# Structural Statistical Quantifiers and Thermal Features of Quantum Systems

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## Abstract

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## 1. Introduction

#### 1.1. LMC Structural Quantifiers

#### 1.2. Thermal Uncertainty Relations (TURs)

## 2. The Thermal Quantum Case

## 3. A Strict Bound Relating D to Quantum Uncertainties

## 4. Extending Bridges to a Semi-Classical Environment

#### 4.1. Introduction: Coherent States and Husimi Distributions

#### 4.2. HO Specialization

## 5. HO–Semi-Classical Thermal Treatment and Uncertainty Relations

## 6. Possible Classical Extension

## 7. Application to a Nuclear Physics Model

#### 7.1. The Model

#### 7.2. Second Quantization Language

#### 7.3. Hamiltonian H for Our Model

#### 7.4. Phase Transitions

#### 7.5. Finite Temperature

#### 7.6. Application Results

## 8. Conclusions

- At the minimum minimorum uncertainty value, the entropy, specific heat, and structural quantifier C all vanish.
- There is a strong connection between the disequilibrium D and the thermal uncertainty (TU). As D grows, the TU decreases. The TU is minimal for pure states where $D=1$.
- Note that all quantities involved in (15) are observable (in principle), so we are dealing with a relation that has its counterpart in nature.

- The Wehrl structural quantifier ${C}_{sc}$ attains its maximum values at the same place at which the quantal structural quantifier C does so.
- This place corresponds to the maximum possible semi-classical localization in phase space.
- Wehrl’s structural quantifier ${C}_{sc}$ grows from zero at null vibrational energy (VE) until the VE attains half of the thermal–kinetic energy, and then remains constant.
- ${\Delta}_{\mu}x\phantom{\rule{0.166667em}{0ex}}{\Delta}_{\mu}p/\u0127$ can be regarded as the phase-space localization error e (in its natural units) that accompanies the Husimi distribution.
- The Wehrl structural quantifier ${C}_{sc}$ becomes a maximum in these circumstances.
- We emphasize that ${C}_{V}$ attains its constant classical value as soon as the thermal energy equals the vibrational one.
- The three different structural quantifiers, C, at play in this work behave in a remarkably similar fashion, as shown in the last graphs.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LMC | Lopez-Rioz, Mancini, and Calbet |

LMCTSQ | Lopez-Rioz, Mancini, and Calbet thermal structural quantifiers |

TUR | Thermal uncertainty relation |

HO | Harmonic oscillator |

MM | Minimum minimorum |

TQF | Thermal quantum quantifiers |

HD | Husimi distributions |

DD | Density distribution |

LM | Lipkin model |

## Appendix A. Mathematics Program

## References

- Esquivel, R.O.; Angulo, J.C.; Antolín, J.; Dehesa, J.S.; Lopez-Rosa, S.; Flores-Gallegos, N. Analysis of complexity measures and information planes of selected molecules in position and momentum spaces. Phys. Chem. Chem. Phys.
**2010**, 12, 7108–7116. [Google Scholar] [CrossRef] [PubMed] - Toranzo, I.V.; Dehesa, J.S. Entropy and complexity properties of the d-dimensional blackbody radiation. Eur. Phys. J. D
**2014**, 68, 316. [Google Scholar] [CrossRef][Green Version] - Bouvrie, P.A.; Angulo, J.C.; Dehesa, J.S. Entropy and complexity analysis of Dirac-delta-like quantum potentials. Physica A
**2011**, 390, 2215–2228. [Google Scholar] [CrossRef] - López-Ruiz, R.; Mancini, H.L.; Calbet, X. A statistical measure of complexity. Phys. Lett. A
**1995**, 209, 321–326. [Google Scholar] [CrossRef][Green Version] - Pennnini, F.; Plastino, A. Disequilibrium, thermodynamic relations, and Rényi’s entropy. Phys. Lett. A
**2017**, 381, 212–215. [Google Scholar] [CrossRef] - López-Ruiz, R. Complexity in Some Physical System. Int. J. Bifurc. Chaos
**2001**, 11, 2669–2673. [Google Scholar] [CrossRef][Green Version] - Anteneodo, C.; Plastino, A.R. Some features of the López-Ruiz-Mancini-Calbet (LMC) statistical measure of complexity. Phys. Lett. A
**1996**, 223, 348–354. [Google Scholar] [CrossRef] - Martin, M.T.; Plastino, A.; Rosso, O.A. Statistical complexity and disequilibrium. Phys. Lett. A
**2003**, 311, 126–132. [Google Scholar] [CrossRef] - Rudnicki, L.; Toranzo, I.V.; Sánchez-Moreno, P.; Dehesa, J.S. Monotone measures of statistical structural quantifier. Phys. Lett. A
**2016**, 380, 377–380. [Google Scholar] [CrossRef][Green Version] - Ribeiro, H.V.; Zunino, L.; Lenzi, E.K.; Santoro, P.A.; Mendes, R.S. Complexity-Entropy Causality Plane as a Complexity Measure for Two-Dimensional Patterns. PLoS ONE
**2012**, 7, e40689. [Google Scholar] [CrossRef][Green Version] - López-Ruiz, R.; Mancini, H.; Calbet, X. A Statistical Measure of structural quantifier in Concepts and recent advances in generalized information measures and statistics. In Bentham Science Books; Kowalski, A., Rossignoli, R., Curado, E.M.C., Eds.; Bentham Science Publishers: New York, NY, USA, 2013; pp. 147–168. [Google Scholar]
- Sen, K.D. (Ed.) Statistical Structural Quantifier, Applications in Electronic Structure; Springer: Berlin, Germany, 2011. [Google Scholar]
- Martin, M.T.; Plastino, A.; Rosso, O.A. Generalized statistical structural quantifier measures: Geometrical and analytical properties. Physica A
**2006**, 369, 439–462. [Google Scholar] [CrossRef] - Ghosh, P.; Nath, D. Complexity analysis of two families of orthogonal functions. Int. J. Quant. Chem.
**2019**, 119, e25964. [Google Scholar] [CrossRef] - Fulop, A. Statistical complexity of the time dependent damped L84 model. Chaos
**2019**, 29, 083105. [Google Scholar] [CrossRef] [PubMed] - Plastino, A.; Moszkowski, S.M. Simplified model for illustrating Hartree-Fock in a Lipkin-model problem. Nuovo Cimento
**1978**, 47, 470–474. [Google Scholar] [CrossRef] - Kruse, M.K.G.; Miller, H.G.; Plastino, A.; Plastino, A.R. Thermodynamics’ third law and quantum phase transitions. Physica A
**2010**, 389, 2533–2540. [Google Scholar] [CrossRef] - Kruse, M.K.G.; Miller, H.G.; Plastino, A.; Plastino, A.R. Thermodynamic Detection of Quantum Phase Transitions. Int. J. Mod. Phys. B
**2010**, 24, 5027–5036. [Google Scholar] [CrossRef] - Cambiaggio, M.C.; Plastino, A.; Szybisz, L. Constrained Hartree-Fock and quasi-spin projection. Nucl. Phys.
**1980**, 344, 233–248. [Google Scholar] [CrossRef] - Zander, C.; Plastino, A.; Plastino, A.R. Quantum entanglement in a many-body system exhibiting multiple quantum phase transitions. Braz. J. Phys.
**2009**, 39, 464–467. [Google Scholar] [CrossRef] - Kruse, M.K.G.; Miller, H.G.; Plastino, A.; Plastino, A.R. Aspects of quantum phase transitions. arXiv
**2008**, arXiv:0809.3514. [Google Scholar] - Lipkin, H.J.; Meshkov, N.; Glick, A.J. Validity of many-body approximation methods for a solvable model: (III). Diagram summations. Nucl. Phys.
**1965**, 62, 211–224. [Google Scholar] [CrossRef] - Plastino, A.R.; Ferri, G.L.; Rocca, M.C.; Plastino, A. Information-Based Numerical Distancesbetween Equilibrium and Non-EquilibriumStates. Angelo Plastino J. Mod. Phys.
**2020**, 11, 1031–1043. [Google Scholar] [CrossRef] - Pennini, F.; Plastino, A. Statistical quantifiers for few-fermion’ systems. Physica A
**2018**, 491, 305–312. [Google Scholar] [CrossRef] - Peltier, S.M.; Plastino, A. A density-matrix approach to critical phenomena. Nucl. Phys.
**1984**, 430, 397–408. [Google Scholar] [CrossRef] - Nagata, S. Linkage between thermodynamic quantities and the uncertainty relation in harmonic oscillator model. Results Phys.
**2016**, 6, 946–951. [Google Scholar] [CrossRef][Green Version] - Rosenfeld, L. Ergodic Theories; Caldirola, P., Ed.; Academic Press: New York, NY, USA, 1961. [Google Scholar]
- Mandelbrot, B. The role of sufficiency and of estimation in thermodynamics. Ann. Math. Stat.
**1962**, 33, 1021–1038. [Google Scholar] [CrossRef] - Mandelbrot, B. An outline of a purely phenomenological theory of statistical thermodynamics–I: Canonical ensembles. IRE Trans. Inform. Theory
**1956**, 2, 190–203. [Google Scholar] [CrossRef] - Mandelbrot, B. On the derivation of statistical thermodynamics from purely phenomenological principles. J. Math. Phys.
**1964**, 5, 164–171. [Google Scholar] [CrossRef] - Lavenda, B.H. Thermodynamic uncertainty relations and irreversibility. Int. J. Theor. Phys.
**1987**, 26, 1069–1084. [Google Scholar] [CrossRef] - Lavenda, B.H. Bayesian approach to thermostatistics. Int. J. Theor. Phys.
**1988**, 27, 451–472. [Google Scholar] [CrossRef] - Lavenda, B.H. On the phenomenological basis of statistical thermodynamics. J. Phys. Chem. Solids
**1988**, 49, 685–693. [Google Scholar] [CrossRef] - Uffink, J.; van Lith, J. Thermodynamic uncertainty relations. Found. Phys.
**1999**, 29, 655–692. [Google Scholar] [CrossRef] - Pennini, F.; Plastino, A.; Plastino, A.R.; Casas, M. How fundamental is the character of thermal uncertainty relations? Phys. Lett. A
**2002**, 302, 156–162. [Google Scholar] [CrossRef][Green Version] - Pathria, R.K. Statistical Mechanics; Pergamon Press: Exeter, UK, 1993. [Google Scholar]
- Dodonov, V.V. Quantum variances. J. Opt. BA
**2001**, 4, S98. [Google Scholar] [CrossRef] - Wehrl, A. General properties of entropy. Rep. Math. Phys.
**1978**, 16, 221. [Google Scholar] [CrossRef] - Gnuzmann, S.; Życzkowski, K. Renyi-Wehrl entropies as measures of localization in phase space. J. Phys. A
**2001**, 34, 101233. [Google Scholar] - Anderson, A.; Halliwell, J.J. Information-theoretic measure of uncertainty due to quantum and thermal fluctuations. Phys. Rev. D
**1993**, 48, 275. [Google Scholar] [CrossRef][Green Version] - Glauber, R.J. Coherent and incoherent states of the radiation field. Phys. Rev.
**1963**, 131, 2766. [Google Scholar] [CrossRef] - Klauder, J.R.; Skagerstam, B.S. Coherent States; World Scientific: Singapore, 1985. [Google Scholar]
- Schnack, J. Thermodynamics of the harmonic oscillator using coherent states. Europhys. Lett.
**1999**, 45, 647. [Google Scholar] [CrossRef][Green Version] - Katz, A. Principles of Statistical Mechanics: The Information Theory Approach; Freeman and Co.: San Francisco, CA, USA, 1967. [Google Scholar]
- Husimi, K. Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn.
**1940**, 22, 264–314. [Google Scholar] - Lieb, E.H. Proof of an Entropy Conjecture of Wehrl. Commun. Math. Phys.
**1978**, 62, 35–41. [Google Scholar] [CrossRef] - Pennini, F.; Plastino, A. Heisenberg-Fisher thermal uncertainty measure. Phys. Rev. E
**2004**, 69, 057101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Scully, M.O. ; Zubairy, Quantum Optics; Cambridge University Press: Cambridge, NY, USA, 1997. [Google Scholar]
- Pennini, F.; Plastino, A. Power-law distributions and Fisher’s information measure. Physica A
**2004**, 334, 132–138. [Google Scholar] [CrossRef][Green Version] - Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Satuła, W.; Dobaczewski, J.; Nazarewicz, W. Odd-even staggering of nuclear masses: Pairing or shape effect? Phys. Rev. Lett.
**1998**, 81, 3599. [Google Scholar] [CrossRef][Green Version] - Dugett, T.; Bonche, P.; Heenen, P.H. Pairing correlations. II. Microscopic analysis of odd-even mass staggering in nuclei. J. Meyer
**2001**, 65, 014311. [Google Scholar] [CrossRef][Green Version] - Ring, P.; Schuck, P. The Nuclear Many-Body Problem; Springer: Berlin, Germany, 1980. [Google Scholar]
- Xu, F.R.; Wyss, R.; Walker, P.M. Mean-field and blocking effects on odd-even mass differences and rotational motion of nuclei. Phys. Rev. C
**1999**, 60, 051301(R). [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**Left panel**): statistical complexity C versus $\u0127\omega /{k}_{B}T$. The maximum C value is detected whenever the thermal energy equals the vibrational one, which happens at $\u0127\omega /{k}_{B}T=1$, as indicated by the vertical line. (

**Right panel**): C versus ${k}_{B}T/\u0127\omega $. In addition, the maximum is located in $\u0127\omega /{k}_{B}T=1$.

**Figure 2.**(

**Left panel**): thermal quantum quantifiers (TQFs) versus $\Delta x\phantom{\rule{0.166667em}{0ex}}\Delta p/\u0127$. Towards the right, we reach the classical limit. We can appreciate how the thermal quantifiers behave along such a route. (

**Right panel**): TQFs versus $\u0127/\Delta x\phantom{\rule{0.166667em}{0ex}}\Delta p$ expressed in ${k}_{B}$-units. Towards the left, we reach the classical limit. These plots should be compared to the corresponding ones displayed in Ref. [26].

**Figure 3.**(

**Left panel**): statistical complexity ${C}_{sc}$ versus $\u0127\omega /{k}_{B}T$. The vertical line indicates equality between the thermal and the vibrational energies. As the frequency $\omega $ grows relative to the temperature, the complexity becomes a constant. (

**Right panel**): structural quantifier ${C}_{sc}$ versus ${\Delta}_{\mu}x\phantom{\rule{0.166667em}{0ex}}{\Delta}_{\mu}p/\u0127$. The vertical line indicates optimal semi-classical localization.

**Figure 4.**(

**Left panel**): structural quantifier ${C}_{class}$ versus $\u0127\omega /{k}_{B}T$. The maximum is attained at $\u0127\omega /{k}_{B}T=1$, that is, equality between thermal and vibrational energies. (

**Right panel**): structural quantifier ${C}_{class}$ versus ${\Delta}_{class}x{\Delta}_{class}p=\u0127$. Remarkably enough, ${C}_{class}$ is maximal at the same uncertainty values that maximize its quantum counterpart.

**Figure 5.**Our three manners of calculating complexities (or structural quantifiers), C versus ${k}_{B}T/\u0127\omega $. The vertical line signals equality between the thermal and vibrational energies. As the temperature grows, the three manners tend to yield identical results.

**Figure 6.**${C}_{V}$ versus $V/T$ for $N=4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}5$.

**Figure 7.**$F)$ versus $V/T$ for $N=4\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}5$.

**Figure 8.**$S\left(J\right)$ versus $V/T$ for different J values. Since the fermion number $N=2J$, we detect a significantly distinct behavior according to whether the fermion number is even or odd.

**Figure 9.**$D\left(J\right)$ versus $V/T$ for different J values. Since the fermion number $N=2J$, we detect a significantly distinct behavior according to whether the fermion number is even or odd.

**Figure 10.**$C\left(J\right)$ versus $V/T$ for different J values. This displays a maximum that signals the phase transition. Since the fermion number $N=2J$, we detect a significantly distinct behavior according to whether the fermion number is even or odd.

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Pennini, F.; Plastino, A.; Plastino, A.R.; Hernando, A. Structural Statistical Quantifiers and Thermal Features of Quantum Systems. *Entropy* **2021**, *23*, 19.
https://doi.org/10.3390/e23010019

**AMA Style**

Pennini F, Plastino A, Plastino AR, Hernando A. Structural Statistical Quantifiers and Thermal Features of Quantum Systems. *Entropy*. 2021; 23(1):19.
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**Chicago/Turabian Style**

Pennini, Flavia, Angelo Plastino, Angel Ricardo Plastino, and Alberto Hernando. 2021. "Structural Statistical Quantifiers and Thermal Features of Quantum Systems" *Entropy* 23, no. 1: 19.
https://doi.org/10.3390/e23010019