In this article, the classical formulations of the second law of thermodynamics as they relate to the evolution of living systems will be presented. Some mistakes in the understanding of the physical meaning of this general law of nature will be noted. It is asserted that many misunderstandings of the second law of thermodynamics are related to terminological confusion and to the underestimation or disregard of the theory developed by Willard Gibbs and other founders of "true thermodynamics", which is impossible to disprove.
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Making no pretensions to perfection, the author offers some advice to researchers dealing with thermodynamics. The author believes that, when considering thermodynamic problems, "ambiguous" terms and definitions should be clarified preliminarily in order to preclude possible misunderstandings. It is also advisable to refer to the classical works of those noted; including textbooks, encyclopedias, and founding articles in each respective historical publication. This will allow the correctness of the results reported to be estimated at least preliminarily.
Willard Gibbs [1839-1903]
Rudolf Clausius [1822-1888]
Classics of science enunciated the second law of thermodynamics, one of general laws of nature, in the first half of the 19th century. Well-known formulations of this law are associated with the names of Sadi Carnot (1824), Rudolf Clausius (1850), and William Thomson (Lord Kelvin) (1851). Although the formulations themselves are different, mainly because of the difference in phrasing, they may be considered equivalent. Many authors have attempted to change or improve the formulations as regards to their physical meaning, yet none have succeeded. The meaning and essence of these formulations has not been disproved to date [1–23].
Sadi Carnot [1796-1832]
LAW OF TEMPORAL HIERARCHIES:
The discovery of the law of temporal hierarchies, which may be considered a new general law of nature, has determined the extension of Gibbs's theory to living systems [17, 22–28]. This law  makes it possible to apply thermodynamics, or more precisely the hierarchic thermodynamics of quasi-closed systems, to all hierarchies of the real world, particularly, living objects and biological systems, to quite a good approximation.
dS ≥ δQ / T
where the equality sign pertains to reversible processes and the inequality (greater-than) sign, to irreversible ones. Expression (1) is suitable for a simple isolated system, which can exchange neither substance nor energy with the environment and whose internal energy (U) and volume (V) are constant. In such systems only the work of expansion or no work at all is performed In this case, the second law of thermodynamics may be written as:
dSU,V ≥ 0
Thus, the entropy of this system increases when irreversible processes occur, and it is maximum in the state of thermodynamic equilibrium.
The second law of thermodynamics according to Thomson, i.e. Thomson's principle, states that: “the process during which work is transformed into heat without any other changes in the system's state is irreversible.” This means that all heat withdrawn from a body cannot be entirely transformed into work unless the system is changed in other respects. This formulation is equivalent to the statement that the perpetuum mobile of the second kind is impossible [7-10].
Carnot's theorem is also equivalent to the impossibility of the perpetuum mobile of the second kind. According to this theorem, no heat engine can have a higher efficiency than that of the Carnot cycle, η = (T1 – T2)/T1, which is determined only by the temperatures of the heater and the cooler (T1 and T2, respectively). Carnot's theorem lays the basis for the absolute temperature scale. Sometimes, the second law of thermodynamics is formulated as the well-known Caratheodory's principle (1908).
where k is the Boltzmann constant. Note that the Boltzmann's substantiating the statistical basis of the second law of thermodynamics, as well as the statistical substantiation of phenomenological thermodynamics suggested by Gibbs, involves ideal models, e.g., a perfect gas. In the case of more complex systems [3-4, 8], where pronounced (especially, strong) electromagnetic interactions between particles (molecules) are observed, it is difficult to perform the calculations. Therefore, it is obvious that these models are unlikely to be effective when studying most natural systems (e.g., biological), i.e., systems that are far from corresponding to ideal or simple models.
S = k ln W
New concepts, however, have extended the possible applications of the second law of thermodynamics to different sciences, especially chemistry and, as it turned out later, biology. This became possible mainly due to J.W. Gibbs' works performed in 1873–1878. To a certain approximation, the methodologies of Gibbs thermodynamics  have been extended to date to all hierarchies of natural systems, which are generally open ones .
Later, Prigogine also supposed, on the basis of previous notions of Boltzmann, Schrödinger, and their followers in the life sciences, that the phenomenon of life is hardly consistent with the second law of thermodynamics. He noted, "During the last decades, an opinion has widely spread that there is the apparent contradiction between biological order and laws of physics—particularly the second law of thermodynamics" (1980). Prigogine also emphasized that:
"The general struggle for existence of animate beings is therefore not a struggle for raw materials - these, for organisms, are air, water and soil, all abundantly available—nor for energy which exists in plenty in any body in the form of heat (albeit unfortunately not transformable), but a struggle for entropy, which becomes available through the transition of energy from the hot sun to the cold earth."
Then, in 1944, Schrödinger wrote:
"the only way a living system stays alive, away from maximum entropy or death is to be continually drawing from its environment negative entropy. Thus the device by which an organism maintains itself stationary at a fairly high level of orderliness (= fairly low level of entropy) really consists in continually sucking orderliness from its environment. …Plants of course have their most powerful supply in negative entropy in sunlight…"
During the 1870s, the second law of thermodynamics was substantiated in the kinetic theory of gases by Ludwig Boltzmann with his H theorem. Here, H is the Boltzmann H function (functional, to be precise) determined from the mean logarithm of the particle distribution function. The Boltzmann H function is proportional to the entropy of a perfect gas. The physical meaning of entropy is revealed in statistical physics. Boltzmann demonstrated that the entropy is related to the logarithm of thermodynamic probability (W):
Ludwig Boltzmann [1844-1906]
The discovery of this law confirms the universality of classical thermodynamic methods, and the name of Josiah Willard Gibbs even more vividly symbolizes the future of science that confirms the validity of general laws of nature as applied to the evolution of all material systems at all organizational levels of our world.
Advances in classical thermodynamics as well as approximate thermodynamics, i.e., the quasi-equilibrium thermodynamics of quasi-closed systems, are described in a number of textbooks and monographs. They are certainly numerous; the reader is encouraged to refer to the works [1–17], which will be very useful for all beginning researchers.
Law of Temporal Hierarchies: any living system of any temporal hierarchical level in a normal state has a thermostat - a surrounding medium that is characterized by slightly changing average values of thermodynamic parameter.
THE SECOND LAW:
Clausius' formulation of the second law of thermodynamics, also known as the Clausius principle, states that: "a process that involves no changes except for the transfer of heat from a warmer body to a colder body is irreversible, i.e., heat cannot spontaneously pass from a colder body to a warmer one"[7-10]. Building on this principle, in 1865, Clausius introduced the concept of entropy(S), a function of state of a system, i.e. a function that has a full differential, according to the Clausius inequality:
PHENOMEOMENOLOGICAL vs. NONEQUILIBRIUM:
The thermodynamics of nonequilibrium processes deals with the rate of increase in or, as it is sometimes called, production of entropy. Therefore, it is sometimes asserted that nonequilibrium thermodynamics provides "the quantitative characteristic of the second law of thermodynamics" . In the given case, however, this statement is reasonable only when applied to transformations in simple isolated systems where all processes are close to equilibrium. Only in a system that is close to equilibrium can the differential of this function of state of the system (entropy) be considered to be a full one, to an acceptable approximation. However, all the aforesaid is usually underestimated; therefore, many works on nonequilibrium thermodynamics, especially the thermodynamics of systems that are far from equilibrium, remain a faint "future hope." Some of these works, we daresay, are near "mathematically trimmed" fantasies useless for real life [22–24].
Historically, the formulations of the second law of thermodynamics were closely associated with the study of heat engines. This approach has been developed by physicists, mainly thermal physicists, and heat engineers. Another trend in the use of the second law of thermodynamics is related to the attempts of some mathematicians and physicists constructing ideal and simple models to explain many natural phenomena in statistical terms. However, since all interactions in real systems are near to impossible to take into account, there is but little hope that calculations in the framework of these models will successfully solve the problem. Hence, only the phenomenological thermodynamics of systems close to equilibrium, i.e. equilibrium or quasi-equilibrium thermodynamics, are likely to ensure the insight into many natural phenomena and make reliable quantitative predictions.
The above formulations of the second law of thermodynamics are, in a sense, somewhat outside the realm of the chemistry of molecular and supramolecular systems. These formulations may seem to be even farther from biology, sociology, and other sciences that are mainly based on chemistry, both molecular chemistry per se and the chemistry of supramolecular structures, which we perceive as "chemistry around us". Therefore, it is not unexpected that a purely physical, rather than a physicochemical, approach to the origin of life, biological evolution, and the aging of living organisms has lead to numerous misunderstandings—one might say, even to tragic errors—in life science. It should suffice to mention L. Boltzmann's, E. Schrödinger's, I. Prigogine’s [29–31], and other researchers' fallacies accounted for by neglecting to some or another extent, Gibbs's works and underestimating the possibilities offered by thermodynamics. The following publications emphasized the substantial misunderstandings in this field [11, 20, 22, 23] that the founders of classical thermodynamics noted long ago [1, 2, 10–13].
To justify these statements, let us make a digression to cite the renowned scientists Boltzmann and Schrödinger who asserted that "living organisms struggle for negative entropy" or, as it is sometimes called, "negentropy". In addition, we will cite some of Prigogine's quotations that appeared even on the cover of his books. The reader will find references to them in the Internet [31, 32]. Here we note that the quotations presented below do not pertain to the second law of thermodynamics in its classical form [2, 9, 10]. Today, they may seem surprising, especially when we take into account that all this was written several years after Gibbs published his works. For example, Boltzmann (1886) wrote:
Erwin Schrodinger [1887-1961]
SECOND LAW vs. DISSIPATIVE STRUCTURES:
In order to solve these "contradictions", Prigogine  developed the theory of dissipative structures, i.e., the structures that appear in systems that are far from equilibrium, the prime example being Bénard cells. Later, it turned out that the theory did not allow for the overcoming of the aforementioned "contradictions." In fact, it made the imbroglio even more intricate. It later became obvious that Prigogine's views do not agree with the second law of thermodynamics [20, 22, 24]. This is so in many respects. Suffice it to say that, in the general case, the Prigogine entropy (S' or Si) has no full differential. Therefore, his theory cannot be regarded as thermodynamic. This is a kinetic theory based on an "entropy" (Prigogine's entropy, S') which can be neither calculated nor measured.
Prigogine considered Gibbs's work to be mainly theoretical and stated that Gibbs's method is inapplicable to studying physicochemical transformations, such as chemical reactions, because the values used in this method are the functions of state pertaining to the whole system or its individual component. Prigogine actively publicized his views via scientific literature and textbooks . As a result, many researchers refer to Prigogine's formulation of the second law of thermodynamics.
This formulation, in actuality, applies to a particular speculative model and may be accepted only on certain premises that cannot be proved. Unfortunately, this concept, which, in a sense, contradicts the principles of science itself [5, 6], was supported by many researchers. Owing to efficient publicity, these colleagues were convinced by hardly comprehensible (in physical terms) formulas and doubtful argumentation. From this perspective, the supporters of Prigogine's theory were, in a sense, deceived.
Ilya Prigogine [1917-2003]
"this contradiction cannot be removed as long as one tries to understand living systems by the methods of equilibrium thermodynamics".
Thus, the aforementioned concepts by Boltzmann, Schrödinger, Prigogine, and their adherents turned out to be at best tentative ones, or even a dead end. They hampered for many decades in the search for the ways of explaining the evolution of living systems in physicochemical terms on the basis of the second law of thermodynamics. As detailed previously, only in recent decades were the principles of hierarchical thermodynamics, i.e. macrothermodynamics, formulated. In this direction, the theoretical structure of Gibbs’ equilibrium methodologies have been extended for use in creating the physical or physicochemical theories of the origin of life, biological evolution, and the aging of living organisms [17, 22–28].
As noted, the physical substantiation of the second law of thermodynamics deals with ideal processes and is based on the concept of statistical entropy. The nonequilibrium thermodynamics of systems that are far from equilibrium tries to study the changes in "kinetic entropy", e.g. Prigogine's entropy (S' or Si), which, as mentioned, has no full differential or even an approximate one and cannot be calculated or applied to biological processes in principle! In addition, the approaches used in nonequilibrium thermodynamics of systems far from equilibrium create difficulties related, e.g., to the notions on the thermodynamics of processes and the thermodynamics of systems.
Although Prigogine's theory proved an impasse, it still has its followers. Nevertheless, no numerical data obtained from either experiment or observations have confirmed the theory even at the qualitative level [20, 22]. Moreover, many physicochemical processes of the formation of spatially periodic structures, which Prigogine and his coauthors regarded as “dissipative”, were explained long ago in terms of the thermodynamic models of quasi-equilibrium systems, without involving the concept of dissipative structures. It is generally known that in 1897 the German chemist Wilhelm Ostwald used the notion of supersaturation to explain the existence of such systems in nature.
Wilhelm Ostwald [1853-1932]
where V is the volume of the system; m is the mass of the identified microvolumes; x, y, and z are coordinates; "—" implies we are using the specific value (relating to the macrovolume) of G; "im" signifies a inter-molecular or supra-molecular structure; and "~" stresses the heterogeneous character of the system.
Note that expression (4) implies that intermolecular or supramolecular interactions in all hierarchical structures of biological tissues, both intracellular and intercellular interactions, are taken into account. This is justified because the structural hierarchy does not necessarily coincide with the temporal one. For example, cells of some types do not divide; like organs, they age simultaneously with the body as a whole. However, for each supramolecular hierarchy (j–1), there exists a higher hierarchy (j+х) such that:
One of the greatest merits of Gibbs and other renowned founders of classical thermodynamics is that they used the works by Joseph Lagrange, Leonhard Euler, and other outstanding mathematicians, in particular the variation principles developed by them, as a basis for the concepts of the functions of state of the system other than entropy, which, like entropy, have full differentials. The functions of state permit the determination of the directions of spontaneous processes and the estimation of the extent of their advancement in individual thermodynamic systems identified in the real world.
In other words, the evolution of systems themselves can now be studied, to a certain approximation, if certain natural (independent) variables are constant. The Gibbs functionG, i.e. the Gibbs free energy or, briefly, the Gibbs energy, can be used for studying equilibrium, i.e. quasi-equilibrium, processes and closed systems, i.e. quasi-closed systems in which quasi-equilibrium transformations occur, at constant temperature and pressure. Similarly, the Helmholtz functionA, is applicable to studying these processes and systems at constant temperature and volume.
Certainly, this is only true on the assumption that the functions of state of the systems studied have actual physical sense at any moment of time. This is true for systems close to equilibrium but not for those far from equilibrium. Here, we emphasize once more that the law of temporal hierarchies gives grounds for the use of the functions of state when the direction and the extent of advancement of the evolutionary processes that occur in quasi-closed systems are estimated at different hierarchical levels of living matter . For clarity, let us make a digression on the law of temporal hierarchies.
The law of temporal hierarchies makes it possible to identify quasi-closed thermodynamic systems and subsystems within open biological systems, thus facilitating the study of individual development (ontogenesis) and evolution (phylogenesis) of these subsystems via the study of the changes in the “specific”, i.e. calculated per unit volume or mass, Gibbs function for the formation of a given higher monohierarchical structure out of lower monohierarchical structures. For example, it has been found that the specific Gibbs function for the formation of supramolecular structures of biological tissues G tends towards its minimum in the course of ontogenesis as well as for phylogenesis and evolution as a whole:
t j-1 << t j+x
where t j-1 and t j+x are the mean lifetimes (or lifespan) of the elementary structures of the respective structural hierarchies in the living system; x = 0, 1, 2, etc. Note that the internal medium and many fragments of nondividing cells are nevertheless renewed due to metabolism. The use of expression (4) actually means that, in the given case, the law of temporal hierarchies assumes the following form:
... << t m << t im << t org << t pop << ...
Here, t m is the mean lifetime of molecules or chemical substance involved in metabolism in the body, t im is the mean lifetime of all intermolecular or supramolecular structures of tissues renewed during individual growth and development, t org is the mean lifetime of individual organisms in a population, and t pop is the mean population lifetime.
Here, we have deliberately excluded the lifetimes of cells and some other complex supramolecular structures from the series of strong inequalities (6) for the reasons indicated above. However, this series certainly represents a general law of nature consistent with reality and reflecting the existence of temporal hierarchies in living systems.
The law of temporal hierarchies is related to the presence of metabolism or other forms of substance transformation at all hierarchical levels. Note that metabolism is an essential characteristic of living organisms. This law, which some refer to as Gladyshev's law, allows for the strict demonstration of the possibility of identifying or discerning quasi-closed monohierarchical systems and subsystems within open polyhierarchical biological systems. This statement entirely agrees with the experience accumulated in theoretical and experimental physics [3, 5]. Certainty demands that this assertion cannot raise any objections.
It is impossible in this short article to list all of the important conditions for the use of each function of state of each respective system. Moreover, we have not noted all of the main "delicate" points that beginners should take into account. Besides, we have referred to just a few publications, those that are most important. It should also be noted that this paper, as well as most publications on thermodynamics, may contain some inaccuracies of wording resulting from the ambiguity of translation. For example, most professional scientists know about inexcusable confusions with the terms isolated system and closed system (originally English). Both terms are sometimes translated into Russian as замкнутая система (literally, closed system). So the terms are often regarded as equivalent or identical.
Other errors result from semantic coincidence of some terms. For example, the Gibbs and Helmholtz free energies are often confused with energy in the ordinary sense. This is why many researchers have attempted to replace this term with the term the Gibbs function . Another example is the term complex system. Here, the word complex has a double meaning. In thermodynamics, a complex system, as opposed to a simple one, usually means a system in which or on which a work other than the work of expansion is done [15, 16]. Sometimes, however, the word complex is used to emphasize a structural or some other heterogeneity of the system itself or the diversity of its elements. This also applies to the term simple system, and so on.
Certainly, these and other such confusions may lead to blunders that escape a nonprofessional's notice. These and other similar errors creep into some textbooks, reference books, and then into the Internet. It is our intention that the above remarks will warn beginners about the erroneous views that may exist in thermodynamics. Also, we encourage all physicists, chemists, biologists, and other specialists that deal with thermodynamics to study the Gibbs phenomenological thermodynamics first of all.
As noted above, this authentic and, in a certain sense, true thermodynamics is based on the notion of full differentials. This approach to understanding the world surrounding us is intrinsically irrefutable. We may only discuss the accuracy of the Gibbs thermodynamics as applied to, e.g., quasi-closed systems the processes in which are close to equilibrium. In accordance with the very essence of the full differential, i.e. its mathematical meaning, as well as the first law of thermodynamics, the change in the function of state of the system accompanying the transition from one equilibrium state to another is independent of the way or mechanisms of this transition.
It is likely that our lack of knowledge on actual complex systems may be partly attributed to the changes in entropy during this transition, being that the entropy cannot be measured directly. The changes in phenomenological entropy accompanying transformations in both simple and complex systems may be calculated only if one has studied the corresponding thermal processes. In statistical terms, the entropy is calculated only for ideal systems or systems close to ideal. It is impossible to perform any precise calculations of this function of state for systems with significant interactions between particles, i.e. molecules and supramolecular structures, on a statistical basis. We would like to emphasize that this applies to complex thermodynamic systems, i.e., the systems in which measurable interactions occur.
Thermodynamics, owing to its impeccably reliable mathematical basis, may be regarded as a "machine" that always yields the right result if the premises are correct. Physical chemistry has repeatedly confirmed this [8–10, 14, 19]. Unfortunately, some physicists, biophysicists, biologists, and, especially, modern "philosophers" are still unaware of this experience of chemists and chemical technologists.
We repeat that the aforementioned ambiguities, which are mainly related to the disregard of the correct use of many terms that are semantically similar but differ in physical meaning, result in confusion and misunderstandings. These misunderstandings discredit, at least in nonprofessionals' opinion, thermodynamics itself and science as a whole. Hence, the numerous incorrect interpretations of the second law of thermodynamics, various dubious "views" on entropy [11, 13, 20, 22], and other far-fetched "functions of state of systems" in the literature are apparent.
Many authors, ignoring classical works in this field, apply different formulations of the second law of thermodynamics to systems where they are inherently inapplicable. Some of these authors suggest their own interpretations of this general law of nature. This debases science and education. Moreover, it can be said that several "second laws of thermodynamics" have appeared, none of which having anything to do with reality. A good example is the aforementioned Prigogine's  interpretation of the second law of thermodynamics. This interpretation "extends" the well-known incorrect and indemonstrable statement by the great Boltzmann , who neglected the important concepts put forward by Clausius and Gibbs.
The interpretation suggested by Prigogine has practically conquered the "scientific" world and still remains one of the trendiest interpretations of the second law of thermodynamics. We are well aware that it would be hopeless to argue with the visionaries that create or support these concepts: they have developed an excellent method for leading such debates. They unfailingly give lots of arguments, which are mostly quotations from published or oral statements made by other visionaries or by insufficiently informed scientists. It is often emphasized that those scientists are well known or even famous. However, the visionaries forget that scientists that are well known and famous in one field are not necessarily professionals in others. The only way to withstand this conjuncture is to refer the readers to classical works and serious textbooks written in a highly professional milieu of world-renowned scientific schools with centuries-long traditions.
Thus, making no pretensions to perfection, we would nevertheless like to offer advice to researchers dealing with thermodynamics, as well as other branches of science, and the editors of scientific periodicals. This advice is the following: when discussing the problems of thermodynamics or using its mathematical tools for calculations, it is necessary to clarify "ambiguous" terms and definitions. It is also advisable to refer to the classical works including textbooks, reference books, and encyclopedias that the authors of the original publications used. In this case, the correctness of the results reported in the publications can be at least preliminarily estimated.
I am grateful to Prof. M.M. Abdildin, Prof. G. Arrhenius, Prof. N.N. Bogolyubov, Jr., Prof. K.G. Denbigh, Prof. V.P. Kazakov, Prof. Yu.S. Lipatov, Prof. A.A. Logunov, Prof. V.V. Sychev, and Prof. V.M. Frolov for the support, systematic fruitful discussions, and valuable advice.
THIMS' HUMAN THERMODYNAMICS:
On the forefront of hierarchical thermodynamics, is the work of chemical engineer Libb Thims who in 2001 published, via local distribution, a short paper entitled “On the application of the Gibbs free energy equation to the human reaction mechanism.” Before this, however, building on the mathematical framework and structure of Gibbsian thermodynamics, beginning in 1995, Thims proposed to investigate the interactions of humans, from a reactionary point of view, within their respective structural hierarchies, based on the essentials of physical chemistry, i.e. Gibbsian thermodynamics, and thus created “human thermodynamics”.
In fact, this application of thermodynamics has been applied similarly to the philosophic reduction principle that was also used in the development of hierarchical thermodynamics. In doing so, Thims extended, i.e. applied, the principles of chemical thermodynamics to the interactions between humans. He has named the elemental structures, i.e. the human organism and their communities in the human hierarchy, “human molecules”. Thims’ model fully corresponds to the principles of hierarchical thermodynamics which allow us to apply the laws of physics and chemistry, primary physical chemistry, to all temporal and structural hierarchies and sub-hierarchies of our world. Thus Thims’ theory has a reliable foundation and is a key step in the human community sciences.