, the correct ensemble is essential for fully-accurate descriptions of fluctuations, dynamics, and the distribution of independent regions inside most samples [15,16,17].

#### 2.2. Standard Statistical Mechanics

^{th}macrostate, or in terms of the average probability of finding the i

^{th}macrostate (${\rho}_{i}$). The Gibbs (or Boltzmann-Gibbs) expression for average entropy is given by $\overline{S}/k=-{\displaystyle \sum}_{i}{\rho}_{i}\mathrm{ln}{\rho}_{i}$, where k is Boltzmann’s constant and the sum is over all possible macrostates. Alternatively, Boltzmann’s expression for the entropy of each macrostate is S

_{i}/k = ln(Ω

_{i}). In the microcanonical ensemble, where all microstates are assumed to be equally likely, both expressions yield identical values for equilibrium average behavior $\overline{S}={\displaystyle \sum}_{i}{S}_{i}$. (Generalized entropies have been introduced to investigate the possibility that all microstates are not equally likely [51], but here we focus on specific non-additive and non-extensive contributions to entropy that arise naturally from finite-size effects in thermodynamics.) Gibbs’ expression has the advantage that it also applies to other macroscopic ensembles, while Boltzmann’s expression has the advantage that it can accommodate non-equilibrium conditions [52], including small systems that may fluctuate. Here, we stress how nanoscale thermal properties must also govern large systems that subdivide into a heterogeneous distribution of subsystems, and how nanothermodynamics impacts the statistical mechanics of specific models.

## 3. An Introduction to Nanothermodynamics

^{2/3}, and recent results have greatly extended these ideas to include shape-dependent terms [53,54,55]. Here, we focus on contributions to $\mathcal{E}$ that are not explicitly contained in the microscopic interactions, instead emerging from nanoscale behavior. One example is the semi-classical ideal gas, where $\mathcal{E}$ comes entirely from entropy due to indistinguishable statistics of particles within regions, with particles in separate regions distinguishable by their locations. Other examples utilize 1-D systems, where the increase in entropy favoring a distribution of localized fluctuations can dominate the decrease in energy favoring uniform interactions, so that thermal equilibrium often involves dynamic heterogeneity. Because $\mathcal{E}$ uniquely accommodates contributions to energy and entropy from local symmetry and internal fluctuations, $\mathcal{E}$ is essential for strict adherence to the laws of thermodynamics across all size scales, especially behavior that emerges on the scale of nanometers.

## 4. Extending Statistical Mechanics to Treat Multiscale Heterogeneity

## 5. Finite-Size Effects in the Thermal Properties of Simple Systems

#### 5.1. Semi-Classical Ideal Gas

^{3}) containing on the order of Avogadro’s number of atoms (N~N

_{A}= 6.022 × 10

^{23}atoms/mole). Assume monatomic atoms at temperature T with negligible interactions (ideal gas), so that the average internal energy comes only from their kinetic energy, $\overline{E}$ = 3N(½kT). Gibbs’ paradox [18,19,20,21] is often used to argue that the entropy of such thermodynamic systems must be additive and extensive. Nanothermodynamics is based on assuming standard thermodynamics in the limit of large systems, while treating non-extensive contributions to thermal properties of small systems in a self-consistent manner. Here, we review and reinterpret several results given in chapters 10 and 15 of Hill’s Thermodynamics of Small Systems [13]. We emphasize that sub-additive entropy, a fundamental property of quantum-mechanics [23,68], often favors subdividing a large system into an ensemble of nanoscale regions, increasing the total entropy and requiring nanothermodynamics for a full analysis.

^{23}atoms) of argon gas (mass m = 6.636 × 10

^{−26}kg) at a temperature of 0 C (T = 273.15 K), yielding the thermal de Broglie wavelength Λ = $h/\sqrt{2\pi mkT}$ ≈ 16.7 pm. At atmospheric pressure (101.325 kPa), the number density of one amagat (N/V = 2.687 × 10

^{25}atoms/m

^{3}) gives an average distance between atoms of ${\left(V/N\right)}^{1/3}={\left(\overline{V}/\overline{N}\right)}^{1/3}\approx $ 3.34 nm, and a mean-free path of $\ell =\left(V/N\right)/\left(\sqrt{2}\pi {d}^{2}\right)\approx $ 59.3 nm (using a kinetic diameter of d = 0.376 nm for argon). Under these conditions the Sackur–Tetrode formula predicts a dimensionless entropy per atom of ${S}_{0}/Nk=5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}\overline{N}/\overline{V}\right]\approx $ 18.39 (equal to 152.9 J/mole-K). In the canonical ensemble the subdivision potential is positive, ${\mathcal{E}}_{c}/kT\approx \mathrm{ln}\left(\sqrt{2\pi N}\right)$, so that when subtracted from the Sackur–Tetrode formula the entropy is reduced. Although the magnitude of this entropy reduction per atom is microscopic, ${\mathcal{E}}_{c}/NkT$ = 4.70 × 10

^{−23}, even such a small reduction is used to justify the standard thermodynamic hypothesis of a single homogeneous system. However, the hypothesis breaks down if subsystems are not explicitly constrained to have a fixed size. Indeed, regions in the nanocanonical ensemble have a sub-additive entropy that increases upon subdivision. Specifically, ${\mathcal{E}}_{nc}/kT=-\mathrm{ln}\left(\overline{N}+1\right)$ is negative when $\overline{N}>0$, confirming that any system of ideal gas atoms favors subdividing into an ensemble of regions whenever the size of each small region is not externally constrained. Thermal equilibrium in the nanocanonical ensemble is usually found by setting${\mathcal{E}}_{nc}=0$ [57], yielding $\overline{N}\to 0$ and an increase in entropy per atom of: $-{\mathcal{E}}_{nc}/\overline{N}kT=\underset{\overline{N}\to 0}{\mathrm{lim}}[\mathrm{ln}\left(\overline{N}+1\right)/\overline{N}]$ = 1, about 5.4% of the Sackur–Tetrode component. However, the Sackur–Tetrode formula has been found to agree with measured absolute entropies of four monatomic gases, with discrepancies (0.07–1.4%) that are always within two standard deviations of the measured values [69]. Thus, the experiments indicate that $\overline{N}$ >> 1 in real gases, presumably due to quantum symmetry on length scales of greater than 10 nm. For example, if quantum symmetry (indistinguishability) occurs for atoms over an average distance of the mean-free path (ℓ = 58.3 nm), then $\overline{N}={\ell}^{3}\left(N/V\right)\approx $ 5600 atoms. Now the subdivision potential per atom yields $-{\mathcal{E}}_{nc}/\overline{N}kT=\mathrm{ln}\left(\overline{N}+1\right)/\overline{N}\approx 0.0015$, well within experimental uncertainty. In any case, nature should always favor maximum total entropy, no matter how small the gain, so that the statistics of indistinguishable particles may apply to semi-classical ideal gases across nanometer-sized regions, but not across macroscopic volumes.

_{1}particles of ideal gas 1, and the other box N

_{2}= N

_{1}particles of ideal gas 2, so that when combined, both specific densities are halved, e.g., ${N}_{1}/\left(V+V\right)=\frac{1}{2}{N}_{1}/V$. The Sackur–Tetrode formula yields an increased entropy from mixing: $\Delta {S}_{0}/k={N}_{1}\{5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}{N}_{1}/\left(V+V\right)\right]\}+{N}_{2}\left\{5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}{N}_{2}/\left(V+V\right)\right]\right\}-{N}_{1}\{5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}{N}_{1}/V)\right]\}-{N}_{2}\left\{5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}{N}_{2}/V\right]\right\}=\left({N}_{1}+{N}_{2}\right)\mathrm{ln}2$. This entropy of mixing dominates all ensembles. In fact, because the subdivision potentials in Table 1 depend on the number of particles in the system, but not on the volume, finite-size effects in the entropy are unchanged by mixing two types of gases. Specifically, for the canonical ensemble: $\Delta {S}_{c}/k\approx \Delta {S}_{0}/k-2\mathrm{ln}\left[\sqrt{2\pi {N}_{1}}\right]+\mathrm{ln}[\sqrt{2\pi {N}_{1}}]+\mathrm{ln}\left[\sqrt{2\pi {N}_{2}}\right]=\Delta {S}_{0}/k$. Similarly, for the nanocanonical ensemble: $\Delta {S}_{nc}/k=\Delta {S}_{0}/k+2\text{}\mathrm{ln}\left(\overline{N}+1\right)-\text{}\mathrm{ln}\left(\overline{N}+1\right)-\text{}\mathrm{ln}\left(\overline{N}+1\right)=\Delta {S}_{0}/k$.

_{1}particles of ideal gas 1. When the boxes are combined, the particle density does not change $\left({N}_{1}+{N}_{1}\right)/\left(V+V\right)={N}_{1}/V$. From the Sackur–Tetrode formula for effectively infinite systems of indistinguishable particles, the total entropy also does not change: $\Delta {S}_{0}/k=\left({N}_{1}+{N}_{1}\right)\left\{5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}({N}_{1}+{N}_{1})/\left(V+V\right)\right]\right\}-2{N}_{1}\left\{5/2-\mathrm{ln}\left[{\mathsf{\Lambda}}^{3}{N}_{1}/V\right]\right\}$ = 0. Adding finite-size effects to the canonical ensemble (row 2 in Table 1), combining identical systems increases the total entropy: $\frac{\Delta {S}_{c}}{k}\approx \frac{\Delta {S}_{0}}{k}-\mathrm{ln}[\sqrt{2\pi \left({N}_{1}+{N}_{1}\right)}]+2\text{}\mathrm{ln}\left[\sqrt{2\pi {N}_{1}}\right]=\mathrm{ln}\left[\sqrt{\pi {N}_{1}}\right].$ Quantitatively, if each box initially contains one mole of particles, then N

_{1}= 6.022 × 10

^{23}yields $\Delta {S}_{c}/k\approx $ 27.95. Although the entropy increase per particle is extremely small, any increase in entropy inhibits heterogeneity in bulk systems, supporting the standard thermodynamic assumption of large homogeneous systems. However, this entropy increase applies only to ensembles having subsystems of fixed size. In contrast, combining boxes in the nanocanonical ensemble decreases the total entropy. Specifically, in thermal equilibrium at constant density, both $\overline{N}$ and $\overline{V}$ remain constant so that $\overline{N}/\overline{V}={N}_{1}/V$, yielding a decrease in total entropy when boxes are combined: $\Delta {S}_{nc}/k=\Delta {S}_{0}/k+\text{}\mathrm{ln}\left(\overline{N}+1\right)-2\text{}\mathrm{ln}\left(\overline{N}+1\right)=-\text{}\mathrm{ln}\left(\overline{N}+1\right)$. The per-particle entropy change is again extremely small for large boxes, but the inverse process of subdividing into small internal regions should proceed until the increase in per-particle entropy reaches its maximum: $\underset{\overline{N}\to 0}{\mathrm{lim}}-\Delta {S}_{nc}/\overline{N}k=1$. As previously discussed (following Figure 2), the fact that real gases do not show such deviations from the Sackur–Tetrode formula [69] implies $\overline{N}\gg 1$; but any increase in entropy is favored by the second law of thermodynamics, and required by a fundamental property of quantum mechanics for sub-additive entropy [23,68]. Moreover, similarly uncorrelated small regions are found to dominate the primary response measured in liquids and solids [36,37,38,39,40,41,42,43,44,45,46,47,48,49].

#### 5.2. Finite Chain of Ising Spins

_{t}), when the number of subsystems increases (dη > 0), the average subsystem size (N) decreases, the energy levels may broaden from finite-size effects due to surface states, interfaces, thermal fluctuations, etc. The subdivision potential in nanothermodynamics uniquely allows systematic treatment of these finite-size effects, thereby ensuring that energy is strictly conserved, even on the scale of nanometers.

_{i}= ±1. Using +J for the energy of anti-aligned neighboring spins, and −J between aligned neighbors. The Hamiltonian is

_{i}= −1), with (N − x) low-energy bonds (b

_{i}= +1), the internal energy is $E=-J\left(N-2x\right)$. The multiplicity of ways for this energy to occur is given by the binomial coefficient

#### 5.3. The Subdivided Ising Model: Ising-Like Spins with a Distribution of Neutral Bonds

**X**), η’ = 3 neutral bonds (

**O**), and N − η’ − x = 5 low-energy bonds (●). It is again convenient to write the energy in terms of the bonds, which may now have three distinct states, yielding the Hamiltonian

^{η’}arises because each neutral bond has two possible states for its neighboring spin. This first sum yields a type of canonical ensemble for the system with fixed η’. A second sum is over all values of η’. The multiplicity is given by the binomial for the number of ways that the neutral bonds can be distributed, which arises from the trinomial after the first summation. The behavior of this model is summarized in Table 3.

#### 5.4. Finite Chains of Effectively Indistinguishable Ising-Like Spins

#### 5.5. Entropy and Heat in an Ideal 1-D Polymer

^{th}unit unconnected. All other units have both ends freely jointed to neighboring units. For simplicity assume all units are uniaxial (1-D), with x segments pointing in the –X direction and N − x segments in the +X direction. The free end of the polymer is at position X(x) = (N − 2x) a. The multiplicity matches that of Equation (18a) for the standard Ising model, and Table 2 gives the resulting microcanonical entropy ${S}_{mc}/k=\mathrm{ln}\left[2\left(\begin{array}{c}N\\ x\end{array}\right)\right]$. The elastic restoring force from the entropy [1] is: F = − T ΔS/ΔX Note that this model involves differences (not differentials) because it is comprised of discrete polymer units in 1-D. Such discrete differences circumvent Stirling’s formula, improving the accuracy, especially for small systems. For an incremental shortening of the polymer ΔX = −2a, using half integers to best represent the average values at each integer, the change in configurational entropy of the polymer is $\Delta {S}_{mc}/k=\mathrm{ln}\left[\frac{N!}{\left(x+\frac{1}{2}\right)!\left(N-x-\frac{1}{2}\right)!}\right]-\mathrm{ln}\left[\frac{N!}{\left(x-\frac{1}{2}\right)!\left(N-x+\frac{1}{2}\right)!}\right]=\mathrm{ln}\left[\frac{N-x+\frac{1}{2}}{x+\frac{1}{2}}\right]$. Solving for the average number of negatively aligned units as a function of F gives $\overline{x}=\frac{N+\frac{1}{2}\left(1-{e}^{2aF/kT}\right)}{1+{e}^{2aF/kT}}$, yielding the equilibrium endpoint of the polymer $X\left(\overline{x}\right)=\left(N+1\right)a\mathrm{tanh}\left(aF/kT\right)$. At high-temperatures X($\overline{x}$) ≈ $\left(N+1\right)\frac{{a}^{2}F}{kT}$, similar to the standard expression for the ideal 1-D polymer if N >> 1.

_{L}(X)) never deviate from a maximum value:

_{L}(X) must increase, and vice versa. Quantitatively, using the microcanonical entropy for the polymer (adapted from Table 2), allowing fast fluctuations that are localized and reversible, the entropy of the local bath is:

#### 5.6. Simulations of Finite Chains of Effectively Indistinguishable Ising-Like Spins

_{L}(X). Note that Equation (22) favors high entropy in the local bath, just as Equation (21) favors high energy (and hence high entropy) in a large reservoir. Additionally, note that Equation (19) gives S

_{L}(X) = S

_{max}− S(X), the offset between the maximum configurational entropy and its current value, not just the change in entropy between initial and final states. Justification comes from the assumption that fast fluctuations involve local properties that do not have time to couple to the large reservoir, and even localized thermal process must not diminish the total entropy. Furthermore, S(X) for finite systems involves nonlinear terms that cannot be reduced to linear differentials.

_{c}(1000). The inset in Figure 7 shows the chain-size dependence of this f

_{c}. Three distinct features shown by the simulations in Figure 7 mimic measured noise in quantum bits [82,83]: 1/f-like noise with a slope of magnitude 0.92 ± 0.02; S(f) in smaller chains with discrete Lorentzian spectra; and white noise at higher frequencies. Figure 7 also shows that there are two ways to reduce low-frequency noise in the orthogonal Ising model. Specifically, at log(f/f

_{0}) = 4 maximum noise occurs when N ≈ 50. Noise decreases for larger N as f

_{c}shifts to lower frequencies, and decreases for smaller N as 1/f-like noise saturates at low frequencies in small systems, avoiding the divergence as f → 0 [76]. Thus, Figure 7 shows that the orthogonal Ising model has fluctuations in alignment that yield measured frequency exponents for 1/f-like noise, a crossover to white noise at higher f, and discrete Lorentzian responses; three distinct features that mimic measured spectra.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch showing four ways of subdividing a sample into η = 9 subsystems, forming various ensembles [17]. For small and fluctuating subsystems, full accuracy requires that the correct ensemble be used for the specific constraints, which may depend on the type and time scale of the dynamics.

**Figure 2.**Crude characterization of a system (top row) and its multiplicities for two types of subdivision. The particles (dots) can be on either side of the system, but must be in separate volumes for fixed N in canonical subsystems (second row). Nanocanonical subsystems have variable N, and variable V, increasing the net entropy.

**Figure 3.**Sketch showing the process of combining two dissimilar semi-classical ideal gases (

**upper left**), or two similar systems of semi-classical ideal gas (

**lower left**). Although total particle density is constant in both cases, entropy increases due to mixing if dissimilar gases are combined.

**Figure 4.**Fundamental equation for conservation of energy in magnetic systems, including finite-size effects, with a sketch of how a three-energy-level system can be changed by various contributions.

**Figure 5.**Sketch of 11 Ising-like spins in a chain connected by N = 10 bonds. Here, η’ = 3 bonds are neutral (O) (yielding η = 4 regions), x = 2 bonds are high-energy (X), and N − η’ − x = 5 bonds are low energy (●).

**Figure 6.**Internal energy distributions from a 1-D Ising model with N = 50 bonds at four temperatures, given in the legend. Symbols show histograms from simulations using the usual Metropolis algorithm that includes only interaction energy (open), and when an offset entropy term is added (solid). Lines are fits to the data using the integrand in the numerator of Equation (18a) (solid) or Equation (18b) (dashed).

**Figure 7.**Frequency dependence of power spectral densities. Solid lines are from several simulations of an orthogonal Ising model about an average temperature of kT/J ~ 200, with system size (N) given in the legend. Solid circles show measured flux noise from a qubit [82]. The characteristic frequency (f

_{0}) and amplitude of the simulations have been offset so that N = 50 (red line) mimics the data, with no other adjustable parameters. The dashed line fitted to the N = 1000 simulation has a slope of 0.92 ± 0.02, consistent with measurements of flux noise in qubits [83]. White noise (dotted line) occurs above the crossover frequency, f > f

_{c}. The inset shows the N dependence of this f

_{c}.

Ensemble | Partition Function | Fundamental Thermodynamic Function and Variables |
---|---|---|

Microcanonical (N,V,E) | ${\Omega}_{1}=V\frac{\pi}{4}{\left(\frac{8m}{{h}^{2}}\right)}^{3/2}\sqrt{E}$ ${\Omega}_{N}\approx \frac{1}{N!}{\left[V{\left(\frac{4\pi mE}{\text{}3N{h}^{2}}e\right)}^{3/2}\right]}^{N}$ | ${S}_{mc}/Nk=\frac{1}{N}\mathrm{ln}\left({\Omega}_{N}\right)\approx \frac{3}{2}+\mathrm{ln}\left(V\right)+\frac{3}{2}\mathrm{ln}\left[\frac{4\pi mE}{\text{}3N{h}^{2}}\right]-\frac{1}{N}\mathrm{ln}\left(N!\right)$ $\approx \frac{5}{2}-\mathrm{ln}\left(\frac{N}{V}\right)+\frac{3}{2}\mathrm{ln}\left[\frac{4\pi mE}{\text{}3N{h}^{2}}\right]-\frac{1}{N}\mathrm{ln}\left(\sqrt{2\pi N}\right)$ ${\mathcal{E}}_{mc}/kT\approx \mathrm{ln}\left(\sqrt{2\pi N}\right)$ |

canonical (N,V,T) | ${\mathrm{Q}}_{1}={{\displaystyle \int}}_{0}^{\infty}{\Omega}_{1}{e}^{-\frac{E}{kT}}dE$ $=V/{\Lambda}^{3}$ $\Lambda =h/\sqrt{2\pi mkT}$ ${\mathrm{Q}}_{N}={Q}_{1}^{N}/N!\text{}$ | $A/kT=-\mathrm{ln}\left({\mathrm{Q}}_{N}\right)$$=-N\mathrm{ln}\left({Q}_{1}\right)+\mathrm{ln}\left(N!\right)$ $\approx $ $N\mathrm{ln}\left(N{\Lambda}^{3}/V\right)-N+\mathrm{ln}\left(\sqrt{2\pi N}\right)$ $\overline{E}=\frac{\partial \mathrm{ln}{Q}_{N}}{\partial \left(-1/kT\right)}=\frac{3}{2}NkT$ $\overline{\mu}/kT\equiv -\mathrm{ln}\left({Q}_{N+1}/{Q}_{N}\right)=\mathrm{ln}\left({\Lambda}^{3}/V\right)+\mathrm{ln}\text{}\left(N+1\right)$${S}_{c}/Nk=\left(\overline{E}-A\right)/NkT\approx \frac{5}{2}-\text{}\mathrm{ln}\left(N{\Lambda}^{3}/V\right)-\frac{1}{N}\mathrm{ln}\left(\sqrt{2\pi N}\right)$ ${\mathcal{E}}_{c}={\mathcal{E}}_{mc}$ |

grand canonical (μ,V,T) | $\Xi ={{\displaystyle \sum}}_{N=0}^{\infty}\frac{1}{N!}{\left[\frac{V}{{\Lambda}^{3}}\right]}^{N}{\lambda}^{N}$ $\lambda ={e}^{\mu /kT}$ | $\Phi /kT=-\mathrm{ln}\left(\Xi \right)=-V\lambda /{\Lambda}^{3}$ $\overline{N}=-\partial \Phi /\partial \mu =V\lambda /{\Lambda}^{3}$ $\lambda ={e}^{\mu /kT}=\overline{N}{\Lambda}^{3}/V$ → $\mu /kT=\mathrm{ln}\left(\overline{N}{\Lambda}^{3}/V\right)$ ${S}_{gc}/\overline{N}k=\left(\overline{E}-\Phi -\mu \overline{N}\right)/\overline{N}kT=\frac{5}{2}-\mathrm{ln}\left(\overline{N}{\Lambda}^{3}/V\right)$ ${\mathcal{E}}_{gc}=0$ |

Nanocanonical (µ,P,T) | ${\rm Y}={{\displaystyle \int}}_{0}^{\infty}{e}^{\frac{V}{{\Lambda}^{3}}\lambda}\text{}{e}^{-\frac{PV}{kT}}\text{}\left[\frac{P}{kT}\right]dV$ | ${\mathcal{E}}_{nc}/kT=-\mathrm{ln}\left(\mathsf{{\rm Y}}\right)=\mathrm{ln}\left[1-kT\lambda /P{\Lambda}^{3}\right]$ $\overline{N}=\lambda \partial \mathrm{ln}\left({\rm Y}\right)/\partial \lambda =\left(kT\lambda /P{\Lambda}^{3}\right)/[1-kT\lambda /P{\Lambda}^{3}$] → $kT\lambda /P{\Lambda}^{3}=\overline{N}/\left(\overline{N}+1\right)$${\mathcal{E}}_{nc}/kT=-\mathrm{ln}\left[\overline{N}+1\right]$ ${S}_{nc}/\overline{N}k=\left(\overline{E}+P\overline{V}-\mu \overline{N}-{\mathcal{E}}_{nc}\right)/\overline{N}kT=\frac{5}{2}-\mathrm{ln}\left(\overline{N}{\Lambda}^{3}/\overline{V}\right)+\frac{1}{\overline{N}}\mathrm{ln}\left(\overline{N}+1\right)$ |

**Table 2.**N + 1 Ising spins (N bonds) in zero field with x high-energy bonds (+J) and (N − x) low-energy bonds (−J).

Ensemble | Partition Function | Fundamental Thermodynamic Function and Variables |
---|---|---|

Microcanonical (N + 1,x) | $\Omega =\frac{2N!}{x!\text{}\left(N-x\right)!\text{}}$ | ${S}_{mc}/k=\mathrm{ln}\left(\Omega \right)=\mathrm{ln}\left[N!\right]+\mathrm{ln}\left(2\right)-\mathrm{ln}\left[x!\right]-\mathrm{ln}\left[\left(N-x\right)!\right]$ $\approx N\mathrm{ln}\left(N\right)-x\mathrm{ln}\left(x\right)-\left(N-x\right)\mathrm{ln}\left(N-x\right)-\mathrm{ln}[\sqrt{2\pi x\left(1-x/N\right)}$$]$ |

canonical (N + 1,T) | $\mathrm{Q}={\displaystyle {\displaystyle \sum}_{x=0}^{N}}\Omega \text{}{e}^{\frac{\left(N-2x\right)J}{kT}}$ | $A/kT=-\mathrm{ln}\left(\mathrm{Q}\right)=-\mathrm{ln}[2{({e}^{J/kT}+{e}^{-J/kT})}^{N}$] $=-N\mathrm{ln}\left[2\mathrm{cosh}\left(J/kT\right)\right]-\mathrm{ln}\left(2\right)$ $\overline{E}/J=\left(2\overline{x}-N\right)=-N\mathrm{tan}\mathrm{h}\left(J/kT\right)$ $\overline{\mu}/kT=-\mathrm{ln}\left[2\mathrm{cos}\mathrm{h}\left(J/kT\right)\right]$ ${S}_{c}/k=\left(\overline{E}-A\right)/kT=-N\left(J/kT\right)\text{}\mathrm{tan}\mathrm{h}\left(J/kT\right)+N\text{}\mathrm{ln}[2\mathrm{cosh}\left(J/kT\right)]+\mathrm{ln}\left(2\right)$ |

Nanocanonical (µ,T) | ${\rm Y}={\displaystyle {\displaystyle \sum}_{N=0}^{\infty}}\mathrm{Q}\text{}{e}^{\frac{\left(N+1\right)\mu}{kT}}$ | ${\mathcal{E}}_{nc}/kT=-\mathrm{ln}\left(\mathsf{{\rm Y}}\right)$ $=\mathrm{ln}\left[\frac{1}{2}{e}^{-\mu /kT}-\mathrm{cosh}\left(J/kT\right)\right]=0$ → $\mu /kT=-\mathrm{ln}\{2[\mathrm{cosh}\left(J/kT\right)+1]\}\text{}$ $\overline{N}+1=\partial \mathrm{ln}\left({\rm Y}\right)/\partial \left(\mu /kT\right)$ $=\frac{1}{2}{e}^{-\mu /kT}=\mathrm{cosh}(J/kT)+1$ → $\mu /kT=-\mathrm{ln}\left\{2\left[\overline{N}+1\right]\right\}\text{}$ ${S}_{nc}/k=\left(\overline{E}-\mu \overline{N}-{\mathcal{E}}_{nc}\right)/kT=-\overline{N}\left(J/kT\right)\text{}\mathrm{tan}\mathrm{h}\left(J/kT\right)+\overline{N}\text{}\mathrm{ln}\{2[\mathrm{cosh}\left(J/kT\right)+1]\}$ |

**Table 3.**N + 1 Ising-like spins with η’ neutral bonds (energy=0), x high-energy bonds (+J), and N − η’ − x low-energy bonds (−J).

Ensemble | Partition Function | Fundamental Thermodynamic Function and Variables |
---|---|---|

microcanonical (N + 1,η’,x) | $\Omega =\frac{2N!{2}^{\eta \prime}}{x!\eta \prime !\left(N-\eta \prime -x\right)!}$ | ${S}_{mc}/k=\mathrm{ln}\left(\Omega \right)=\mathrm{ln}\left[N!\right]+\mathrm{ln}\left({2}^{\eta \prime +1}\right)-\mathrm{ln}\left[x!\right]-\mathrm{ln}\left[\eta \prime !\right]-\mathrm{ln}\left[\left(N-\eta \prime -x\right)!\right]$ $\approx N\mathrm{ln}\left(N\right)-x\mathrm{ln}\left(x\right)-\eta \prime \mathrm{ln}\left(\eta \prime /2\right)-\left(N-\eta \prime -x\right)\mathrm{ln}\left(N-\eta \prime -x\right)$ |

quasi-canonical (N + 1,η’,T) | $Z={\displaystyle {\displaystyle \sum}_{x=0}^{N-\eta \prime}}\Omega {e}^{\frac{\left(N-\eta \prime -2x\right)J}{kT}}$ | $Z={2}^{N+1}\left[\mathrm{cosh}(\frac{J}{kT}\right){]}^{N-{\eta}^{\prime}}N!/\left[\eta \prime !\left(N-\eta \prime \right)!\right]$ $N-\eta \prime -2\overline{x}=\partial \mathrm{ln}Z/\partial (J/kT)=\left(N-\eta \prime \right)\mathrm{tanh}(J/kT)$ $E/J=-\left(N-\eta \prime -2\overline{x}\right)=-\left(N-\eta \prime )\mathrm{tan}\mathrm{h}\text{}(J/kT\right)$ |

canonical (N + 1,T) | $Q={\displaystyle {\displaystyle \sum}_{\eta \prime =0}^{N}}\frac{N!}{\eta \prime !\left(N-\eta \prime \right)!}Z$ | $A/kT=-\mathrm{ln}\left(\mathrm{Q}\right)=-\left(N+1\right)\mathrm{ln}\left(2\right)-N\mathrm{ln}\left[1+\mathrm{cos}\mathrm{h}\left(J/kT\right)\right]$ $N-\overline{\eta \prime}=\mathrm{cosh}\left(J/kT\right)\partial \mathrm{ln}\mathrm{Q}/\partial \mathrm{cos}\mathrm{h}\left(J/kT\right)=N\mathrm{cosh}(J/kT)]/[1+\mathrm{cosh}(J/kT)]$ $\overline{E}/J=-\left(N-\overline{\eta \prime})\mathrm{tan}\mathrm{h}\text{}(J/kT\right)=-N\mathrm{sinh}(J/kT)/\left[1+\mathrm{cosh}(J/kT\right)]$ ${S}_{c}/k=-N\left(J/kT\right)\mathrm{sinh}(J/kT)/\left[1+\mathrm{cosh}(J/kT\right)]+N\text{}\mathrm{ln}[1+\mathrm{cosh}\left(J/kT\right)]$ |

**Table 4.**N + 1 Ising-like spins, N indistinguishable bonds: x high-energy bonds (+J) and (N − x) low-energy bonds (−J).

Ensemble | Partition Function | Fundamental Thermodynamic Function and Variables |
---|---|---|

microcanonical (N,x) | $\Omega =2$ | ${S}_{mc}/k=\mathrm{ln}\left(\Omega \right)=\mathrm{ln}\left(2\right)$ |

canonical (N,T) | $\mathrm{Q}={\displaystyle {\displaystyle \sum}_{x=0}^{N}}\Omega {e}^{\frac{\left(N-2x\right)J}{kT}}$ | $A/kT=-\mathrm{ln}\left(\mathrm{Q}\right)=-\mathrm{ln}\{2\mathrm{sinh}\left[\left(N+1\right)J/kT\right]/\mathrm{sinh}\left(J/kT\right)\}$ $\overline{E}/J=\left(2\overline{x}-N\right)=-\left(N+1\right)\mathrm{cot}\mathrm{h}\left[\left(N+1\right)J/kT\right]$$+\mathrm{coth}\left(J/kT\right)$ $\overline{\mu}/kT=\mathrm{ln}[\mathrm{sinh}\left(NJ/kT\right)]-\mathrm{ln}\left\{\mathrm{sin}\mathrm{h}\left[\left(N+1\right)J/kT\right]\right\}$ ${S}_{c}/k=\left(\overline{E}-A\right)/kT$ |

nanocanonical (µ,T) | ${\rm Y}={\displaystyle {\displaystyle \sum}_{N=0}^{\infty}}\mathrm{Q}{e}^{\frac{\left(N+1\right)\mu}{kT}}$ | ${\mathcal{E}}_{nc}/kT=-\mathrm{ln}\left(\mathsf{{\rm Y}}\right)=\mathrm{ln}[\mathrm{cosh}\left(\mu /kT\right)-\mathrm{cosh}\left(J/kT\right)]$$=0$ $\mu /kT=-\mathrm{acosh}[\mathrm{cosh}(J/kT)+1]$ $\overline{N}+1=\sqrt{{\left[\mathrm{cosh}\left(J/kT\right)+1\right]}^{2}-1}$ ${S}_{nc}/k=\left(\overline{E}-\mu \overline{N}-{\mathcal{E}}_{nc}\right)/kT$ |

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Chamberlin, R.V.; Clark, M.R.; Mujica, V.; Wolf, G.H.
Multiscale Thermodynamics: Energy, Entropy, and Symmetry from Atoms to Bulk Behavior. *Symmetry* **2021**, *13*, 721.
https://doi.org/10.3390/sym13040721

**AMA Style**

Chamberlin RV, Clark MR, Mujica V, Wolf GH.
Multiscale Thermodynamics: Energy, Entropy, and Symmetry from Atoms to Bulk Behavior. *Symmetry*. 2021; 13(4):721.
https://doi.org/10.3390/sym13040721

**Chicago/Turabian Style**

Chamberlin, Ralph V., Michael R. Clark, Vladimiro Mujica, and George H. Wolf.
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https://doi.org/10.3390/sym13040721