and work interaction between system and reservoir. Therefore, making reference to the statement formulated by Gyftopoulos and Beretta  , the Highest-Generalized-Entropy Principle implies that among all states of a system characterized by given value of energy, number of constituents and parameters such as volume, there exist a unique stable equilibrium, according to Second Law statement here addressed to, and generalized entropy of this state is larger than the generalized entropy of any other state with the same value of energy, number of constituents and parameters.

On the other hand, if all states with same value of generalized entropy are considered, and assuming the Stable-Equilibrium-State Principle as the relationship between energy and entropy as reported in previous Section 3, one can infer the Lowest-Generalized-Energy Principle implying that among all states of a system characterized by given value of entropy, number of constituents and parameters, the generalized energy of the unique stable equilibrium state is lower than the generalized energy of any other state with the same value of generalized entropy, number of constituents and parameters.

8. Necessity and Sufficiency of Generalized Potential Equality

The present section describes the proof that generalized potential equality is a condition necessary and sufficient for stable equilibrium (or that stable equilibrium is a condition sufficient and necessary for generalized potential equality). This proof is included in a paper already published by the author  nevertheless is here again reported for sake of completeness and consistency as well as to better clarify the rationale behind the generalization of properties and principles here proposed.

According to the proof theory, deriving a thesis from a hypothesis implies the logical proof of hypothesis sufficiency and, vice versa, deriving the hypothesis from the thesis implies the logical proof of hypothesis necessity. The opposite is valid if thesis is replaced by hypothesis or, consistently, if hypothesis is replaced by thesis. The proof that equality of temperatures, potentials and pressures within the whole composite system-reservoir are necessary condition of stable equilibrium, hence “Equilibrium => Equality”, is described by Gyftopoulos and Beretta  who adopt the Highest-Entropy Principle to prove that temperature-potential-pressure equality is the consequence of subsystems individual-and-interacting stable equilibrium. With an opposite logical inference, the proof that individual-and-interacting stable equilibrium is inferred from temperature-potential-pressure equality is achieved by Gaggioli by adopting the Lowest-Energy Principle  . Therefore, in both cases, stable equilibrium is a sufficient condition for equality, hence, once again, “Equilibrium => Equality”. As sufficiency of equilibrium (or necessity of equality) is the sole condition established and proved and, on the other hand, the necessity of stable equilibrium (or sufficiency of equality) is not proved, then the system-reservoir composite can experience equality of temperature, potential and pressure while the composite itself is not in a stable equilibrium state since the equilibrium is not necessary as well (or equality is not sufficient as well) in contradiction to the assumed stable equilibrium. To resolve this contradiction, stable equilibrium has to be sufficient and necessary condition for equality of generalized potential or, vice versa, equality of generalized potential has to be necessary and sufficient condition for stable equilibrium. Reference can be made to the thermal, chemical, mechanical contributions of entropy, which is an additive property, so that the sum of these contributions constitutes generalized entropy which is the base of Highest-Generalized-Entropy Principle. To prove the necessity and sufficiency, without disproving the proofs already provided in the literature, one has to demonstrate that equality (or equilibrium) is necessary and sufficient conditions, thus Gaggioli’s statement is also necessary and Gyftopoulos and Beretta’s statement is also sufficient, both implying that the inference “Equality => Equilibrium” is complementary to the inference “Equilibrium => Equality” so that both equilibrium and equality are necessary and sufficient conditions for each other. In different terms, stable equilibrium is true if and only if generalized potential equality is true and vice versa generalized potential equality is true if and only if stable equilibrium is true. Figure 1 represents the hierarchical structure of the statement of necessity and sufficiency conditions.

Thus, Gaggioli adopts the Lowest-Energy Principle and Gyftopoulos and Beretta adopt the Highest-Entropy Principle, and since entropy depends on the difference between energy and generalized available energy which express the First Law and Second Law respectively, then entropy remains the suitable quantity to attain the proofs of both sufficiency and necessity. Therefore entropy property should account for equality of potential and equality of pressure, in addition to equality of temperature, in order to constitute the procedure for such a proof. First and Second principles are mutually correlated to each other: Lowest-Energy Principle ó Highest-Entropy Principle. In fact, the Stable-Equilibrium-State Principle establishes the mathematical relationship among all system properties at stable equilibrium. This relationship also exists between the Lowest-Energy Principle and the Highest-Entropy Principle which are intrinsic to the Stable-Equili- brium-State Principle as stated by the above fundamental relations  .

The proof of the necessity of stable equilibrium (or the sufficiency of generalized potential equality) consists of deriving stable equilibrium from equality and may be established through the “reductio ad absurdum” assuming that temperature, potential and pressure of system and reservoir are equal while the system-reservoir composite is not in stable equilibrium. Indeed, this condition is admitted by the sufficiency of stable equilibrium as the only

Figure 1. Hierarchical structure of logical relationship between stable equilibrium and generalized potential equality between system and reservoir.

condition which does not “necessarily” imply that stable equilibrium is a consequence of equality of temperature, potential and pressure in the system-reservoir composite so that the equality may be compatible with non-equi- librium. This equality of generalized potential would thus be able to move the system into a non-equilibrium state without undergoing any net change of the environment, or, would be able to generate a weight process by means of a PMM2 which is impossible according to the Second Law statement based on stable equilibrium as assumed. The consequence is that equality must imply stable equilibrium, that is, equality must be a sufficient condition for stable equilibrium (or stable equilibrium must be a necessary condition for equality). The proof of this sufficiency can be based on the Highest-Generalized-Entropy Principle where generalized entropy depends on thermal, chemical and mechanical contributions associated to heat, mass and work interactions between system and reservoir. Indeed, entropy property depends on temperature, potential and pressure, however this relationship does not “suffice” to prove stable equilibrium. For instance, adiabatic reversible process, namely isoentropic, causes changes in pressure as well as in temperature but equality of temperature does not in turn imply equality of pressure between system and reservoir. Instead, equality of temperature, potential and pressure associated to heat, mass and work interactions respectively, each individually identified as an additive contribution, ensures stable equilibrium. Adopting once again the paradigm of “reductio ad absurdum”, if equality of generalized potential is not associated to stable equilibrium, then the system can undergo a process due to whatever interaction, bringing it to stable equilibrium with an increase of entropy due to even one type of interaction moved by not equal potential that is impossible as equality is an assumption and being stable equilibrium associated to the highest generalized entropy. Having assumed the equality of generalized potential between system and reservoir, then generalized entropy has to assume the highest value with respect to any other state with non-null difference of temperature, potential and pressure. Therefore, each thermal, chemical and mechanical contribution of generalized entropy, has to be individually the highest. To do so, each individual heat, mass or work interaction, determined by the difference of temperature, potential and pressure respectively, has to be able to bring the system itself to the stable equilibrium state.

Finally, on the basis of the bi-univocal logical inference between stable equilibrium and generalized potential equality, and also considering the validity of Stable-Equilibrium-State Principle which correlates stable equilibrium with all properties characterizing the system in each thermodynamic state  , the Second Law statement can be enunciated in terms of existence and uniqueness of generalized potential equality state for each given value of energy content compatible with a given composition of constituents and compatible with a given set of parameters of any system. This statement can be extended to and remains valid also for neutral equilibrium  .

9. Conclusions and Future Developments

The result of the present study is twofold. Firstly, it consists of a proposal to encompass the Second Law statement by a bi-univocal logical inference between stable equilibrium, assumed as a hypothesis, and generalized potential equality derived as the thesis of a theorem of necessity and sufficiency of stable equilibrium for generalized potential equality in many-particle systems. The second result is that the proof is achieved by means of the definition of Highest-Generalized-Entropy Principle in which the generalized entropy is derived from generalized exergy property.

A consequence emerging from the generalized definition of entropy is that chemical entropy is an intrinsic and independent property of any system in any state and represents additional contribution, with respect to thermal entropy, determined by potential of the system itself characterized by its atomic and molecular configuration, regardless of the content of thermal entropy. Moreover, chemical entropy characterizes mass interactions of a system with a reservoir even in case no heat interactions are occurring while mass is entering or leaving the system. On the other hand, work interaction, representing the mechanical exergy, can also be correlated to the chemical exergy associated to an amount of mass interaction released to the system. As far as chemical entropy is concerned, future developments may be envisaged for atomic and molecular systems undergoing non-equili- brium phenomena determining specific geometrical configurations within the systems. In particular, Entropy Generation Minimization (EGM) methodology  - can include the chemical entropy to provide a system’s evolution schema suitable to describe organized structures and their self-assembling and self-organizing processes especially in living systems at molecular and cellular level  - . The definition of generalized entropy can be also extended to Quantum Physics domain to account for the contribution of quantum entropy  within the framework of Unified Quantum Theory of Mechanics and Thermodynamics.

Future developments may concern further discussions about generalized entropy, characterizing heat, mass and work interactions depending on temperature, potential and pressure here restricted to many-particle systems. An extension of this generalized paradigm to definitions, properties and theorems relating to few-particle systems and non-equilibrium states will be based on, and account for, recent treatments of thermodynamic entropy property   . This recent rigorous and formalized framework of definitions and theorems does not need the concept of reservoir to define thermodynamic entropy for each given value of energy content of the system which depends solely on the temperature. The extension to few-particle systems in non-equilibrium states would involve potential and pressure of the systems, in addition to temperature, aimed at generalizing the definition of thermodynamic entropy to all kinds and forms of potential characterizing matter at molecular, atomic and sub- nuclear level.

Nomenclature

: system

: system-reservoir composite

: energy, J

: exergy, J

: mass interaction, J

: chemical constituent, mol

: pressure, Pa

: heat interaction, J

: reservoir

: universal gas constant, J/mol×K

: entropy, J/K

: temperature, K

: internal energy, J

: volume, cubic m

: work interaction, J

: mole

Greek Symbols

: chemical potential, mol×K

: available energy, J

Subscripts and Superscripts

: composite system-reservoir

: chemical

: generalized

: irreversible

: mechanical

maximum

: minimum

: net

: reservoir

: reversible

: thermal

: initial state

: final state

: outward interaction flow

: inward interaction flow

Conflicts of Interest

The authors declare no conflicts of interest.

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