# Hierarchical Structure of Generalized Thermodynamic and Informational Entropy

## Abstract

**:**

## 1. Premise

## 2. Introduction

- (i)
- Thermodynamic foundations framework in the conception of Hatsopoulos, Gyftopoulos and Beretta [1,2,3,4] claim that thermodynamic entropy is an inherent property of matter in its broader sense related to any system, large or small, in any state, equilibrium or non-equilibrium, even at macroscopic non-statistical level with no need for any microscopic statistical rationale [5,6].
- (ii)
- The inherent character of entropy extends its validity to any scale of physical dimensions, hence classical and quantum mechanics equations of any particle are in compliance with the inherent essence and physical meaning of entropy including non-statistical and statistical methods of mechanics and thermodynamics [7]. In addition, quantum thermodynamics and the unified quantum theory of mechanics and thermodynamics [8,9,10,11] have demonstrated that irreducible uncertainties and probabilistic nature of phenomena are the ultimate root causes of irreversibility existing in microscopic dynamics.
- (iii)
- Nevertheless, according to an information-based conception, a different school of thought has devised proofs that information and Shannon informational entropy [12,13] are in turn inherently associated to physic states of matter, as demonstrated by Jaynes [14,15], Landauer [16,17], and Karnani, Paakkonen and Annila [18]. Therefore, Boltzmann and Gibbs statistical entropy are correlated to Shannon entropy and this relationship is not only a formal correspondence and homology. Both thermodynamic aspect and informational aspects are inherent to any system in any state and the implication of quantum mechanics in quantum information theory advocates this principle [19]. Indeed, informational entropy is in turn an inherent property of matter as any physical state is characterized by an amount of information and a corresponding amount of uncertainty that depends on the scale of the system up to quantum where Heisenberg indetermination principle constitutes a physical fundamental. The statement that information is a physical entity does not disprove that entropy is an inherent property of matter. Instead, both represent different expressions of a unique fundamental and elementary characteristic of the phenomenological physical reality.
- (iv)
- Information is an inherent property of any system in any state since it is associated to the state of properties. Consequently, the relationship between thermodynamic and informational viewpoints represent an intrinsic property of any system in any state being the two viewpoints coexisting and complementary. Any microscopic up to macroscopic scale of classical (and non-statistical) thermodynamics is affected by this correlation and generalization of theorems or properties can be adopted in the domain of information theory, as explained by Kafri [20].

## 3. Considerations on Physical Aspect of Second Law and Thermodynamic Entropy

- (1)
- The Second Law statement is based on the existence and uniqueness of stable equilibrium.
- (2)
- Stable equilibrium implies thermal equilibrium, chemical equilibrium and mechanical equilibrium.
- (3)
- Corollary of stable equilibrium is the impossibility of Perpetual Motion Machine of the Second Kind (PMM2).
- (4)
- PMM2 is adopted in the proof of the entropy definition related to temperature, hence it is the definition of a thermal entropy property.
- (5)
- Highest-(thermal)-entropy principle is applied to prove that stable equilibrium implies the equality of temperature, potential and pressure while thermal entropy determines thermal energy and heat interaction only, this representing a logical incompleteness and inconsistency thus introducing an incongruity [29,30,31].
- (6)
- To remove the incongruity, equality of temperature, potential and pressure have to imply thermal, chemical and mechanical equilibria and this opposite proof needs chemical entropy and mechanical entropy, in addition to thermal entropy, to assert a highest-generalized-entropy principle to be used in the proof [29,30,31].

## 4. Second Law Statements Related to Thermal or Chemical Potentials

## 5. Perpetual Motion Machines of Second Kind (PMM2) as a Corollary of Second Law

#### 5.1. Thermal-Mechanical PMM2

- 1.
- Mechanical aspect of non-existence of PMM2 performing an ideal direct heat-to-work conversion cycle implies that it is not possible to convert a given amount of thermal energy at high temperature into mechanical energy with no production of thermal energy at lower temperature (Kelvin–Planck and Poincaré); in this case, the PMM2 canonical efficiency is ${\eta}^{\begin{array}{l}DIRECT\\ THERMAL\end{array}}=1-\frac{{T}_{R}}{T}$.
- 2.
- Thermal aspect of non-existence of PMM2 performing an ideal inverse work-to-heat conversion cycle implies that it is not possible to convert a given amount of thermal energy at low temperature into thermal energy at high temperature with no contribution of mechanical energy input (Clausiu and Thompson).
- 3.
- Mechanical aspect of non-existence of PMM2 performing an ideal inverse work-to-heat conversion cycle implies that it is not possible to convert a given amount of mechanical energy at high-pressure-low-volume into thermal energy with no production of mechanical energy at low-pressure-high-volume.
- 4.
- Thermal aspect of non-existence of PMM2 performing an ideal direct heat-to-work conversion cycle implies that it is not possible to convert a given amount of mechanical energy at low-pressure-high-volume into mechanical energy at high-pressure-low-volume with no contribution of thermal energy input.

#### 5.2. Chemical-Mechanical PMM2

- 5.
- Mechanical aspect of non-existence of PMM2 performing an ideal direct mass-to-work conversion cycle implies that it is not possible to convert a given amount of chemical energy at high potential into mechanical energy with no production of chemical energy at lower potential; in this case, the PMM2 canonical efficiency is ${\eta}^{\begin{array}{l}DIRECT\\ CHEMICAL\end{array}}=1-\frac{{\mu}_{R}}{\mu}$.
- 6.
- Chemical aspect of non-existence of PMM2 performing an ideal inverse work-to-mass conversion cycle implies that it is not possible to convert a given amount of chemical energy at low potential into chemical energy at high potential with no contribution of mechanical energy input.
- 7.
- Mechanical aspect of non-existence of PMM2 performing an ideal inverse work-to-mass conversion cycle implies that it is not possible to convert a given amount of mechanical energy at high-pressure-low-volume into chemical energy with no production of mechanical energy at low-pressure-high-volume.
- 8.
- Chemical aspect of non-existence of PMM2 performing an ideal direct mass-to-work conversion cycle implies that it is not possible to convert a given amount of mechanical energy at low-pressure-high-volume into mechanical energy at high-pressure-low-volume with no contribution of chemical energy input.

#### 5.3. Physical Meaning of PMM2 Impossibility

## 6. Hierarchical Configuration and Levels of Multiscale Mesoscopic Systems

#### 6.1. Maxwell’s Demon and Degrees of Freedom

#### 6.2. Degrees of Freedom and Hierarchical Levels

## 7. Generalized Thermodynamic Entropy and Exergy Properties

#### 7.1. Thermodynamic Entropy Components

#### 7.2. Exergy Contributions

## 8. Hierarchical Structure of Thermodynamic Entropy and Exergy Properties

- $P$: determined by the kinetic energy and potential energy of particles per unit of volume;
- $T$: kinetic energy per unit of particles; and
- $\mu $: potential energy per unit of mole.

- (a)
- Zeroth Hierarchical Level (HL0): The system is considered as a macroscopic rigid whole.
- (b)
- First Molecular Hierarchical Level (HL1): Phase-Constituent, the macroscopic system is considered as a set of atoms and/or molecules; $-PV$ (mechanical internal energy); translational–rotational kinetic energy; and translational–rotational potential energy.
- (c)
- Second Sub-Molecular Hierarchical Level (HL2). Atoms and sub-molecules, as component elements of molecules, constitute a microscopic system: $+TS+\mu n$ (vibration-translational/vibration-rotational kinetic energy and vibration-translational/vibration-rotational potential energy).
- (d)
- Third Nuclear Hierarchical Level (HL3). Nucleons (protons and neutron) and electrons constitute atoms or group of atoms of group of molecules: ${E}_{HL3}^{KINETIC}+{E}_{HL3}^{POTENTIAL}$(vibration-translational/vibration-rotational kinetic energy and vibration-translational/vibration-rotational potential energy).
- (e)
- Fourth Sub-Nuclear Hierarchical Level (HL4). Sub-nuclear hadrons and particles constitute a nucleus: ${E}_{HL4}^{KINETIC}+{E}_{HL4}^{POTENTIAL}$ (vibration-translational/vibration-rotational kinetic energy and vibration-translational/vibration-rotational potential energy).

#### 8.1. Macroscopic Level

#### 8.2. Microscopic Level

## 9. Non-Equipartition Theorem of Entropy

- Equipartition theorem of energy: Reversible and irreversible conversion processes and maximum entropy production principle $\Rightarrow $ multi-scale configuration of systems emerging from energy dissipation along hierarchical levels.
- Non-equipartition theorem of entropy: Reversible and irreversible conversion processes and entropy generation minimization paradigm $\Rightarrow $ constructive evolution of systems through self-organizing capability and shaping of optimized hierarchical configurations.

## 10. Conclusions

## Funding

## Conflicts of Interest

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Palazzo, P. Hierarchical Structure of Generalized Thermodynamic and Informational Entropy. *Entropy* **2018**, *20*, 553.
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Palazzo, Pierfrancesco. 2018. "Hierarchical Structure of Generalized Thermodynamic and Informational Entropy" *Entropy* 20, no. 8: 553.
https://doi.org/10.3390/e20080553