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

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## Abstract

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## 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

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**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 $\mathcal{U}$.

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**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