# Statistical Mechanics at Strong Coupling: A Bridge between Landsberg’s Energy Levels and Hill’s Nanothermodynamics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ‘Temperature-Dependent’ Energy Levels and the Hamiltonian of Mean Force

## 3. Strong Coupling and Hill’s Nanothermodynamics

#### 3.1. Hill’s Subdivision Potential

#### 3.2. Statistical Mechanics at Strong Coupling

#### Generalization to Other Ensembles

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Campa, A.; Dauxois, T.; Fanelli, D.; Ruffo, S. Physics of Long-Range Interacting Systems; Oxford University Press: Oxford, UK, 2014. [Google Scholar]
- Latella, I.; Peréz-Madrid, A.; Campa, A.; Casetti, L.; Ruffo, S. Thermodynamics of Nonadditive Systems. Phys. Rev. Lett.
**2015**, 114, 230601. [Google Scholar] [CrossRef] [PubMed][Green Version] - de Miguel, R.; Rubí, J.M. Strong Coupling and Nonextensive Thermodynamics. Entropy
**2020**, 22, 975. [Google Scholar] [CrossRef] [PubMed] - Rushbrooke, G.S. On the statistical mechanics of assemblies wose energy-levels depend on temperature. Trans. Faraday Soc.
**1940**, 36, 1055. [Google Scholar] [CrossRef] - Landsberg, P.T. Statitical Mechanics of Teperature-Dependent Energy Levels. Phys. Rev.
**1954**, 95, 643. [Google Scholar] - Elcock, E.W.; Landsberg, P.T. Temperature Dependent Energy Levels in Statistical Mechanics. Proc. Phys. Soc. Lond. Sect. B
**1957**, 70, 161. [Google Scholar] [CrossRef] - Cuden, C.B. The Temperature Dependence of the Energy Gaps in Semiconductors. Ph.D. Thesis, University of British Columbia, Vancouver, BC, Canada, 1969. [Google Scholar] [CrossRef]
- Bendt, P.J.; Cowan, R.D.; Yarnell, J.L. Excitations in Liquid Helium: Thermodynamic Calculations. Phys. Rev.
**1959**, 113, 1386. [Google Scholar] [CrossRef] - Donnelly, R.J.; Roberts, P.H. A theory of temperature-dependent energy levels: Thermodynamic properties og He II. Low Temp. Phys.
**1977**, 27, 687. [Google Scholar] [CrossRef] - Allen, P.V.; Heine, V. Theory of the temperature dependence of electronic band structures. J. Phys. C Solid State Phys.
**1976**, 9, 2305. [Google Scholar] [CrossRef] - Patrick, C.E.; Giustino, F. Unified theory of electron-phonon renormalization and phono-assisted optical absorption. J. Phys. Condens. Matter
**2014**, 26, 365503. [Google Scholar] [CrossRef][Green Version] - Dykman, M.I.; Kono, K.; Kostantinov, D.; Lea, M.J. Ripplonic Lamb Shift for Electrons on Liquid Helium. Phys. Rev. Lett.
**2017**, 119, 256802. [Google Scholar] [CrossRef][Green Version] - Erez, A.; Meir, Y. Effect of amplitude fluctuations on the Berezinskii-Kosterlitz-Thouless transition. Phys. Rev. B
**2013**, 88, 184510. [Google Scholar] [CrossRef][Green Version] - Kolář, M.; Ryabov, A.; Filip, R. Optomechanical oscillator controlled by variatioons in its heat bath temperature. Phys. Rev. A
**2017**, 95, 042105. [Google Scholar] [CrossRef][Green Version] - Yamano, T. Efficiencies of thermodynamics when temperature-dependent energy levels exist. Phys. Chem. Chem. Phys.
**2016**, 18, 7011. [Google Scholar] [CrossRef] - Yamano, T. Effect of temperature-dependent energy levels on exergy. J. Phys. Commun.
**2017**, 1, 055007. [Google Scholar] [CrossRef] - de Miguel, R.; Rubí, J.M. Thermodynamics far from the thermodynamic limit. J. Phys. Chem. B
**2017**, 121, 10429–10434. [Google Scholar] [CrossRef] [PubMed][Green Version] - Shental, O.; Kanter, I. Shannon meets Carnot: Generalized second thermodynamic law. Europhys. Lett.
**2009**, 85, 10006. [Google Scholar] [CrossRef][Green Version] - Hill, T.L. Thermodynamics of Small Systems. J. Chem. Phys.
**1962**, 36, 3182. [Google Scholar] [CrossRef] - Hill, T.L. Thermodynamics of Small Systems, Parts I & II; Dover: New York, NY, USA, 2013. [Google Scholar]
- Hill, T.L. Perspective: Nanothermodynamics. Nano Lett.
**2001**, 1, 111. [Google Scholar] [CrossRef] - Hill, T.L. A different Approach to Nanothermodynamics. Nano Lett.
**2001**, 1, 273. [Google Scholar] [CrossRef] - Bedeaux, D.; Kjelstrup, S.; Schnell, S.K. Nanothermodynamics—General Theory; PoreLab: Trondheim, Norway, 2020. [Google Scholar]
- Rubí, J.M.; Bedeaux, D.; Kjelstrup, S. Thermodynamics for Single-Molecule Stretching Experiments. J. Phys. Chem. B
**2006**, 110, 12733. [Google Scholar] [CrossRef][Green Version] - Quian, H. Hill’s small systems nanothermodynamics: A simple macromolecular partition problem with a statistical perspective. J. Biol. Phys.
**2012**, 38, 201. [Google Scholar] [CrossRef][Green Version] - Chamberlin, R.V. The BigWorld of Nanothermodynamics. Entropy
**2015**, 17, 52. [Google Scholar] [CrossRef][Green Version] - Galteland, O.; Bedeaux, D.; Hafskjold, B.; Kjelstrup, S. Pressures Inside a Nano-Porous Medium. The Case of a Single Phase Fluid. Front. Phys.
**2019**, 7, 60. [Google Scholar] [CrossRef][Green Version] - Rauter, M.T.; Galteland, O.; Erdõs, M.; Moultos, O.A.; Vlugt, T.J.H.; Schnell, S.; Bedeaux, D.; Kjelstrup, S. Two-Phase Equilibrium Conditions in Nanopores. Nanomaterials
**2020**, 10, 608. [Google Scholar] [CrossRef] [PubMed][Green Version] - Strøm, B.A.; He, J.; Bedeaux, D.; Kjelstrup, S. When Thermodynamic Properties of Adsorbed Films Depend on Size: Fundamental Theory and Case Study. Nanomaterials
**2020**, 10, 1691. [Google Scholar] [CrossRef] - Talkner, P.; Hanggi, P. Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical. Rev. Mod. Phys.
**2020**, 92, 041002. [Google Scholar] [CrossRef] - Kolář, M.; Ryabov, A.; Filip, R. Heat capacities of thermally manipulated mechanical oscillator at strong coupling. Sci. Rep.
**2019**, 9, 10855. [Google Scholar] [CrossRef] [PubMed] - de Miguel, R.; Rubí, J.M. Negative thermophoretic force in the strong coupling regime. Phys. Rev. Lett.
**2019**, 123, 200602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Whitfield, G.; Engineer, M. Temperature dependence of the polaron. Phys. Rev. B
**1975**, 12, 5472. [Google Scholar] [CrossRef] - Pathria, R.K.; Beale, P.D. Statistical Mechanics, 3rd ed.; Elsevier: Oxford, UK, 2011. [Google Scholar]
- Talkner, P.; Hanggi, P. Open system trajectories specify fluctuating work but not heat. Phys. Rev. E
**2016**, 94, 022143. [Google Scholar] [CrossRef][Green Version] - Seifert, U. First and Second Law of Thermodynamics at Strong Coupling. Phys. Rev. Lett.
**2016**, 116, 020601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jarzynski, C. Stochastic and Macroscopic Thermodynamics of Strongly Coupled Systems. Phys. Rev. X
**2017**, 7, 011008. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Illustration of the main features of the three small-system theories. In Landsberg’s theory, the energy levels ${E}_{1}$ of the system (blue sphere) are modified into ${\mathcal{E}}_{1}$ due to its interaction with the (red) environment. Statistical mechanics at strong coupling starts with a total Hamiltonian consisting of the environment (red), the system (blue) and the interaction (green); averaging the system and the interaction over the environment results in a temperature-dependent Hamiltonian of mean force ${\mathcal{H}}_{1}$. Hill’s nanothermodynamics considers the system of interest as if it were surrounded by a macroscopic set of interacting replicas; the interaction causes the effective energy to depart from U and become $\mathcal{U}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

de Miguel, R.; Rubí, J.M. Statistical Mechanics at Strong Coupling: A Bridge between Landsberg’s Energy Levels and Hill’s Nanothermodynamics. *Nanomaterials* **2020**, *10*, 2471.
https://doi.org/10.3390/nano10122471

**AMA Style**

de Miguel R, Rubí JM. Statistical Mechanics at Strong Coupling: A Bridge between Landsberg’s Energy Levels and Hill’s Nanothermodynamics. *Nanomaterials*. 2020; 10(12):2471.
https://doi.org/10.3390/nano10122471

**Chicago/Turabian Style**

de Miguel, Rodrigo, and J. Miguel Rubí. 2020. "Statistical Mechanics at Strong Coupling: A Bridge between Landsberg’s Energy Levels and Hill’s Nanothermodynamics" *Nanomaterials* 10, no. 12: 2471.
https://doi.org/10.3390/nano10122471