Supplementary commentary (V) on my thermodynamic draft paper

Supplementary commentary (V) on my thermodynamic draft paper for Thermodynamic time determination under thermodynamic spacetime along with mathematically interpreted classical thermodynamics (as Draft V) Minoru Myojo Takatsuki Research Site for Thermal Science (Date: February 25, 2024) The discipline of physical chemistry, strongly influenced by the academic achievements of thermodynamics and quantum mechanics, has developed as a tool to guide theoretical verification of quantum mechanics in the right direction. Physical chemistry has two sides: one deals with classical particles, and the other deals with elementary particles such as electrons in accordance with quantum theory. However, the distinction between the two is not so clear. When dealing with the concept of temperature in physical chemistry, it is inevitable that it is defined within a large number of particles in thermal equilibrium, and that it is a particle population that allows for the stochastic element of the Boltzmann distribution and for the statistical processing. If these arguments are to be considered in connection with the “quantification of thermodynamic time” of special relativity in the field of thermodynamics, it will be necessary to sort out some issues. 1. Regarding the temperature concept brought about by physical chemistry When dealing with the concept of temperature in the discipline of physical chemistry, it seems most appropriate to introduce the following chart from the author of Reference #21 in my thermodynamics draft, which itemizes the points made by the author, W.J. Moore, in his textbook and leads to the following discussion. Eventually, it comes down to the important issue that the Boltzmann distribution gives a molecular definition of temperature. Statistical thermodynamics (§12 in Basic Physical Chemistry 21)) Kinetic properties of molecule i Statistical mechanical properties of the system position xi, yi, zi temperature T momentum pxi, pyi, pzi pressure P mass mi mass M kinetical energy Eki entropy S [statistical mechanics] potential energy Epij internal energy U between molecules i and j gibbs free energy G - According to W.J. Moore’s textbook of physical chemistry, the third law of statistical thermodynamics provides S0 = k ln g0 by applying the canonical ensemble partition functions Z (V, T) (= gi exp(‒Ei/kT)) as sums of energy levels with degeneracy factors and 1 the relation of lim → = 0 in the limit T → 0. - Many polyatomic molecules that are different from monoatomic molecules have internal and vibrational degrees of freedom, and the energy of the molecule is classified into translational, rotational, or vibrational types. - Regarding the relationship between the Boltzmann distribution and temperature, it is important to pay attention to the electron distribution in addition to the translational, rotational, and vibrational distributions mentioned above (see Sections 2 and 4). 2. Quantum perspective in physical chemistry In physical chemistry, it has become mainstream to describe molecular energy with the help of quantum theory. - Physical chemistry requires a quantum-mechanical interpretation of the energy of molecules, including the classical treatment of heat capacity measurements. - Physical chemistry provides a quantum-mechanical explanation for the energy levels of molecules and demonstrates how experimental data obtained by spectroscopic measurements can be explained by this theory. - Understanding how individual molecules are distributed in their energy levels and how the concept of heat capacity is interpreted in molecular terms will lead to an understanding of the “concept of thermal equilibrium,” a fundamental concept in thermodynamics. - Internal energy includes not only thermal energy, which refers to the energy generated by the movement of atoms, molecules, and ions that constitute a system, but also intermolecular energy, which acts between molecules involved in such as the evaporation of liquid, and chemical energy, which changes chemical bonds due to the interaction between electrons in atoms, molecules, and ions and atomic nuclei. The study of the interaction between light and matter is called spectroscopy, and it is expressed by the energy levels of molecules as seen through experiments, that is, molecular spectroscopy. - The quantum level of a molecule can be clearly expressed as the energy level of a molecule seen experimentally through spectroscopy. - Moore’s textbook clearly shows the “relationship between vibrational energy and molecular dissociation (in his Figure 12⋅4)” using the rotational quantum number J in spectroscopy. 3. Ergodic hypothesis: Relationship between ensemble average and time average The Moore’s textbook sets two assumptions regarding the relationship between ensemble average and time average. Here, let me show you the first assumption of them: 2 - The average value of the kinetic variable M taken over a long time for a real system is equal to the ensemble average of M if and only if the set of systems reproduces exactly the thermodynamic state of the real system in question and its surrounding systems. Strictly speaking, this assumption is correct only in the limit N → ∞. The advantage of this set idea is that the unfeasible infinite time average is replaced by a feasible ensemble average. However, in practice, it is said that it cannot be proven that the ensemble average is equal to the time average. The author of the thermodynamics draft visually illustrated the perspective of this ergodic hypothesis below. Understanding the ergodic hypothesis by applying the following assumptions Assumption 1 Let us assume that almost all of the particles that make up the Maxwellian distribution are invisible, but that only one particle is visible, and we will follow its changes over time. Assumption 2 View each of the three planes, x-y, y-z, and z-x, on the origin from the time axis direction after a certain amount of time has elapsed. (This presentation was held in LS15: the 15th International Symposium on the Science and Technology of Lighting in Kyoto in May 22-27, 2016.) This approach may still be far from equivalence between ensemble and time averages. Depending on how the hypothesis is formulated, the “proof approach” based on visual verification has neither been confirmed nor denied. It can be said that the existence of the ergodic hypothesis increases the value of expressing infinity using the number of classical particles (N → ∞), and further increases the value of applying the Boltzmann relation S = k lnWB that he initiated. 4. A commentary of W.J. Moore left in his textbook “§5⋅6 Molecular interpretation of temperature” of “§5 Boltzmann distribution and temperature” in W.J. Moore’s textbook describes his thought as an essential interpretation of temperature as a molecular theory. It is shown as follows: - This is a final note on the essential interpretation of the temperature. It is a quantity defined only for a large number of particles in thermal equilibrium. The conditions highlighted in bold are important here. The concept of temperature is based on the Boltzmann distribution. This distribution shown in §5⋅4 has statistical significance 3 because it was derived using the idea of probability. Thus, it applies only to large ensembles of particles that are amenable to statistical treatment. (This expression comes from Japanese translation.) This idea of Moore will be discussed in Section 5 below. 5. Discussion - Background of the author’s involvement in physical chemistry We have seen how the field of physical chemistry has been supported by quantum theory and has focused on describing physical and chemical phenomena involving electrons. The author of the thermodynamics draft himself was familiar with Moore’s textbook (Physical Chemistry, the predecessor of Reference #21) when he was a student and was overwhelmed by the contents of the textbook. However, on the other hand, he became growing dissatisfied with the fact that the “thermodynamic time” that Eddington had ever proposed, which should be described in the field of thermodynamics, had never been taken up. The purpose of my personal approach was to perform microscopic statistical processing to clarify Fig 2 of the thermodynamic draft. - Regarding the difference between the handling of electrons in physical chemistry and the conclusion in Fig 2 in thermodynamics When classical thermodynamics deals with classical particles, the number N of particles is expressed as 1 ≤ N ≤ ∞, but in physical chemistry there is no descriptive convention regarding the number of particles when it deals with classical particles and electrons. The effect of this seems to depend largely on the operation that converts the classical particle in Fig 2(a) to the quantum particles in Fig 2(b) in quantum thermodynamics. The point to be noted here is that the vertical axis in Fig 2 has been changed according to the particle correspondence for each purpose, but the horizontal axis in Fig 2 has kept the number of classical particles for statistical processing fixed. The size of elementary particles such as electrons is given as 0 < N < 1 and the number of elementary particles can be estimated as that in corresponding to the size of classical particles in Fig 2(b). It seems that such efforts have not been discussed in physical chemistry so far, but it can be understood that they are necessary to count the “initial number” using the special relativity for the quantification of thermodynamic time. - Other elements deemed necessary for discussion. As is mentioned in §5⋅6 of W.J. Moore’s textbook, the temperature concept is required to be a quantity defined only for a large number of particles in thermal equilibrium. Namely, it is understood that it applies only to a large population of particles for which the Boltzmann distribution allows for the statistical treatment. Such an understanding seems to be closely linked to: • The ergodic hypothesis continues to exist as a hypothesis, • The value of treating the infinite limit with the number of classical particles (N → ∞), 4 and • Significance of existence of the Boltzmann relation S = k lnWB (defined by Boltzmann). 6. Conclusion While the number N of classical particles is expressed as 1 ≤ N ≤ ∞, the size of elementary particles such as electrons is expressed as 0 < N < 1, and at the same time, the number of elementary particles corresponds to the number of classical particles appearing in Fig 2(b). When considering introducing the concept of quantifying thermodynamic time into the fields of thermodynamics and physical chemistry, the Boltzmann relation S = k lnWB gives the thermodynamic limit state pV = TS and the statistical mechanical “initial number” in thermodynamics. This allows for a discussion of the special relativity to quantify the thermodynamic time. My previous first proposal on the commentary (IV) on Dec. 21, 2023 In this updated commentary, we will discuss two matters. The first point concerns that the difference between a Boltzmann-like description and a Plancklike one in the Boltzmann relation, which deals with probabilistic elements, must be emphasized in the thermodynamics draft paper. We need to explain the Boltzmann relation S = k lnW, which shows the difference between a Boltzmann-like description and a Planck-like one. The first public indication of that was in the 2014 material. However, I would like to point out here that the material presented at one workshop not only contained many errors, but also had problems with the content of that abstract. No matter how much progress has been made in the mathematical interpretation of thermodynamic theory, the value of the empirical knowledge accumulated so far by classical thermodynamic theory remains intact. On the contrary, it also seems that one perspective of the lack of progress in the mathematical interpretation of thermodynamic theory appeared in the abstract of the 2014 article. The real value of this knowledge was mentioned at the end of the conclusion of the thermodynamics draft paper. A revised version of the schematic diagram describing the Boltzmann relation S = k lnW and its related matters, which describes the difference between a Boltzmann-like description and a PlanckAppendix C like one, is shown in the second sheet below (see Footnote #3 in the draft). After showing a revised version of Figure 15 in the 2014 article this time, the posting of that article would be dropped. The relationship between the 2014 article the author recently posted for article introduction and his way of thinking about classical thermodynamics If only the abstract of the 2014 article is applied, it may cause misunderstandings. The author's real intention at that time was simply to point out the lack of efforts to approach thermodynamic theory from a mathematical scientific perspective. There has been no slight change in my respect for the empirical knowledge of classical thermodynamics. 5 Please allow me to introduce here the last part of “V. CONCLUSION” of the thermodynamics draft paper again: “Those contents were summarized as follows in light of the claims of this paper. The basic part of the thermodynamic theory, which is supported by empirical knowledge such as thermodynamic principles and laws, has been successfully explained using mathematical interpretation. This success shows that the mathematical interpretation in natural scientific field can contribute to the further development of thermodynamic research, not by denying the empirical knowledge, but by further respecting it.” A revised version of the schematic diagram describing the Boltzmann relation S = k lnWB and its related matters shown in Figure15 of the 2014 article pV = nRT pV = NkT Nothing When N = 0 Macroscopic When N = NA n = 1, then pV = RT Microscopic When N = 1 (T n = 0, then pV = 0 pV = 0 (Everything is nothing, including energy.) pV = NAkT = RT pV = (1/NA)RT = kT pV = kT (There exists energy when N = 1.) 0) T (N 0 The third law of 1) thermodynamics lim S = S0 ≡ 0 T→0 WP = 1 WB = 1 S = k ln WP S=0 N S=k N = 1, WB = 1 (T Limit point N = 1 (N, WB, Z) = (1, 1, 1) S = k ln(e‒1+WB) (see Figure 1(d) in the thermodynamics draft) Quasistatic (N = 5.5) p = (N/V)kT 0) 0) pV = ST (quasistatic) (S = 5.5k) S = k ln(e‒1+WB) (in the initial number region N) 1, WP = 1 (T S = k lnWB (macroscopic) pV N = 5.5, ln(e‒1+WB) = 5.5 N ln e 1 WB ST Both Boltzmann relations, S = k lnWB and S = k lnWP, allow for probabilistic descriptions. The former Boltzmann relation S = k lnWB allows for probability expression that excel in macroscopic statistical analysis over countable classical particle numbers. On the other hand, the latter Boltzmann relation S = k lnWP makes it possible to describe zero entropy (S → 0; the third law of thermodynamics) depending on the probability of WP = 1 from an uncountable temperature element that tends toward zero (T → 0). The author of the thermodynamics draft paper believed that statistical analysis would be possible even by using the small number of classical particles in the microscopic world in the same way as in the macroscopic world. This was made possible by adopting the Boltzmann relation S = k lnWB and presenting the quasistatic condition pV = TS (i.e., lnW = N). The Boltzmann relation, S = k lnWB, which is distinguished from S = k lnWP, may help us make statistical analysis even in microscopic regions of thermodynamics more freely applicable. My previous second proposal on the commentary (IV) on Dec. 21, 2023 The second point concerns why F.W. Ostwald, who was one of researchers taking a skeptical 6 stand on the existence of atoms and molecules, came to appear as a translator of Gibbs' paper, which contributed to the development of statistical mechanics after Maxwell and Boltzmann. The material for the discussion is based on an English "afterword", translated from the Japanese, involving two volumes (Gas Molecular Kinetics and Statistical Mechanics) of a series of classical physical papers by Maxwell, Loschmidt, Zelomelo, and Boltzmann, in which the location of the original articles is specified. It is hoped that the relevant thermodynamic paper will provide useful perspectives that encourage further discussion. Afterword "Commentary of Gas Molecular Kinetics" in Volume 5 of a series of Classical Physical Papers (Tokai University Press) The papers in this volume consist of three papers by Maxwell on molecular motion and five papers created by Loschmidt, Zelomelo, and Boltzmann around two papers by Boltzmann on kinetic thermal theory (i.e., a 1872 paper on the H-theorem and others, and a 1877 paper on the probabilistic interpretation of statistical mechanics, now expressed in the relation S = k logW (same as S = k lnW), both of which are included in Volume 6 of this series). Therefore, by reading this volume together with Volume 6 (Statistical Mechanics), it is possible to trace the transition and development of the theory of gas molecular motion up to the completion of Gibbs' statistical mechanics. The theory of gas molecular motion traces dynamically a group of mass points or rigid bodies subject to the classical mechanical laws of motion, which are exerting various interactions. The main problems that have appeared in the transition and development of the group are the determination of the distribution function at the equilibrium state and the guarantee of reaching the equilibrium state. What is known today as the Maxwell-Boltzmann distribution law relates to the former, and the Boltzmann’s H-theorem and the theory of probabilistic interpretation discussed above relate to the latter. The first paper of this volume is the first to give the distribution function in gas molecular kinetics. The so-called Maxwell's velocity distribution law is also given here. However, the velocity distribution law is derived here based on the spatial isotropy of the distribution of a group consisting of elastic hard spheres. The dynamic determination of the distribution function corresponding to the steady state of a mass point or a group of rigid bodies moving while exerting interaction can be seen in the second paper. In the year of the publication of the second paper (1866), Boltzmann published the first paper of his kinetic thermal theory (included in Volume 6 of this series), and after that, his theory was further developed. The derivation of the Maxwell-Boltzmann distribution, which generalizes the Maxwell distribution, is also a result of this period, and Boltzmann's theorem mentioned in the title of the third paper refers to Boltzmann's work on the distribution of kinetic energy at equilibrium. Maxwell formulated this work more generally and more beautifully in the third paper. In the course of this formulation, the concept of ensemble was introduced for the first time, giving a clear distinction between dynamical and statistical considerations, which, of course, were inherited by Gibbs and become the basic concepts of his statistical mechanics. Statistical mechanics at equilibrium was completed to a certain extent by Gibbs' "Elementary Principles in Statistical Mechanics" (1902). The meaning of the fact that it gave the basic principle is that it gave the general relational expression that should be satisfied when a group consisting of fundamental particles such as atoms and molecules is in equilibrium in a form that is independent of the properties of specific particles. In other words, the completion of Gibbs' statistical mechanics of equilibrium provides a basic principle that does not require or is not related to various assumptions made in the course of the development of the gas molecular motion theory regarding the interaction between particles and others. In this sense, Maxwell's third paper directed the gas molecular motion theory toward a more general principle. On the other hand, Boltzmann's paper on the H-theorem (1872, in Volume 6 of this series) was intended to provide a dynamical guarantee that a population of particles in a nonequilibrium state should reach an equilibrium state irreversibly, not to mention Boltzmann's ambitious work aimed 7 at the dynamical deduction of the second law. The fourth paper is a rebuttal by Loschmidt to this paper. Loschmidt disproved the generality of Boltzmann's H-theorem by specifically pointing out instances where the H-theorem does not hold, which meant that the H-theorem for Boltzmanns state, although fairly general, still contained some special assumptions. Boltzmann's answer to this is in his 1877 paper (included in Volume 6 of this series), in which he introduced the concept of Komplexionen and gave the expression for entropy, given today as S = k logW (same as S = k lnW). This equilibrium is a state realized with overwhelmingly high frequency, and the distribution functions at it are described in correspondence with that overwhelmingly high frequency state. The sixth to eighth papers are the counterargument by Zelomelo to Boltzmann's theory and Boltzmann's argument on the counterargument, which examined the problem on the irreversible process of the dynamical state of the system based on Poincare's theory of three body problem and criticized Boltzmann's explanation of recursion. As mentioned above, with the establishment of Gibbs' statistical mechanics, statistical mechanics at equilibrium is formally completed in a form corresponding to thermodynamic theory at equilibrium, and since the theoretical form is independent of individual molecular processes, the molecular kinetic theory before Gibbs does not appear on the surface as far as problems at equilibrium are concerned. However, molecular kinetic theory has reappeared on the surface of various problems belonging to the series of development of thermodynamics and statistical mechanics of nonequilibrium state, and the significance of molecular kinetic theory with respect to plasma and other theories is rather increasing in this sense today, and I believe that a compilation of the classic representative papers of molecular kinetic theory can give considerable suggestions. (Commentator: W. Taniguchi) References in Volume 5 (Gas Molecular Kinetics) J. C. Maxwell, “Illustrations of the Dynamical Theory of Gases”, Phil. Mag., (4)19, 1932(Part I), 20, 21-33(Part II), 33-37(Part III), (1860); Scientific Papers, 1, 377-409 (Dover pub.) J. C. Maxwell, “On the Dynamical Theory of Gases”, Phil. Trans. R. Soc. London, 157, 4988,(1867); Phil. Mag., (4) 35, 129-145, 185-217, (1868); Scientific Papers, 2, 26-78 (Dover pub.) J. C. Maxwell, “On Boltzmann's Theorem on the average distribution of energy in a system of material points”, Cambridge Philosophical Society's Transaction., XII., (1879); Scientific Papers, 2, 713-741 (Dover pub.) J. Loschmidt, “Über den Zustant des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft”, Wiener Berichte 73, 128-142 (1876), 366-372 (1877), 287298 (1877) L. Boltzmann, “Bemerkungen über einige Probleme der mechanischen Wärmetheorie”, Wiener Berichte 75, 62-100 (1877). Wissendhaftche Abhandlungen von Ludeig Boltznann, 112-148, Leipzig, (1909). E. Zermelo, “Ueber einen Satz der Dynamik und die mechanische Wärmetheorie”, Wiedemann Annalen 57, 485-494 (1896). L. Boltzmann, “Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo”, Wiedemann Annalen 57, 773-784 (1896). Wissendhaftche Abhandlungen von Ludeig Boltznann, 567-578, Leipzig, (1909). E. Zermelo, Ueber mechanische Erklärungen irreversibler Vorgänge. Eine Autwork auf Hrn. Boltzmann’s „Entgegnung”, Wiedemann Annalen 59, 793-801 (1896). References in Volume 6 (Statistical Mechanics) L. Boltzmann, “Über die mechanishe Bedentung des zweiten Hauptsatzes der Wärmetheorie”, Wiener Berichte 53, 195-220 (1866). L. Boltzmann, “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”, Wiener Berichte 63, 275-370 (1872). L. Boltzmann, “Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht”, Wiener Berichte 76, 373-435 (1877). (J.W. Gibbs' references that have presented as #16‒#18 in my thermodynamics draft should be added 8 here.) The author would address the questions and other issues raised in our scientific matters according to the above commentary. After confirming almost all points, the author would show you the significant viewpoints that need to be summarized, along with useful messages. Summary of the above explanation • The transition and development of the theory of gas molecular motion up to the completion of Gibbs' statistical mechanics - The Maxwell’s first paper (1860) gave the distribution function in gas molecular kinetics. - The Maxwell’s second paper (1866) gave the dynamic determination of the distribution function corresponding to the steady state of a mass point or a group of rigid bodies. - Boltzmann published the first paper (1866) of his kinetic thermal theory, and after that, his theory was further developed. - Boltzmann’s paper on the H-theorem (1872) was intended to provide a dynamical guarantee that a population of particles in a nonequilibrium state should reach an equilibrium state irreversibly, not to mention Boltzmann’s ambitious work aimed at the dynamical deduction of the second law. - Boltzmann's answer to the Loschmidt’s question is in his 1877 paper, in which he introduced the concept of Komplexionen and gave the expression for entropy, given today as S = k logW (same as S = k lnW). - The Maxwell’s third paper (1879) directed the gas molecular motion theory toward a more general principle. The derivation of the Maxwell-Boltzmann distribution, which generalizes the Maxwell distribution, is also a result of this period, and Boltzmann’s theorem mentioned in the title of the third paper refers to Boltzmann’s work on the distribution of kinetic energy at equilibrium. - In the course of the formulation of the Maxwell-Boltzmann distribution by Maxwell, the concept of ensemble was introduced for the first time, giving a clear distinction between dynamical and statistical considerations, which, of course, were inherited by Gibbs and become the basic concepts of his statistical mechanics. • What the completion of Gibbs' statistical mechanics brought about - Statistical mechanics at equilibrium was completed to a certain extent by Gibbs' "Elementary Principles in Statistical Mechanics" (1902). - The meaning of the fact that it gave the basic principle is that it gave the general relational expression that should be satisfied when a group consisting of fundamental particles such as atoms and molecules is in equilibrium in a form that is independent of the properties of specific particles. In other words, the completion of Gibbs' statistical mechanics of equilibrium provides a basic principle that does not require or is not related to various assumptions made in the course of the development of the gas molecular motion theory regarding the interaction between particles and others. - As mentioned above, with the establishment of Gibbs' statistical mechanics, statistical mechanics at equilibrium is formally completed in a form corresponding to thermodynamic theory at equilibrium, and since the theoretical form is independent of individual molecular processes, the molecular kinetic theory before Gibbs does not appear on the surface as far as problems at equilibrium are concerned. - Under such situations where Maxwell and Boltzmann studied the fundamental theory of statistical mechanics from the point of view of gas molecular kinetics and Maxwell presented the ensemble concept, Gibbs established statistical mechanics at equilibrium, and he contributed to the expansion of the field of physical chemistry and its theoretical establishment by utilizing the graphical or geometric representation of thermodynamics in equilibrium systems. 9 Useful messages (related matters or Q & A) • The author wonder why Ostwald could have been an English translator into German. - If you know Gibbs' work as a contributor to physical chemistry theory, you wouldn't question the fact that the physical chemist Ostwald was the translator of the Gibbs paper into German. Before contributing to physical chemistry theory, however, Gibbs served as a sort of successor to Boltzmann, who believed in the existence of atoms and molecules and studied statistical mechanics. Ostwald was part of a group that did not believe in the existence of atoms and molecules. Such questions may arise when you look at Figure 1 of the thermodynamics draft paper. • Has it been understood that Maxwell's and Boltzmann's work on gas molecular kinetics and Gibbs' work on equilibrium statistical mechanics are in part of a series of academic systems? - Both studies have concepts connected to statistical mechanics of equilibrium systems, but Gibbs' one has the advantage of being constructed with a theory that does not depend on the interactions required by the other, and that it provides basic principles. However, the ensemble concept emerged from the equilibrium concept of particle populations deriving to the Maxwell-Boltzmann distribution, which might have helped Gibbs for his theoretical construction. - It gave the impression that Gibbs' work on equilibrium statistical mechanics was separate from Maxwell's and Boltzmann's work on gas molecular kinetics, and the effect seemed to extend to Ostwald. The fundamental principles acquired by Gibbs may have concealed Boltzmann's academic troubles, while on the other hand they contributed to the development of the theory of physical chemistry. • Can we say that the (static) tangent planes shown in Figures 1(a) and 1(c) of the thermodynamics draft paper follow Gibbs' descriptive format of statistical mechanics? - At least Figure 1(c) is in an equilibrium state under isothermal and isobaric conditions, which satisfies Gibbs' description conditions for statistical analysis. - Original Boltzmann relation S = k lnWB (that is initiated by Boltzmann), which describes a countable element in probability, would allow us to express the thermodynamic microworld in Figure 1(a), which allows us to interpret mathematically quasistatic behaviors pV = TS (i.e., lnWB = N) in quasistatic processes handled in classical thermodynamics. - In the thermodynamic draft paper, the thermodynamic lower limit to this completion of Gibbs' equilibrium statistical mechanics points to the "static tangent plane of Figure 1(a)," which helps explain the finite-length magnitude of the thermodynamic infinitesimal of Figure 5. • How many thermodynamic limits are there? - The first fundamental thermodynamic limit lies in the special concept of infinity (N → ∞). - The second one lies in the thermodynamic lower limit (N = 5.5) that allows for providing a finiteness of thermodynamic (thermodynamically functionalized) infinitesimal. That can be verified through a statistical analysis using S = k lnWB. - There is a limit having a minimized countable classical particle (N = 1) just before disappearing. The statistical analysis up to N → 1 using S = k lnWB (together with ln(e‒1+WB) = N) will be enabled. - There is a separate limit where the entropy is zero (S → 0), which is unrelated to the size of the number of classical particles and is based on the absolute zero temperature (T → 0) as an uncountable element. 10 My previous commentary (III) on December 2, 2023 After revising my thermodynamic draft and posting it on November 27, 2023, I created a document like a cover letter to a journal. All changes made since the original document are shown in green text. I will also leave the previous two commentaries for reference. In this supplementary commentary, let me show you how the quantification of thermodynamic time using the Boltzmann relation S = k lnW in the statistical analyses of Appendix A has been Appendix C carried out (see footnote #3 and the relevant part of Appendix B). After that, the author will show the difference between the special relativity handled in the quantum theory and the special relativity handled under the extreme region in thermodynamics, and he will take the attitude of watching the future evolution of the thermodynamic theory with reference to the completion of the quantum theory. Before the development and perfection of quantum theory, quantum physicists apparently viewed statistical mechanics, including the Boltzmann relation in thermodynamics, as a rival discipline in finding the special relativity for the study to microscopic regions. It is not certain that the same idea existed on the part of the thermodynamics scholars of the time, but at least they seem to have been aware of the term “thermodynamic time” that emanated from A.S. Eddington. Appendix C Footnote #3 Related parts “Relationship between the Boltzmann relation S = k lnW and quantum mechanics” in Appendix B Useful message (if necessary) ScIII-1 One of the thermodynamic limit expressions based on the relation S = k lnW, “N/lmW = 1”, is an indispensable condition when dealing with microscopic statistics. Under this condition, an “initial number” as classical particles emerges, and the existence of a “finite length” of the thermodynamic infinitesimal highlighted in Figure 5 has been clarified. The quantum special relativity and the thermodynamic special relativity will be compatible, and the quantification of thermodynamic time will be realized. ScIII-2 No matter how well thermodynamics can be mathematically interpreted to fit the microscopic condition N/lnW = 1, it is unlikely that the fact has been accepting by almost all researchers. It seems that there was a time when it was difficult to give complete accuracy to the means to prove the theorization of quantum theory. I heard that the legitimacy of quantum theory was completely guaranteed through the fact that the consideration by quantum mechanics gave a result that broke Bell’s inequality (CHSH inequality). In the case of thermodynamic theory, I do not know whether it is appropriate to give the same approach as Bell’s inequality (CHSH inequality), but it is desirable to acquire a new theoretical judgment means for the microscopic condition of N/lnW = 1 in 11 thermodynamics. Related parts Useful message From a completely different perspective, I have heard that “a researcher’s (if necessary) work” leading to an aspect of the chaotic system of “disturbance prediction” derived from differential equations comes from hints from mathematically literate philosophers or pure philosophers. I would be happy if visionary philosophers, and researchers with a wealth of knowledge about the integrity of quantum mechanical theory, through their independent work, could answer my questions about how or to what extent thermodynamics should be interpreted mathematically. My previous commentary (II) on November 13, 2023 After posting my revised thermodynamic draft paper in this site, I prepared this commentary as a document like a cover letter to a journal. I have indicated the corrections from my previous draft in green text so that you can see where they are. Another form of entropy S = k ln(e−1+W) is provided mathematically for being more exact lower limit point. (Line 16 of ABSTRACT) Figures 2(b) and 2(c) Related parts Sec. IV. A-7 (In this discussion, the new relation ln(e‒1+W) = N in (N, Z) = (1, 1) is obtained by combining ln(e) = N = 1 with ‒1+W = 0 (W = 1).) Useful message Footnotes #1 and #2 → Appendix C (if necessary) ScII-1 The difference between Figures 2(b) and 2(c) helps us present a new concept that approaches closer to the thermodynamic lower limit (N = 5.5). In our usual understanding, the slope of the line given in macroscopic regions is determined by an isothermal isobaric condition at that time, and the slope is not “1” in most cases (p.4, Line 3). Although the slope of the line for the number density N/V given in the macroscopic region was not (but less than) “1”, as shown in Figure 1(d), the slope of the line becomes “1” when the relation S = k lnW is substituted into the equation of state pV = nRT (p.4, Line 8). Why does this happen? Related parts Figures 1(a)‒1(d) This question (↑) is related to ScII-1 above. ScII-2 Useful message It may come from the microscopic description using S = k lnW. When approaching the thermodynamic lower limit according to Figures 6(c) (if necessary) and 6(d), two finite values of pV and TS appear. In addition, a rotational concept in Figure 6(e) helps us clarify the presence of the thermodynamic limit state pV = TS (i.e., lnW = N), together with the thermodynamically expressive equation (30) (p.4, Line 5). Figures 6(a)‒6(e) The Boltzmann constant and its relation, which are usually said to have been first adopted by L.E. Boltzmann, sit at the gateway to the microscopic world. 12 The difference between Figures 2(a) and 2(b) clarifies the fundamental difference in concept between Bose statistics and multiplicity functions by applying the small number of classical particles. Figures 2(a) and 2(b) Related parts The fact in Figure 2(a) that the multiplicity function, which is good at handling a large number of classical particles, has difficulty in describing the microscopic energy state suggests the importance of clarifying a quantum-like description in this region. In the quantum statistical thermodynamics region described in Figure 2(b), quantum Bose statistics for classical particles becomes the focus of attention, representing energies Z of the number of quanta that correspond to the initial number N of classical particles. At this point, an inflection point like M values in Figure 2(a) does not appear in the energy Z value shown in Figure 2(b) (p.5, Line 12). Useful message Sec. IV. A-7 (if necessary) Footnotes #1 and #2 → Appendix C ScII-3 The importance of applying a small number of classical particles was realized in understanding the relationship between Bose statistics and multiplicity functions. Through my past studies, I have had the experience of deepening discussions on microscopic studies of thermodynamics and physical chemistry with the help of quantum theory and the thermal statistical physics. In that sense, I remain a believer in this academic field. At the same time, there seems a little problem. The problem is that even after A.S. Eddington first proposed the concept of the “arrow of time” along with the concept of thermodynamic time, and even after I. Prigogine emphasized the need for the concept of thermodynamic time, any discussion that allow us to explore the concept of thermodynamic time have never emerged in the above-mentioned academic fields. The author divides this discussion into two types: “everything is in a microscopic world” and “it is the microscopic world itself,” leaving room for particle number analysis through thermal statistical physics, especially for the latter. The “number of classical particles” that appeared there can now be expressed as an “initial number” that allows for the existence of Minkowski spacetime to be replaced with thermodynamic spacetime. Footnotes #1 and #2 → Appendix C Related parts (These two footnotes were applied for the explanation in the above.) Figures 8(a)‒8(c) C on p.17 is introduced here again. Useful message Appendix Footnote #2 (if necessary) The efforts so far to study the microscopic world of thermodynamics and physical chemistry ScII-4 using quantum theory and thermal statistical physics have focused on investigating microscopically by replacing classical particles with Bose particles. On the other hand, it seems that there has been no opportunity to take up the microscopic world itself as a research theme of thermodynamics and physical chemistry. The former approach of replacing classical particles with Bose particles and investigating them microscopically will continue to be needed in thermodynamics and physical chemistry research, and the latter, research on the microscopic world itself, will also be necessary in the future. The difference between being in the microscopic world and watching from the sidelines was 13 discussed. I think that both approaches are important. In Sec. III, key elements for mathematical descriptions of functionalized infinitesimal finiteness are noted. I am not aware of any legitimate statement in the original concept of mathematics regarding the finiteness of functionalized infinitesimals and their classification. However, based on the math-like description of Clausius’ definition of entropy in Figure 5, we can understand that the finite length of such infinitesimals is not at least the same as that in mathematical (geometric) one. When trying to mathematically estimate the finite length of a functionalized infinitesimal, it is reasonable to use the origin of equation (4), which has nothing to do with mathematical functions, as the reference dot, as shown in Figure 3(a). Even if the geometric pattern changes due to the axes exchange in Figure 4(a), the meaning of physical pictures does not change. Through this operation, we can recognize the difference in infinitesimal finite length between mathematics and physics. Using this reference dot and the table in Figure 4(b), it is permissible to outline descriptions of finite lengths of mathematical (geometric) infinitesimals, general physics infinitesimals, and thermodynamic infinitesimals. The concept of differential quotient, in which I. Newton is said to have been involved, seems to have been established on the premise of assigning functionalization to infinitesimals. Figures 5(a) and 5(b) Related parts Figures 3(a) and 3(b) Figures 4(a) and 4(b) Useful message (if necessary) ScII-5 I believe that the differential quotient-like infinitesimal concept has played a major role not only in the field of geometry in mathematics, but also in the fields of general physics and thermodynamics. From the Clausius’ entropy definition in Figure 5 and the quasistatic behaviors, which are hold by total differential condition, the maintenance of the thermodynamic state quantity is guaranteed. On the other hand, when the total differential condition is violated, the visual second law of thermodynamics ΔS > 0 shown in Figures 7(d) and 7(e) emerges. Figures 5(a) and 5(b) Related parts Figures 7(a)‒7(f) Useful message When giving credibility to this description, the reliability of #Sc-5's explanation will increase. (if necessary) ScII-6 It was argued in this draft that long awaited Clausius’ discovery of the state quantitative entropy dS (= δqrev/T) using a non-state quantitative heat δqrev and the explanation of the increasing entropy ΔS > 0 of the second law of thermodynamics can all be described mathematically by “the 14 maintenance continuation of the total differential condition and its violation of the condition.” With the progress of quantum theory [27], there is no longer any doubt about the coexistence of particle nature of electrons and wave nature by electrons. When this subject is revisited, features of two time-references emerge. Namely, thermodynamic time, which has a finite reference time longer than Planck time, allows us to grasp the flow of time more clearly, making it easier to handle wave functions such as electromagnetic waves (p.26, Line 9 of Appendix B). Related parts ScII-7 Useful message (The author’s view and observations) (if necessary) The combination of two time-references, short Planck time and long thermodynamic time, contributed to the progress of quantum theory. The reality of Planck time is expressed as the limit at which the general relativity, which describes gravity, disappears, while the reality of thermodynamic time is expressed in a form that allows special relativity to work as the initial number just before the classical thermodynamic world disappears. The two time-references, Planck time and thermodynamic time, contributed to the argument of this draft, referring to Zeno’s paradoxes and Aristotle’s argument. Both time-references are valuable, and it is difficult to describe the thermodynamic world using them alone. Furthermore, it is reaffirmed that the existence of both criteria (time-references) is essential even when describing quantum theory. My initial commentary on July 9, 2023 I focused on the term in my Research Interests “reversible logic” first, paying attention to the mathematical operations of total and partial derivatives that thermodynamics makes heavy use of. Time-independent reversibility in dVdp = 0 (dT = 0) of quasistatic processes allows for the mutually complementary indirect proof between Clausius’ and Thomson’s principles. Related figures Figure 5(a) #1 Related equations Equations (10) and (11) Among the expressions that describe the total differential, especially the condition “dVdp = 0 (dT = 0) in equation (7)” plays an important role. Another significance for this is to understand whether the thermodynamic time concept is disappeared or not in microscopic regions of quantum thermodynamics and classical thermodynamics. From this point of view, we should recognize where the notable “initial number” of classical particles in the results of statistical analysis lies (see Figure 2). In other words, being able to confirm the presence or absence of the concept of time in the presence of classical particles shows that the special relativity is established under the microscopic thermodynamic theory. 15 Next, let us think visually about the existence of thermodynamic infinitesimal concepts. #2 A theoretical approach emerging in thermodynamic functions The following is a scientific term from “academically recognized existing theories”. Thermodynamic limit (N → ∞) N/V = const. A new perspective Thermodynamic limit (N = 5.5) (Upper limit) The decrease in the number of classical particles (Lower limit) 5.5) G=U pV = TS Statistical analysis LS = LS (N (lnW = N) in G = U + pV TS V = V(N)p,T pV/TS − 1 0 1 − TS/pV 0 Related figures Figures 1(a)(b) and (d), Figure 2, Figures 6(c)(d), Figures 8(a)(b) Related equations Equations (26) and (27) Such a V = V(N)p,T process maintains a constant number density N/V under thermal equilibrium conditions with dT = 0, dp = 0. In the lower thermodynamic limit given by thermodynamic functions, the volume concept V disappears along with the temperature and pressure concepts. In special relativity, however, the volume concept V = V(N)p,T can be converted to the spatial “length” concept LS = LS(N), and the “spatial density” can be replaced by the “linear density”. The “constant number density of N/LS” itself does not disappear even at the limit point. In such thermodynamic spacetimes, the first step replacement shown in Figures 8(a) and 8(b), V → LS, allows us to understand the leading to the quantification of thermodynamic time. The rationale for that is based on the special relativity initiated by A. Einstein. Next, let us touch on the form of mathematical infinitesimal that geometry claims. #3 The world of mathematical limits expressed by differential quotients Mathematically functionalized infinitesimal lim ∆ ∆ → ( ) (algebraical finiteness) along with lim ∆ = d ∆ → ( ) = d ∗∆ + ∆ d = d ∗d =d d (algebraical finiteness) can create and clarify a geometrical shape emerged by their differential quotients. Related figures Figure 3(b), Figure 4(b) Related equations Equations (5) and (6) As for the mathematical point of view, at present, mathematical infinitesimals are rarely discussed, but Figure 3(b), which allows for the description of geometrical patterns, suggests that such infinitesimals may exist. In fact, the finiteness of the tangent slope or algebraic infinitesimals has already been described in equations (5) and (6), respectively. The notation of mathematical infinitesimals, which can be written as L ∼ 0 (or t ∼ 0) but not as L = 0 nor t = 0, seems to indicate 16 the finiteness for them. It is considered here that an excellent mathematical analysis with the (ε, δ)-definition of limit provides continuity of mathematical functions and will allow for the coexistence with the above notion of infinitesimals. As for the physical point of view, it seems to say that in Figure 4(b) the gravitational limit in general relativity provides physical infinitesimals. In the case of this draft paper, the concept of functionalization for such various infinitesimals is clarified through limit correspondence to thermodynamic functions in #2. Historically, when I. Newton dealt with differential quotients by what he called the fluxion method, the groundwork for the emergence of functionalization concepts for infinitesimals may have been laid. Regarding the finiteness in magnitude of thermodynamic infinitesimals. #4 The relationship between the state quantity U, and non-state quantities that are made of quasistatic heat δqrev and work δwrev. The idea of quasistatic heats, d(ST)p,T and d(pV)p,T in equations (28) and (29), which corresponds to d(pV)T, can be treated in a time-independent isothermal line system of dUT = ∮d(pV)T = 0. Such the relation, dUT = ∮d(pV)T = 0, can make clear the relationship of equation (11) between the state quantity U, and non-state quantities of quasistatic heat δqrev and work δwrev. Related figures Figure 5(a)(b), Figure 7(a)(b) Related equations Equation (10), Equation (11) The quantity “δqrev = ‒ δwrev” in equation (11) given by dVdp = 0 in the total differential condition tells us that the finite length of the entropy dS (= δqrev / T) is much longer than that of other kinds of infinitesimals. This estimate was validated by the statistical analysis shown in Figure 2. In equation (10), it is important to monitor whether the operating point in the thermodynamic function f = f(V, p, T, N) continues to maintain the total derivative condition. The quickest confirmation method for that is to make sure that the distance of the operating point from the origin, (∆ ) + (∆ ) , is to be within a constant value so that total differentiation works at quasistatic regions. There are few thermodynamics textbooks that take a similar point of view, and the Fermi’s thermodynamics textbook is one of them. There is a convention in thermodynamics that the energy added to a system is positive and the energy taken away from the system is negative. What we should be careful about here is not to overlook the relationship between the input and output of energies when starting discussions on multiple systems. Furthermore, when dealing with events that extend into the macroscopic regions of thermodynamics, such as Benjamin Thompson’s “heat generation by a work”, special care must be taken. Thank you for your understanding, sir. Minoru MYOJO 17 

Supplementary commentary (IV) on my thermodynamic draft paper for Thermodynamic time determination under thermodynamic spacetime along with mathematically interpreted classical thermodynamics (as Draft III) Minoru Myojo Takatsuki Research Site for Thermal Science (Date: December 21, 2023) In this updated commentary, we will discuss two matters. The first point concerns that the difference between a Boltzmann-like description and a Plancklike one in the Boltzmann relation, which deals with probabilistic elements, must be emphasized in the thermodynamics draft paper. My first proposal on this commentary (IV) We need to explain the Boltzmann relation S = k lnW, which shows the difference between a Boltzmann-like description and a Planck-like one. The first public indication of that was in the 2014 material. However, I would like to point out here that the material presented at one workshop not only contained many errors, but also had problems with the content of that abstract. No matter how much progress has been made in the mathematical interpretation of thermodynamic theory, the value of the empirical knowledge accumulated so far by classical thermodynamic theory remains intact. On the contrary, it also seems that one perspective of the lack of progress in the mathematical interpretation of thermodynamic theory appeared in the abstract of the 2014 article. The real value of this knowledge was mentioned at the end of the conclusion of the thermodynamics draft paper. A revised version of the schematic diagram describing the Boltzmann relation S = k lnW and its related matters, which describes the difference between a Boltzmann-like description and a Plancklike one, is shown in the second sheet below (see Footnote #3 in the draft). After showing a revised version of Figure 15 in the 2014 article this time, the posting of that article would be dropped. The relationship between the 2014 article the author recently posted for article introduction and his way of thinking about classical thermodynamics If only the abstract of the 2014 article is applied, it may cause misunderstandings. The author's real intention at that time was simply to point out the lack of efforts to approach thermodynamic theory from a mathematical scientific perspective. There has been no slight change in my respect for the empirical knowledge of classical thermodynamics. Please allow me to introduce here the last part of “V. CONCLUSION” of the thermodynamics draft paper again: “Those contents were summarized as follows in light of the claims of this paper. The basic part of the 1 thermodynamic theory, which is supported by empirical knowledge such as thermodynamic principles and laws, has been successfully explained using mathematical interpretation. This success shows that the mathematical interpretation in natural scientific field can contribute to the further development of thermodynamic research, not by denying the empirical knowledge, but by further respecting it.” A revised version of the schematic diagram describing the Boltzmann relation S = k lnWB and its related matters shown in Figure15 of the 2014 article pV = nRT Nothing When N = 0 Macroscopic When N = NA n = 1, then pV = RT Microscopic When N = 1 (T n = 0, then pV = 0 pV = 0 (Everything is nothing, including energy.) pV = NAkT = RT pV = (1/NA)RT = kT pV = kT (There exists energy when N = 1.) 0) T (N 0 The third law of 1) thermodynamics lim S = S0 ≡ 0 T→0 WP = 1 WB = 1 S = k ln WP (N, WB, Z) = (1, 1, 1) S = k ln(e‒1+WB) S=0 N S=k N = 1, WB = 1 (T (see Figure 1(d) in the thermodynamics draft) Quasistatic (N = 5.5) p = (N/V)kT pV = NkT 0) 0) pV = ST (quasistatic) (S = 5.5k) S = k ln(e‒1+WB) (in the initial number region N) 1, WP = 1 (T N = 5.5, ln(e‒1+WB) = 5.5 S = k lnWB (macroscopic) pV N ln e 1 WB ST Both Boltzmann relations, S = k lnWB and S = k lnWP, allow for probabilistic descriptions. The former Boltzmann relation S = k lnWB allows for probability expression that excel in macroscopic statistical analysis over countable classical particle numbers. On the other hand, the latter Boltzmann relation S = k lnWP makes it possible to describe zero entropy (S → 0; the third law of thermodynamics) depending on the probability of WP = 1 from an uncountable temperature element that tends toward zero (T → 0). The author of the thermodynamics draft paper believed that statistical analysis would be possible even by using the small number of classical particles in the microscopic world in the same way as in the macroscopic world. This was made possible by adopting the Boltzmann relation S = k lnWB and presenting the quasistatic condition pV = TS (i.e., lnW = N). The Boltzmann relation, S = k lnWB, which is distinguished from S = k lnWP, may help us make statistical analysis even in microscopic regions of thermodynamics more freely applicable. My second proposal on this commentary (IV) The second point concerns why F.W. Ostwald, who was one of researchers taking a skeptical stand on the existence of atoms and molecules, came to appear as a translator of Gibbs' paper, which contributed to the development of statistical mechanics after Maxwell and Boltzmann. The 2 material for the discussion is based on an English "afterword", translated from the Japanese, involving two volumes (Gas Molecular Kinetics and Statistical Mechanics) of a series of classical physical papers by Maxwell, Loschmidt, Zelomelo, and Boltzmann, in which the location of the original articles is specified. It is hoped that the relevant thermodynamic paper will provide useful perspectives that encourage further discussion. Afterword "Commentary of Gas Molecular Kinetics" in Volume 5 of a series of Classical Physical Papers (Tokai University Press) The papers in this volume consist of three papers by Maxwell on molecular motion and five papers created by Loschmidt, Zelomelo, and Boltzmann around two papers by Boltzmann on kinetic thermal theory (i.e., a 1872 paper on the H-theorem and others, and a 1877 paper on the probabilistic interpretation of statistical mechanics, now expressed in the relation S = k logW (same as S = k lnW), both of which are included in Volume 6 of this series). Therefore, by reading this volume together with Volume 6 (Statistical Mechanics), it is possible to trace the transition and development of the theory of gas molecular motion up to the completion of Gibbs' statistical mechanics. The theory of gas molecular motion traces dynamically a group of mass points or rigid bodies subject to the classical mechanical laws of motion, which are exerting various interactions. The main problems that have appeared in the transition and development of the group are the determination of the distribution function at the equilibrium state and the guarantee of reaching the equilibrium state. What is known today as the Maxwell-Boltzmann distribution law relates to the former, and the Boltzmann’s H-theorem and the theory of probabilistic interpretation discussed above relate to the latter. The first paper of this volume is the first to give the distribution function in gas molecular kinetics. The so-called Maxwell's velocity distribution law is also given here. However, the velocity distribution law is derived here based on the spatial isotropy of the distribution of a group consisting of elastic hard spheres. The dynamic determination of the distribution function corresponding to the steady state of a mass point or a group of rigid bodies moving while exerting interaction can be seen in the second paper. In the year of the publication of the second paper (1866), Boltzmann published the first paper of his kinetic thermal theory (included in Volume 6 of this series), and after that, his theory was further developed. The derivation of the Maxwell-Boltzmann distribution, which generalizes the Maxwell distribution, is also a result of this period, and Boltzmann's theorem mentioned in the title of the third paper refers to Boltzmann's work on the distribution of kinetic energy at equilibrium. Maxwell formulated this work more generally and more beautifully in the third paper. In the course of this formulation, the concept of ensemble was introduced for the first time, giving a clear distinction between dynamical and statistical considerations, which, of course, were inherited by Gibbs and become the basic concepts of his statistical mechanics. Statistical mechanics at equilibrium was completed to a certain extent by Gibbs' "Elementary Principles in Statistical Mechanics" (1902). The meaning of the fact that it gave the basic principle is that it gave the general relational expression that should be satisfied when a group consisting of fundamental particles such as atoms and molecules is in equilibrium in a form that is independent of the properties of specific particles. In other words, the completion of Gibbs' statistical mechanics of equilibrium provides a basic principle that does not require or is not related to various assumptions made in the course of the development of the gas molecular motion theory regarding the interaction between particles and others. In this sense, Maxwell's third paper directed the gas molecular motion theory toward a more general principle. On the other hand, Boltzmann's paper on the H-theorem (1872, in Volume 6 of this series) was intended to provide a dynamical guarantee that a population of particles in a nonequilibrium state should reach an equilibrium state irreversibly, not to mention Boltzmann's ambitious work aimed at the dynamical deduction of the second law. The fourth paper is a rebuttal by Loschmidt to this 3 paper. Loschmidt disproved the generality of Boltzmann's H-theorem by specifically pointing out instances where the H-theorem does not hold, which meant that the H-theorem for Boltzmanns state, although fairly general, still contained some special assumptions. Boltzmann's answer to this is in his 1877 paper (included in Volume 6 of this series), in which he introduced the concept of Komplexionen and gave the expression for entropy, given today as S = k logW (same as S = k lnW). This equilibrium is a state realized with overwhelmingly high frequency, and the distribution functions at it are described in correspondence with that overwhelmingly high frequency state. The sixth to eighth papers are the counterargument by Zelomelo to Boltzmann's theory and Boltzmann's argument on the counterargument, which examined the problem on the irreversible process of the dynamical state of the system based on Poincare's theory of three body problem and criticized Boltzmann's explanation of recursion. As mentioned above, with the establishment of Gibbs' statistical mechanics, statistical mechanics at equilibrium is formally completed in a form corresponding to thermodynamic theory at equilibrium, and since the theoretical form is independent of individual molecular processes, the molecular kinetic theory before Gibbs does not appear on the surface as far as problems at equilibrium are concerned. However, molecular kinetic theory has reappeared on the surface of various problems belonging to the series of development of thermodynamics and statistical mechanics of nonequilibrium state, and the significance of molecular kinetic theory with respect to plasma and other theories is rather increasing in this sense today, and I believe that a compilation of the classic representative papers of molecular kinetic theory can give considerable suggestions. (Commentator: W. Taniguchi) J. C. Maxwell, “Illustrations of the Dynamical Theory of Gases”, Phil. Mag., (4)19, 19References 32(Part I), 20, 21-33(Part II), 33-37(Part III), (1860); in Volume 5 Scientific Papers, 1, 377-409 (Dover pub.) (Gas Molecular J. C. Maxwell, “On the Dynamical Theory of Gases”, Phil. Trans. R. Soc. London, 157, 49Kinetics) 88,(1867); Phil. Mag., (4) 35, 129-145, 185-217, (1868); Scientific Papers, 2, 26-78 (Dover pub.) J. C. Maxwell, “On Boltzmann's Theorem on the average distribution of energy in a system of material points”, Cambridge Philosophical Society's Transaction., XII., (1879); Scientific Papers, 2, 713-741 (Dover pub.) J. Loschmidt, “Über den Zustant des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft”, Wiener Berichte 73, 128-142 (1876), 366-372 (1877), 287298 (1877) L. Boltzmann, “Bemerkungen über einige Probleme der mechanischen Wärmetheorie”, Wiener Berichte 75, 62-100 (1877). Wissendhaftche Abhandlungen von Ludeig Boltznann, 112-148, Leipzig, (1909). E. Zermelo, “Ueber einen Satz der Dynamik und die mechanische Wärmetheorie”, Wiedemann Annalen 57, 485-494 (1896). L. Boltzmann, “Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo”, Wiedemann Annalen 57, 773-784 (1896). Wissendhaftche Abhandlungen von Ludeig Boltznann, 567-578, Leipzig, (1909). E. Zermelo, Ueber mechanische Erklärungen irreversibler Vorgänge. Eine Autwork auf Hrn. Boltzmann’s „Entgegnung”, Wiedemann Annalen 59, 793-801 (1896). References in Volume 6 (Statistical Mechanics) L. Boltzmann, “Über die mechanishe Bedentung des zweiten Hauptsatzes der Wärmetheorie”, Wiener Berichte 53, 195-220 (1866). L. Boltzmann, “Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”, Wiener Berichte 63, 275-370 (1872). L. Boltzmann, “Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht”, Wiener Berichte 76, 373-435 (1877). (J.W. Gibbs' references that have presented as #16‒#18 in my thermodynamics draft should be added here.) 4 The author would address the questions and other issues raised in our scientific matters according to the above commentary. After confirming almost all points, the author would show you the significant viewpoints that need to be summarized, along with useful messages. Summary of the above explanation • The transition and development of the theory of gas molecular motion up to the completion of Gibbs' statistical mechanics - The Maxwell’s first paper (1860) gave the distribution function in gas molecular kinetics. - The Maxwell’s second paper (1866) gave the dynamic determination of the distribution function corresponding to the steady state of a mass point or a group of rigid bodies. - Boltzmann published the first paper (1866) of his kinetic thermal theory, and after that, his theory was further developed. - Boltzmann’s paper on the H-theorem (1872) was intended to provide a dynamical guarantee that a population of particles in a nonequilibrium state should reach an equilibrium state irreversibly, not to mention Boltzmann’s ambitious work aimed at the dynamical deduction of the second law. - Boltzmann's answer to the Loschmidt’s question is in his 1877 paper, in which he introduced the concept of Komplexionen and gave the expression for entropy, given today as S = k logW (same as S = k lnW). - The Maxwell’s third paper (1879) directed the gas molecular motion theory toward a more general principle. The derivation of the Maxwell-Boltzmann distribution, which generalizes the Maxwell distribution, is also a result of this period, and Boltzmann’s theorem mentioned in the title of the third paper refers to Boltzmann’s work on the distribution of kinetic energy at equilibrium. - In the course of the formulation of the Maxwell-Boltzmann distribution by Maxwell, the concept of ensemble was introduced for the first time, giving a clear distinction between dynamical and statistical considerations, which, of course, were inherited by Gibbs and become the basic concepts of his statistical mechanics. • What the completion of Gibbs' statistical mechanics brought about - Statistical mechanics at equilibrium was completed to a certain extent by Gibbs' "Elementary Principles in Statistical Mechanics" (1902). - The meaning of the fact that it gave the basic principle is that it gave the general relational expression that should be satisfied when a group consisting of fundamental particles such as atoms and molecules is in equilibrium in a form that is independent of the properties of specific particles. In other words, the completion of Gibbs' statistical mechanics of equilibrium provides a basic principle that does not require or is not related to various assumptions made in the course of the development of the gas molecular motion theory regarding the interaction between particles and others. - As mentioned above, with the establishment of Gibbs' statistical mechanics, statistical mechanics at equilibrium is formally completed in a form corresponding to thermodynamic theory at equilibrium, and since the theoretical form is independent of individual molecular processes, the molecular kinetic theory before Gibbs does not appear on the surface as far as problems at equilibrium are concerned. - Under such situations where Maxwell and Boltzmann studied the fundamental theory of statistical mechanics from the point of view of gas molecular kinetics and Maxwell presented the ensemble concept, Gibbs established statistical mechanics at equilibrium, and he contributed to the expansion of the field of physical chemistry and its theoretical establishment by utilizing the graphical or geometric representation of thermodynamics in equilibrium systems. 5 Useful messages (related matters or Q & A) • The author wonder why Ostwald could have been an English translator into German. - If you know Gibbs' work as a contributor to physical chemistry theory, you wouldn't question the fact that the physical chemist Ostwald was the translator of the Gibbs paper into German. Before contributing to physical chemistry theory, however, Gibbs served as a sort of successor to Boltzmann, who believed in the existence of atoms and molecules and studied statistical mechanics. Ostwald was part of a group that did not believe in the existence of atoms and molecules. Such questions may arise when you look at Figure 1 of the thermodynamics draft paper. • Has it been understood that Maxwell's and Boltzmann's work on gas molecular kinetics and Gibbs' work on equilibrium statistical mechanics are in part of a series of academic systems? - Both studies have concepts connected to statistical mechanics of equilibrium systems, but Gibbs' one has the advantage of being constructed with a theory that does not depend on the interactions required by the other, and that it provides basic principles. However, the ensemble concept emerged from the equilibrium concept of particle populations deriving to the Maxwell-Boltzmann distribution, which might have helped Gibbs for his theoretical construction. - It gave the impression that Gibbs' work on equilibrium statistical mechanics was separate from Maxwell's and Boltzmann's work on gas molecular kinetics, and the effect seemed to extend to Ostwald. The fundamental principles acquired by Gibbs may have concealed Boltzmann's academic troubles, while on the other hand they contributed to the development of the theory of physical chemistry. • Can we say that the (static) tangent planes shown in Figures 1(a) and 1(c) of the thermodynamics draft paper follow Gibbs' descriptive format of statistical mechanics? - At least Figure 1(c) is in an equilibrium state under isothermal and isobaric conditions, which satisfies Gibbs' description conditions for statistical analysis. - Original Boltzmann relation S = k lnWB (that is initiated by Boltzmann), which describes a countable element in probability, would allow us to express the thermodynamic microworld in Figure 1(a), which allows us to interpret mathematically quasistatic behaviors pV = TS (i.e., lnWB = N) in quasistatic processes handled in classical thermodynamics. - In the thermodynamic draft paper, the thermodynamic lower limit to this completion of Gibbs' equilibrium statistical mechanics points to the "static tangent plane of Figure 1(a)," which helps explain the finite-length magnitude of the thermodynamic infinitesimal of Figure 5. • How many thermodynamic limits are there? - The first fundamental thermodynamic limit lies in the special concept of infinity (N → ∞). - The second one lies in the thermodynamic lower limit (N = 5.5) that allows for providing a finiteness of thermodynamic (thermodynamically functionalized) infinitesimal. That can be verified through a statistical analysis using S = k lnWB. - There is a limit having a minimized countable classical particle (N = 1) just before disappearing. The statistical analysis up to N → 1 using S = k lnWB (together with ln(e‒1+WB) = N) will be enabled. - There is a separate limit where the entropy is zero (S → 0), which is unrelated to the size of the number of classical particles and is based on the absolute zero temperature (T → 0) as an uncountable element. My previous commentary (III) on December 2, 2023 6 After revising my thermodynamic draft and posting it on November 27, 2023, I created a document like a cover letter to a journal. All changes made since the original document are shown in green text. I will also leave the previous two commentaries for reference. In this supplementary commentary, let me show you how the quantification of thermodynamic time using the Boltzmann relation S = k lnW in the statistical analyses of Appendix A has been carried out (see footnote #3 and the relevant part of Appendix B). After that, the author will show the difference between the special relativity handled in the quantum theory and the special relativity handled under the extreme region in thermodynamics, and he will take the attitude of watching the future evolution of the thermodynamic theory with reference to the completion of the quantum theory. Before the development and perfection of quantum theory, quantum physicists apparently viewed statistical mechanics, including the Boltzmann relation in thermodynamics, as a rival discipline in finding the special relativity for the study to microscopic regions. It is not certain that the same idea existed on the part of the thermodynamics scholars of the time, but at least they seem to have been aware of the term “thermodynamic time” that emanated from A.S. Eddington. Footnote #3 Related parts “Relationship between the Boltzmann relation S = k lnW and quantum mechanics” in Appendix B Useful message (if necessary) ScIII-1 One of the thermodynamic limit expressions based on the relation S = k lnW, “N/lmW = 1”, is an indispensable condition when dealing with microscopic statistics. Under this condition, an “initial number” as classical particles emerges, and the existence of a “finite length” of the thermodynamic infinitesimal highlighted in Figure 5 has been clarified. The quantum special relativity and the thermodynamic special relativity will be compatible, and the quantification of thermodynamic time will be realized. No matter how well thermodynamics can be mathematically interpreted to fit the microscopic condition N/lnW = 1, it is unlikely that the fact has been accepting by almost all researchers. It seems that there was a time when it was difficult to give complete accuracy to the means to prove the theorization of quantum theory. I heard that the legitimacy of quantum theory was completely guaranteed through the fact that the consideration by quantum mechanics gave a result that broke Bell’s inequality (CHSH inequality). In the case of thermodynamic theory, I do not know whether it is appropriate to give the same approach as Bell’s inequality (CHSH inequality), but it is desirable to acquire a new theoretical judgment means for the microscopic condition of N/lnW = 1 in thermodynamics. Related parts ScIII-2 7 Useful message From a completely different perspective, I have heard that “a researcher’s (if necessary) work” leading to an aspect of the chaotic system of “disturbance prediction” derived from differential equations comes from hints from mathematically literate philosophers or pure philosophers. I would be happy if visionary philosophers, and researchers with a wealth of knowledge about the integrity of quantum mechanical theory, through their independent work, could answer my questions about how or to what extent thermodynamics should be interpreted mathematically. My previous commentary (II) on November 13, 2023 After posting my revised thermodynamic draft paper in this site, I prepared this commentary as a document like a cover letter to a journal. I have indicated the corrections from my previous draft in green text so that you can see where they are. Another form of entropy S = k ln(e−1+W) is provided mathematically for being more exact lower limit point. (Line 16 of ABSTRACT) Figures 2(b) and 2(c) Related parts Sec. IV. A-7 (In this discussion, the new relation ln(e‒1+W) = N in (N, Z) = (1, 1) is obtained by combining ln(e) = N = 1 with ‒1+W = 0 (W = 1).) Useful message Footnotes #1 and #2 (if necessary) ScII-1 The difference between Figures 2(b) and 2(c) helps us present a new concept that approaches closer to the thermodynamic lower limit (N = 5.5). In our usual understanding, the slope of the line given in macroscopic regions is determined by an isothermal isobaric condition at that time, and the slope is not “1” in most cases (p.4, Line 3). Although the slope of the line for the number density N/V given in the macroscopic region was not (but less than) “1”, as shown in Figure 1(d), the slope of the line becomes “1” when the relation S = k lnW is substituted into the equation of state pV = nRT (p.4, Line 8). Why does this happen? Related parts Figures 1(a)‒1(d) This question (↑) is related to ScII-1 above. ScII-2 Useful message It may come from the microscopic description using S = k lnW. When approaching the thermodynamic lower limit according to Figures 6(c) (if necessary) and 6(d), two finite values of pV and TS appear. In addition, a rotational concept in Figure 6(e) helps us clarify the presence of the thermodynamic limit state pV = TS (i.e., lnW = N), together with the thermodynamically expressive equation (30) (p.4, Line 5). Figures 6(a)‒6(e) The Boltzmann constant and its relation, which are usually said to have been first adopted by L.E. Boltzmann, sit at the gateway to the microscopic world. ScII-3 The difference between Figures 2(a) and 2(b) clarifies the fundamental difference in concept between Bose statistics and multiplicity functions by applying the small 8 number of classical particles. Figures 2(a) and 2(b) Related parts The fact in Figure 2(a) that the multiplicity function, which is good at handling a large number of classical particles, has difficulty in describing the microscopic energy state suggests the importance of clarifying a quantum-like description in this region. In the quantum statistical thermodynamics region described in Figure 2(b), quantum Bose statistics for classical particles becomes the focus of attention, representing energies Z of the number of quanta that correspond to the initial number N of classical particles. At this point, an inflection point like M values in Figure 2(a) does not appear in the energy Z value shown in Figure 2(b) (p.5, Line 12). Useful message Sec. IV. A-7 (if necessary) Footnotes #1 and #2 The importance of applying a small number of classical particles was realized in understanding the relationship between Bose statistics and multiplicity functions. Through my past studies, I have had the experience of deepening discussions on microscopic studies of thermodynamics and physical chemistry with the help of quantum theory and the thermal statistical physics. In that sense, I remain a believer in this academic field. At the same time, there seems a little problem. The problem is that even after A.S. Eddington first proposed the concept of the “arrow of time” along with the concept of thermodynamic time, and even after I. Prigogine emphasized the need for the concept of thermodynamic time, any discussion that allow us to explore the concept of thermodynamic time have never emerged in the above-mentioned academic fields. The author divides this discussion into two types: “everything is in a microscopic world” and “it is the microscopic world itself,” leaving room for particle number analysis through thermal statistical physics, especially for the latter. The “number of classical particles” that appeared there can now be expressed as an “initial number” that allows for the existence of Minkowski spacetime to be replaced with thermodynamic spacetime. Footnotes #1 and #2 Related parts (These two footnotes were applied for the explanation in the above.) Figures 8(a)‒8(c) Useful message Footnote #2 on p.17 is introduced here again. The efforts so far to study the microscopic world of thermodynamics and physical chemistry (if necessary) ScII-4 using quantum theory and thermal statistical physics have focused on investigating microscopically by replacing classical particles with Bose particles. On the other hand, it seems that there has been no opportunity to take up the microscopic world itself as a research theme of thermodynamics and physical chemistry. The former approach of replacing classical particles with Bose particles and investigating them microscopically will continue to be needed in thermodynamics and physical chemistry research, and the latter, research on the microscopic world itself, will also be necessary in the future. The difference between being in the microscopic world and watching from the sidelines was discussed. I think that both approaches are important. 9 In Sec. III, key elements for mathematical descriptions of functionalized infinitesimal finiteness are noted. I am not aware of any legitimate statement in the original concept of mathematics regarding the finiteness of functionalized infinitesimals and their classification. However, based on the math-like description of Clausius’ definition of entropy in Figure 5, we can understand that the finite length of such infinitesimals is not at least the same as that in mathematical (geometric) one. When trying to mathematically estimate the finite length of a functionalized infinitesimal, it is reasonable to use the origin of equation (4), which has nothing to do with mathematical functions, as the reference dot, as shown in Figure 3(a). Even if the geometric pattern changes due to the axes exchange in Figure 4(a), the meaning of physical pictures does not change. Through this operation, we can recognize the difference in infinitesimal finite length between mathematics and physics. Using this reference dot and the table in Figure 4(b), it is permissible to outline descriptions of finite lengths of mathematical (geometric) infinitesimals, general physics infinitesimals, and thermodynamic infinitesimals. The concept of differential quotient, in which I. Newton is said to have been involved, seems to have been established on the premise of assigning functionalization to infinitesimals. Figures 5(a) and 5(b) Related parts Figures 3(a) and 3(b) Figures 4(a) and 4(b) Useful message (if necessary) ScII-5 I believe that the differential quotient-like infinitesimal concept has played a major role not only in the field of geometry in mathematics, but also in the fields of general physics and thermodynamics. From the Clausius’ entropy definition in Figure 5 and the quasistatic behaviors, which are hold by total differential condition, the maintenance of the thermodynamic state quantity is guaranteed. On the other hand, when the total differential condition is violated, the visual second law of thermodynamics ΔS > 0 shown in Figures 7(d) and 7(e) emerges. Figures 5(a) and 5(b) Related parts Figures 7(a)‒7(f) Useful message When giving credibility to this description, the reliability of #Sc-5's explanation will increase. (if necessary) ScII-6 It was argued in this draft that long awaited Clausius’ discovery of the state quantitative entropy dS (= δqrev/T) using a non-state quantitative heat δqrev and the explanation of the increasing entropy ΔS > 0 of the second law of thermodynamics can all be described mathematically by “the maintenance continuation of the total differential condition and its violation of the condition.” 10 With the progress of quantum theory [27], there is no longer any doubt about the coexistence of particle nature of electrons and wave nature by electrons. When this subject is revisited, features of two time-references emerge. Namely, thermodynamic time, which has a finite reference time longer than Planck time, allows us to grasp the flow of time more clearly, making it easier to handle wave functions such as electromagnetic waves (p.26, Line 9 of Appendix B). Related parts ScII-7 Useful message (The author’s view and observations) (if necessary) The combination of two time-references, short Planck time and long thermodynamic time, contributed to the progress of quantum theory. The reality of Planck time is expressed as the limit at which the general relativity, which describes gravity, disappears, while the reality of thermodynamic time is expressed in a form that allows special relativity to work as the initial number just before the classical thermodynamic world disappears. The two time-references, Planck time and thermodynamic time, contributed to the argument of this draft, referring to Zeno’s paradoxes and Aristotle’s argument. Both time-references are valuable, and it is difficult to describe the thermodynamic world using them alone. Furthermore, it is reaffirmed that the existence of both criteria (time-references) is essential even when describing quantum theory. My initial commentary on July 9, 2023 I focused on the term in my Research Interests “reversible logic” first, paying attention to the mathematical operations of total and partial derivatives that thermodynamics makes heavy use of. Time-independent reversibility in dVdp = 0 (dT = 0) of quasistatic processes allows for the mutually complementary indirect proof between Clausius’ and Thomson’s principles. Related figures Figure 5(a) #1 Related equations Equations (10) and (11) Among the expressions that describe the total differential, especially the condition “dVdp = 0 (dT = 0) in equation (7)” plays an important role. Another significance for this is to understand whether the thermodynamic time concept is disappeared or not in microscopic regions of quantum thermodynamics and classical thermodynamics. From this point of view, we should recognize where the notable “initial number” of classical particles in the results of statistical analysis lies (see Figure 2). In other words, being able to confirm the presence or absence of the concept of time in the presence of classical particles shows that the special relativity is established under the microscopic thermodynamic theory. Next, let us think visually about the existence of thermodynamic infinitesimal concepts. 11 #2 A theoretical approach emerging in thermodynamic functions The following is a scientific term from “academically recognized existing theories”. Thermodynamic limit (N → ∞) N/V = const. A new perspective Thermodynamic limit (N = 5.5) (Upper limit) The decrease in the number of classical particles (Lower limit) 5.5) G=U pV = TS Statistical analysis LS = LS (N (lnW = N) in G = U + pV TS V = V(N)p,T pV/TS − 1 0 1 − TS/pV 0 Related figures Figures 1(a)(b) and (d), Figure 2, Figures 6(c)(d), Figures 8(a)(b) Related equations Equations (26) and (27) Such a V = V(N)p,T process maintains a constant number density N/V under thermal equilibrium conditions with dT = 0, dp = 0. In the lower thermodynamic limit given by thermodynamic functions, the volume concept V disappears along with the temperature and pressure concepts. In special relativity, however, the volume concept V = V(N)p,T can be converted to the spatial “length” concept LS = LS(N), and the “spatial density” can be replaced by the “linear density”. The “constant number density of N/LS” itself does not disappear even at the limit point. In such thermodynamic spacetimes, the first step replacement shown in Figures 8(a) and 8(b), V → LS, allows us to understand the leading to the quantification of thermodynamic time. The rationale for that is based on the special relativity initiated by A. Einstein. Next, let us touch on the form of mathematical infinitesimal that geometry claims. #3 The world of mathematical limits expressed by differential quotients Mathematically functionalized infinitesimal lim ∆ ∆ → ( ) (algebraical finiteness) along with lim ∆ = d ∆ → ( ) = d ∗∆ + ∆ d = d ∗d =d d (algebraical finiteness) can create and clarify a geometrical shape emerged by their differential quotients. Related figures Figure 3(b), Figure 4(b) Related equations Equations (5) and (6) As for the mathematical point of view, at present, mathematical infinitesimals are rarely discussed, but Figure 3(b), which allows for the description of geometrical patterns, suggests that such infinitesimals may exist. In fact, the finiteness of the tangent slope or algebraic infinitesimals has already been described in equations (5) and (6), respectively. The notation of mathematical infinitesimals, which can be written as L ∼ 0 (or t ∼ 0) but not as L = 0 nor t = 0, seems to indicate the finiteness for them. It is considered here that an excellent mathematical analysis with the (ε, δ)-definition of limit provides continuity of mathematical functions and will allow for the 12 coexistence with the above notion of infinitesimals. As for the physical point of view, it seems to say that in Figure 4(b) the gravitational limit in general relativity provides physical infinitesimals. In the case of this draft paper, the concept of functionalization for such various infinitesimals is clarified through limit correspondence to thermodynamic functions in #2. Historically, when I. Newton dealt with differential quotients by what he called the fluxion method, the groundwork for the emergence of functionalization concepts for infinitesimals may have been laid. Regarding the finiteness in magnitude of thermodynamic infinitesimals. #4 The relationship between the state quantity U, and non-state quantities that are made of quasistatic heat δqrev and work δwrev. The idea of quasistatic heats, d(ST)p,T and d(pV)p,T in equations (28) and (29), which corresponds to d(pV)T, can be treated in a time-independent isothermal line system of dUT = ∮d(pV)T = 0. Such the relation, dUT = ∮d(pV)T = 0, can make clear the relationship of equation (11) between the state quantity U, and non-state quantities of quasistatic heat δqrev and work δwrev. Related figures Figure 5(a)(b), Figure 7(a)(b) Related equations Equation (10), Equation (11) The quantity “δqrev = ‒ δwrev” in equation (11) given by dVdp = 0 in the total differential condition tells us that the finite length of the entropy dS (= δqrev / T) is much longer than that of other kinds of infinitesimals. This estimate was validated by the statistical analysis shown in Figure 2. In equation (10), it is important to monitor whether the operating point in the thermodynamic function f = f(V, p, T, N) continues to maintain the total derivative condition. The quickest confirmation method for that is to make sure that the distance of the operating point from the origin, (∆ ) + (∆ ) , is to be within a constant value so that total differentiation works at quasistatic regions. There are few thermodynamics textbooks that take a similar point of view, and the Fermi’s thermodynamics textbook is one of them. There is a convention in thermodynamics that the energy added to a system is positive and the energy taken away from the system is negative. What we should be careful about here is not to overlook the relationship between the input and output of energies when starting discussions on multiple systems. Furthermore, when dealing with events that extend into the macroscopic regions of thermodynamics, such as Benjamin Thompson’s “heat generation by a work”, special care must be taken. Thank you for your understanding, sir. Minoru MYOJO 13