# Ensemble Equivalence for Distinguishable Particles

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Concepts

- For distinguishable particles any state can be written as an unrestricted linear combination of elements of the product basis $|{l}_{1}\rangle |{l}_{2}\rangle \cdots |{l}_{N}\rangle \equiv |{l}_{1},{l}_{2},\cdots ,{l}_{N}\rangle $. For an ideal system it is:$$\begin{array}{ccc}\hfill \mathcal{H}|{l}_{1},{l}_{2},\cdots ,{l}_{N}\rangle & =& ({\u03f5}_{{l}_{1}}+\cdots +{\u03f5}_{{l}_{N}})|{l}_{1},{l}_{2},\cdots ,{l}_{N}\rangle .\hfill \end{array}$$
- For indistinguishable particles one can use instead the second-quantization basis $\left|\right|{n}_{0},{n}_{1},{n}_{2},\cdots \rangle \phantom{\rule{-1.2pt}{0ex}}\rangle $ in terms of the occupation numbers ${n}_{\ell}$ of individual levels ℓ. In this representation, the wave function is always invariant under particle exchange, and one avoids an explicit symmetrization or anti-symmetrization process. For an ideal system it is:$$\begin{array}{ccc}\hfill \mathcal{H}\left|\right|{n}_{0},{n}_{1},\cdots \rangle \phantom{\rule{-1.2pt}{0ex}}\rangle & =& ({n}_{0}{\u03f5}_{0}+{n}_{1}{\u03f5}_{1}+\cdots )\left|\right|{n}_{0},{n}_{1},\cdots \rangle \phantom{\rule{-1.2pt}{0ex}}\rangle .\hfill \end{array}$$

- (i)
- In the first example, we consider a non-relativistic gas of non-interacting identical particles without any internal or rotational degrees of freedom and not subject to any external field. The Hamiltonian can be written as$$\mathcal{H}=\sum _{i=1}^{N}\frac{{{\overrightarrow{p}}_{i}}^{\phantom{\rule{2.84544pt}{0ex}}2}}{2m}.$$
- (ii)
- The second example is the previous ideal gas but particles have different masses. This classifies the particles as non-localized, non-identical and, hence, distinguishable both in the classical and quantum versions. The Hamiltonian is$$\mathcal{H}=\sum _{i=1}^{N}\frac{{{\overrightarrow{p}}_{i}}^{\phantom{\rule{2.84544pt}{0ex}}2}}{2{m}_{i}}.$$
- (iii)
- The third example is a set of harmonic oscillators, each one oscillating around a different position ${\overrightarrow{a}}_{i}$$$\mathcal{H}=\sum _{i=1}^{N}\left(\right)open="["\; close="]">\frac{{{\overrightarrow{p}}_{i}}^{\phantom{\rule{2.84544pt}{0ex}}2}}{2m}+\frac{m{\omega}^{2}}{2}{({\overrightarrow{r}}_{i}-{\overrightarrow{a}}_{i})}^{2}$$
- (iv)
- The final example is the statistics of paramagnetism, where we have a set of localized particles with magnetic moments $\left\{{\overrightarrow{\mu}}_{i}\right\}$ in a magnetic field $\overrightarrow{B}$$$\mathcal{H}=\sum _{i=1}^{N}\left(\right)open="["\; close="]">-{\overrightarrow{\mu}}_{i}\xb7\overrightarrow{B}+{h}_{\text{loc}}^{\left(i\right)}$$

## 3. Ensemble Nonequivalence

## 4. Correct Partition Function

#### 4.1. Ideal Gas of Identical Non-Localized Particles

#### 4.2. Ideal Gas of Non-Identical Non-Localized Particles

#### 4.3. Localized Particles

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Pathria, R.K.; Beale, P.D. Statistical Mechanics, 3rd ed.; Butterworth-Heinemann: Oxford, UK, 2011. [Google Scholar]
- Huang, K. Statistical Mechanics; John Wiley & Sons: New York, NY, USA, 1963. [Google Scholar]
- Ehrenfest, P.; Trkal, V. Deduction of the dissociation-equilibrium from the theory of quanta and a calculation of the chemical constant based on this. Proc. K. Ned. Akad. Wet.
**1921**, 23, 162–183. [Google Scholar] - Van Kampen, N. The Gibbs Paradox. In Essays in Theoretical Physics; Parry, W., Ed.; Pergamon Press: Bergama, Turkey, 1984; pp. 303–312. [Google Scholar]
- Swendsen, R.H. Statistical Mechanics of Classical Systems with Distinguishable Particles. J. Stat. Phys.
**2002**, 107, 1143–1166. [Google Scholar] [CrossRef] - Nagle, J.F. Regarding the Entropy of Distinguishable Particles. J. Stat. Phys.
**2004**, 117, 1047–1062. [Google Scholar] [CrossRef] - Swendsen, R.H. Response to Nagle’s Criticism of My Proposed Definition of the Entropy. J. Stat. Phys.
**2004**, 117, 1063–1070. [Google Scholar] [CrossRef] - Swendsen, R.H. Gibbs’ Paradox and the Definition of Entropy. Entropy
**2008**, 10, 15–18. [Google Scholar] [CrossRef] - Cheng, C. Thermodynamics of the System of Distinguishable Particles. Entropy
**2009**, 11, 326–333. [Google Scholar] [CrossRef] - Nagle, J.F. In Defense of Gibbs and the Traditional Definition of the Entropy of Distinguishable Particles. Entropy
**2010**, 12, 1936–1945. [Google Scholar] [CrossRef] - Peters, H. Statistics of Distinguishable Particles and Resolution of the Gibbs Paradox of the First Kind. J. Stat. Phys.
**2010**, 141, 785–828. [Google Scholar] [CrossRef] - Versteegh, M.A.M.; Dieks, D. The Gibbs paradox and the distinguishability of identical particles. Am. J. Phys.
**2011**, 79, 741–746. [Google Scholar] [CrossRef] - Swendsen, R.H. Choosing a definition of entropy that works. Found. Phys.
**2012**, 42, 582–593. [Google Scholar] [CrossRef] - Dieks, D. Is There a Unique Physical Entropy? Micro versus Macro. In New Challenges to Philosophy of Science; Springer: Dordrecht, The Netherlands, 2013; Volume 4, pp. 23–34. [Google Scholar]
- Swendsen, R.H. Statistical mechanics of colloids and Boltzmann’s definition of the entropy. Am. J. Phys.
**2006**, 74, 187–190. [Google Scholar] [CrossRef] - Cates, M.E.; Manoharan, V.N. Celebrating Soft Matter’s 10th Anniversary: Testing the Foundations of Classical Entropy: Colloid Experiments. Soft Matter
**2015**, 11, 6538–6546. [Google Scholar] [CrossRef] [PubMed] - Park, J.; Newman, M.E.J. Statistical mechanics of networks. Phys. Rev. E
**2004**, 70, 066117. [Google Scholar] [CrossRef] [PubMed] - Bianconi, G. Entropy of network ensembles. Phys. Rev. E
**2009**, 79, 036114. [Google Scholar] [CrossRef] [PubMed] - Bianconi, G. The entropy of randomized network ensembles. Europhys. Lett.
**2008**, 81, 28005. [Google Scholar] [CrossRef] - Sagarra, O.; Pérez Vicente, C.J.; Díaz-Guilera, A. Statistical mechanics of multiedge networks. Phys. Rev. E
**2013**, 88, 062806. [Google Scholar] [CrossRef] [PubMed] - Sagarra, O.; Pérez Vicente, C.J.; Díaz-Guilera, A. Role of adjacency-matrix degeneracy in maximum-entropy-weighted network models. Phys. Rev. E
**2015**, 92, 052816. [Google Scholar] [CrossRef] [PubMed] - Fernández-Peralta, A. Statistical Mechanics of Multilayer Networks. Master’s Thesis, University of the Balearic Islands, Palma, Spain, 2015. [Google Scholar]
- Balescu, R. Equilibrium and Non-Equilibrium Statistical Mechanics; John Wiley & Sons: New York, NY, USA, 1975. [Google Scholar]

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Fernández-Peralta, A.; Toral, R.
Ensemble Equivalence for Distinguishable Particles. *Entropy* **2016**, *18*, 259.
https://doi.org/10.3390/e18070259

**AMA Style**

Fernández-Peralta A, Toral R.
Ensemble Equivalence for Distinguishable Particles. *Entropy*. 2016; 18(7):259.
https://doi.org/10.3390/e18070259

**Chicago/Turabian Style**

Fernández-Peralta, Antonio, and Raúl Toral.
2016. "Ensemble Equivalence for Distinguishable Particles" *Entropy* 18, no. 7: 259.
https://doi.org/10.3390/e18070259