The Common Extremalities in Biology and Physics: Maximum Energy Dissipation Principle in Chemistry, Biology, Physics and Evolution

By Adam Moroz

The Common Extremalities in Biology and Physics is the first unified systemic description of dissipative phenomena, taking place in biology, and non-dissipative (conservative) phenomena, which is more relevant to physics. Fully updated and revised, this new edition extends our understanding of nonlinear phenomena in biology and physics from the extreme / optimal perspective.

• The first book to provide understanding of physical phenomena from a biological perspective and biological phenomena from a physical perspective
• Discusses emerging fields and analysis
• Provides examples

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 14, 2011
• ISBN: 9780123851888

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Table of Contents

Cover image

Front-matter

Copyright

Preface

1. Extreme Energy Dissipation

1.1. Hierarchy of the Energy Transformation

1.2. Extreme Properties of Energy Dissipation

1.3. Optimal-Control-Based Framework for Dissipative Chemical Kinetics

1.4. Conclusions

2. Some General Optimal Control Problems Useful for Biokinetics

2.1. Extreme Dissipation, Optimal Control, and the Least Action Principle

2.2. Some One-Dimensional Examples of Biokinetics and Optimal Control

2.3. General Multidimensional Examples of the Introduction of Optimal Control into Biokinetics

2.4. Conclusions

3. Variational and the Optimal Control Models in Biokinetics

3.1. Optimal Control Model of Binding Cooperativity

3.2. Enzyme Kinetics and Optimal Control

3.3. Optimal Control, Variational Methods, and Multienzymatic Kinetics

3.4. Optimal Control in Hierarchical Biological Systems: Organism and Metabolic Hierarchy

4. Extreme Character of Evolution in Trophic Pyramid of Biological Systems and the Maximum Energy Dissipation/Least Action Principle

4.1. Acceleration of Dissipation in Molecular Processes is the Cause of Emergence of Trophic Pyramid of Biological Systems

4.2. Maximum Energy Dissipation Principle and Evolution of Biological Systems

4.3. The Pinnacle of Trophic Pyramid of Biological Systems—Symbiosis of Biological and Nonbiological Accelerating Loops: Technological Accelerating Loop

5. Phenomenological Cost and Penalty Interpretation of the Lagrange Formalism in Physics

5.1. Fusing Mechanics and Optimal Control

5.2. Finiteness of the Propagation Velocity of Physical Interactions and Physical Penalty

5.3. Phenomenology of the Nonmechanical Penalty for Free Fields

5.4. Internal Symmetry of the Physical Penalty

5.5. Physical Interactions and Penalty

5.6. Physical Evolution in Light of Maximum Energy Dissipation Principle

5.7. Conclusion: Physical Phenomena from the Point of View of Biological Ones

6. Conceptual Aspects of the Common Extrema in Biology and Physics

6.1. Self-Sufficiency of Extreme Transformations

6.2. Intensive and Extensive Property of Displaying of Material Instability

6.3. Natural and Biotic Things—Lethal Gap or Irrational Compromise

Main Conclusions and Remaining Questions

Front-matter

The Common Extremalities in Biology and Physics

The Common Extremalities in Biology and Physics

Maximum Energy Dissipation Principle in Chemistry, Biology, Physics and Evolution

Second Edition

Adam Moroz

De Montfort University

Leicester, UK

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Preface

The science of living nature is known as biology. Biology, in the modern sense of the word, encompasses the entire hierarchy of life from the atomic-molecular level to the global biogeocenosis. Furthermore, biology also formulates all temporal laws of relationships in this complicated and, indeed, trophic hierarchy. In other words, biology formulates evolution since life is not only a form of existence but also, in a sense, a triumphal progression towards perfection.

Nevertheless, biology does not provide a satisfactory explanation for the origin of life. How do we account for the emergence of biological processes in this immense universe of dust, stars, planets and vacuum? Is it merely down to random chance? Or, if life is not accidental, what does this signify? Biology does not explain the transition from inorganic objects to organic life perhaps because the reasons are too broad to be understood in purely biological terms. Moreover, the concept of evolution has infiltrated and now permeates physics, that other ancient vision of mankind and nature. A complex question arises: to which laws does life owe its existence? Essentially, the answer lies partially within the realm of physics–a science which is fundamentally concerned with non-living nature–and partially within the realm of biology. It seems that the answer to this question leads to a deep unity between physics and biology.

A non-evolutionary theory of the origin of life (‘the Creation’) centres on the involvement of a ‘super-essence’ (a super-individuality or a super-civilisation) responsible for kick-starting the processes on Earth into life. The theory is reliant on the inevitable and necessary emergence of the ‘super-essence’, preceded by the appearance of primitive or increasingly sophisticated beings in nature at intermediate stages. Therefore the question of the origin of life can be reformulated in various ways: To what extent do the laws of inorganic nature and of physics derive from, produce and require the emergence of biological processes? Is it possible to deduce biological laws from physical and chemical laws? How do we define the relationships between physical and biological processes? According to which law are physical processes transformed into biological processes? To what degree are biological regularities governed by physical regularities? Success in answering these questions, even at an elementary level, might well enable the development of a conceptual methodology that would generate biological laws based on physical laws. Physics and biology would, then, be united by a uniform concept resulting in a scientific ideology more accurately reflecting the interconnectivity of nature.

Therefore, this work represents an attempt to evaluate the feasibility of such a mode of thinking that could be considered to allow some additional steps on the path to better understanding the relationship between biotic and physical processes.

However, one should note that any concept about nature, whether a simple mental picture or a complex formal mathematical scheme, is only one of many models relating to matter. Concepts such as these are produced within the social forms of informational mapping, cognition or information reflection. Mathematical science (including the theory of models and the theory of systems) is itself merely one form of information reflection, mapping and modelling. It can be characterised by a dissociation from the material world (from supporting material messengers and processes), creating an ideal, almost spiritual, models, and sometimes could be thought that nature itself moves according to these models.

Nevertheless, mathematics, though eloquent in its description of nature, is simply a tool. It minimises the materiality of biosocial informational mapping systems, creating sophisticated matter-less models of nature to a somewhat abstract level. One can say that these mathematical models are the most formalised of models and have the most information and functional capacity per least structural-energy cost. This is one reason for the high efficiency of mathematical modelling. And yet, it is an idealisation that could be considered to be rather two-dimensional paper form and recently appears to have taken on a distinctly electronic character.

It is well known that the formal mathematical modeling has achieved the greatest success in explanation, description, and the forecasting of physical phenomena, as well as in formal reconstruction of processes that take place within physical systems. At the highest level, the description of physical systems and processes proceeds from an extreme ideology to enable the formal mapping of physical interaction or dynamics. This ideology is based on the least action principle employing the variational method. The methodology of this approach contains the following stages:

• There is a physical value called the action, which has the dimensional representation of the product of energy by time.

• The action, set as some value on all possible motions of a system, aims at minimum value at any rather small interval of movement of a system.

• From the principle using variational technique, one can obtain equations of movement of a physical system (the Euler–Lagrange equations).

• The trajectories, or the laws of movement of the system, can be obtained from the Euler–Lagrange equations.

As follows from the first stage above, as early as the highest level of formalism, physical modeling implies the energy sense of physical interaction and, as it turns out, physical evolution. It is only at the final stage of the modeling process that the outcome appears as a purely kinematical result—the movement trajectories. The last stage also represents another sort of system behavior model—a model of states of a physical system, on which it is possible to forecast the behavior of a real system.

From this point of view, the formal mathematical description in biology has significant methodological difference, possibly a halfway policy. Here one can initially proceed from concepts and terms of a dynamic system (also of some formal design), and in the majority of classical cases, from a system of differential equations and hybrid systems for more complex models. The solution of such a system represents the law of movement or trajectory, providing information on the location of a real system at any moment of time in the multidimensional phase space of parameters of a biological system.

We shall note that in contrast to the physical way of formal modeling, the energetic sense, as the most formalized scheme of phenomena occurring in a biological system, escapes. However, this sense, indeed, is well verified by the whole logic of physical formalism, and this sense in itself is not less important in the conception of the nature of biological phenomena.

This argument proceeds from the suggestion that it is the energy sense that can initiate the level of formalism, similar to top-level variational formalism in physical description, and consequently, it is ideology of a common and unified approach in biology and physics.

In connection with the above, it is important to look at the most common energy laws of biological phenomena (which, in fact, are the thermodynamic laws) in order to mathematically formalize, with the purpose of development on the basis of these laws, a universal, informative, and formal scheme generalizing the laws of biology. We expect that such ideas could result in a formalism, similar to variational formalism in physics, and that it could be a basis for the ideological unification of biology and physics.

One may also bring to mind that the determining difference of biotic processes is that they carry out the utilization or dissipation of energy, with the qualitatively irreversible transformation of free energy to the thermal form. It is this that hinders a direct introduction of the ideology of the least action principle into biology and in biological kinetics.

Therefore, we could initially consider the interpretation of the variational approach with reference to the processes with explicit dissipation, i.e., to relaxation processes in chemical and biological kinetics.

In this connection, it is expedient to reflect on the energy sense of the phenomena related to these areas, i.e., the hidden dynamic reason of one or another biological processes and the form of their representation (mapping) in the corresponding formal models. In a sense, it would be similar to the solution of the reverse problem of variational calculus for biological kinetics—when the variational function of the corresponding under-integral function, the Lagrange function, needs to be found from the equations of motion, from a dynamic system or a system of differential equations. The solution of such a problem would enable us to analyze in an explicit form the energy properties of the phenomena initially presented within the parameters of a dynamic system. However, the reverse variational problem could be solved for a very limited range of cases, and there is little optimism about finding the successful solution as far as biokinetics is concerned.

Thus, it is possible to follow two different approaches in the formal mathematical and deterministic descriptions of these rather opposite groups of phenomena—biology and physics. The first is related to physics, with an explicit energy sense outgoing from energy properties of the physical phenomena, from the least action principle, leading through the Euler–Lagrange equations to the laws of motion or trajectories. And the second, more widespread in biology, likely begins with a comparison of a physical description, directly from so-called dynamic models, of the systems of differential or other kinds of equations, and it finally results in the same stage—the laws of motion, or trajectories.

We expected that the mutual penetration of both approaches could to a great extent promote mutual development as well as the technical and ideological enrichment of physics and biology.

We shall emphasize that the undertaken consideration concerns rather classical models—the models presented by systems of differential equations; however, even such a phenomenological consideration is difficult to implement consecutively within the frameworks of these two broad and opposing phenomena—the biological and the physical.

1. Extreme Energy Dissipation

In this chapter, thermodynamics as a science that connects physics and biology by the phenomenology of energetic transformations is considered. However, thermodynamics is the simplest model of physical systems, with only two levels of hierarchy: the macroscopic and microscopic. The macroscopic nature of basic thermodynamic quantities—energy, heat and work, free energy, and thermodynamic entropy—is discussed together with the main properties of free energy dissipation or entropy production. The biological multileveled hierarchy and thermodynamic two-leveled hierarchy are compared. A growing role of phenomenological outline in the unified description of the multiple/complex hierarchy of biological systems is suggested. The role of extreme principles in physical description is illustrated in the second section. Two opposite concepts—the Prigogine principle of the minimum entropy production and the Ziegler principle of the maximum rate of energy dissipation—are discussed. The relation of the Ziegler principle to the least action principle has been suggested. The dimensionality idea is used to interpret the least action principle of thermodynamics. In the third section, the dynamic optimal control formulation of mechanics is illustrated. Using this example, the dynamic optimal control approach to formulate the variational framework for nonequilibrium thermodynamics has been proposed. Traditional Lagrange and Hamilton formulations of the variational problem for nonequilibrium thermodynamics are illustrated, as is the Hamilton–Jacobi equation. Within this optimal control-based variational approach, the role of free energy in the cost (penalty) function is discussed, as well as the energetical cost/penalty interpretation of the Lagrangian and Hamiltonian functions and the thermodynamic momenta. Relaxation in an RC circuit in terms of proposed approach is illustrated as simple physical example.

Keywords

Thermodynamics, biology, hierarchy, free energy, maximum energy dissipation principle, least action principle, variational calculus, optimal control, Lagrangian, Hamiltonian

1.1. Hierarchy of the Energy Transformation

1.1.1. Thermodynamics—A Science That Connects Physics and Biology

The general laws connecting biology and physics are particularly related to energy transformations, since thermodynamics is the phenomenological science that describes the energetical macroscopic characteristics of systems. Thermodynamics, which directly relates to biology, is known as biological thermodynamics. It covers subjects connected to the interconversions of different forms of energy, ranging from those in the simplest chemical reactions and ending with energy complex trophic changes of the biomass of different species. The energy and structure conversions in these complex changes eventually end and, can be saying in a different way, transfer to another quality in the large number of social processes.

Evolutionary and methodologically biological thermodynamics begins with the thermodynamics of chemical reactions. The latter are known to have produced a huge variety of far from equilibrium (and also from steady state) phase-separated biochemical systems, which are actually biotic cells. One can, therefore, imply that the thermodynamic (energetical transformation) laws of biology begin with the thermodynamic laws of chemical reactions. The study of these laws is termed chemical kinetics. For example, the thermodynamic fluxes are the velocities of chemical reactions, and chemical forces are no more than the affinity for chemical reactions. It is, therefore, evident that the subjects of chemical thermodynamics and chemical kinetics overlap to a large extent.

One can also say that biotic organisms are complex, phase-separated, chemical reactions that contain very specific molecular forms of informational support processes. It can be said that these reactions, in the process of evolution, have allowed organisms to acquire not only mechanical but also the development of more complex high-adaptive degrees of freedom—informational. On some stages of the evolution, these complex reactions significantly enhanced the role of thermodynamic regulatory feedback loops, regulating for instance the heat balance in the process of cellular respiration or maintaining the temperature of the body and so on.

However, thermodynamic systems operate with some characteristics that reflect the hierarchy of the physical quantities in the process of energy transformation. Biological thermodynamics, in turn, mirrors the hierarchy of the complex biological world. It is, therefore, useful to remind ourselves of the construction of the hierarchal thermodynamic terms and the definition of these with respect to the crucial differences in the organizational hierarchy—a central point in the difference between pure thermodynamic and biological phenomena.

1.1.2. Hierarchy of the Processes and Parameters in Thermodynamics

Thermodynamics is known as a phenomenological science. Thermodynamics represents a classical and historical example of a macroscopic description of the energetic transformations in various macrosystems. However, it is important to note that the understanding of macroscopic and particularly microscopic phenomena has steadily been changing with time.

Thermodynamics, as we know, deals with the systems containing a large number of particles (around 10 ¹⁰–10 ³⁰). As we mentioned, such macroscopic systems can be characterized by two kinds of variables:

1. Macroscopic parameters—characterizing the system in relation to the neighboring macroscopic world, or the system as a whole. Two classic examples of these variables are volume and pressure.

2. Microscopic parameters—characterizing the properties of the particles that make up the system (mass of the particles, their velocities, momenta, and so on). Now, it seems obvious that in any study of processes and systems, it is possible to set at least two fundamentally different edge levels for these processes, i.e., macroscopic and microscopic levels. The former is known as the phenomenological level, which can be heavily characterized by thermodynamics.

Let us note, therefore, that the concept of a thermodynamic system, as studied in thermodynamics, is more complicated than the concept of a mechanical system, due to the dynamic nature of the values at both of these levels. Clearly, these two levels of variables are interrelated, although they have their own dynamism. The inconvenience of describing a one level (macro), which employs the microscopic description of the states of all components of a system of microparticles that carry the microscopic parameters, leads to a statistical interpretation of these quantities, which connects them to the macroscopic parameters. The fundamental relationships involved are closely related to thermodynamics—a form of statistical mechanics. Thermodynamic consideration deals only with the macroscopic parameters of the systems, i.e., those of clear phenomenological character.

Therefore, the distinctive feature of thermodynamics (as a phenomenological, macroscopic description) relative to mechanics (microscopic description) is that for the thermodynamic systems the concept of two types of processes is considered. In some sense, thermodynamics is the first hierarchical science within physics. If in mechanics the reversible character of processes is the rule, and the irreversibility in some way is an exception, in thermodynamics, perhaps, reversibility of processes is the exception, and irreversibility is the rule. Thermodynamics, therefore, requires specific fundamental law to take account of its macroscopic nature—the second law of thermodynamics. The apparent dominance of irreversible processes in the macroworld is associated with the peculiarity of the dynamic nature of the relationship of microstates and macrostates of the thermodynamic system. Reversible processes are understood as taking place in such a way that all the macroscopic parameters can be changed in the opposite direction, without any other macroscopic changes, even outside the system. The irreversible processes occur so that they can run in the opposite direction, just when connected with other macroscopic changes, such as the environment. Reversibility and irreversibility, which manifest themselves macroscopically, are closely linked with the microscopic characteristics of particles, i.e., their own dynamism. Due to the dynamic nature of these macroparameters and the large range of energy that characterizes (changes/transformations) the system, these values have a certain hierarchy.

1.1.3. Macroparameters: Energy and the Forms of Its Exchange

In consideration of the physical interactions in thermodynamics, the nature of interaction is explicitly emphasized as the exchange of energy through two distinct processes—it is the result of work or heat transfer. However, as we mentioned in thermodynamics, there are two levels of hierarchical processes—the microscopic and macroscopic. These and, therefore, the energy exchange (or thus, the interaction) involved in thermodynamics are different and have the appropriate hierarchy. Energy, traditionally, is distinguished in several forms.

The internal energy of a system takes all the available energy into account, without regard to the hierarchy of interactions at the macrolevel or microlevel. This energy includes the energy of all microscopic particles, at all levels of the hierarchy of the system, and includes the energy of all known interactions between them, as well as the macroscopic part of energy (related to the system macroparameters, like pressure, volume). It should be emphasized that because of this broad concept of internal energy, it is impossible to establish its full value for any system, because it includes a large number of constituents that are difficult to take into account. Therefore, we often deal only with the change in internal energy of the system between any of the states of the system.

Heat, also referred to as thermal energy, is the kinetic energy of the microparticles that make up the system. This energy is transmitted through the exchange of the microscopic kinetic energy of the microparticles during their collisions. Therefore, thermal energy (heat) has macroscopic properties due to the large numbers of particles involved in the kinetic motion and the large amount of transferred energy. This type of energy exchange is not linked to the exchange of the energy of a system in the process of work.

Because nonequilibrium states are characteristic of macrosystems, the energy in thermodynamics acquires one other property. The energy can also be considered as a measure that characterizes the aspiration of processes and systems to reach their equilibrium. In other words, it can be considered as the measure of the relationship between the relatively nonequilibrium degrees of freedom and the equilibrium. In a certain sense, the nonequilibrated degrees of freedom can be interpreted as overcrowded by motion. To some extent, the energy is a measure of the overflow by the motion of degrees of freedom (a measure of the nonequilibrium structural state). Therefore, the apparent micro- and macrodifferentiation dominates when considering the hierarchy of forms of energy in thermodynamics.

1.1.4. Macroparameters: Heat as a Nonmechanical Method to Change the Macrostate of Thermodynamic Systems

Thermodynamics, in the first instance, studies the range of phenomena that are related to heat (thermal heat). Heat, the thermal energy Q, is primarily a macroscopic materialization of the mechanical motion of a large number of microparticles. Actually, the energy of this motion is characterized as thermal energy.

Paradoxically, heat is a macroscopic manifestation of microscopic changes, and at the same time, it is a microscopic form of energy exchange, having a macroscopic effect. However, it should be more rigorously understood that heat is the microscopic form of energy transfer that is related to the change of macroscopic parameters, like temperature, which has both a macroscopic and microscopic sense. Therefore, heat transfer is only a microscopic form of change in internal energy.

The temperature reflects the macroscopic manifestation of the intensity of the microscopic motion. Temperature is the molar heat of the kinetic energy per one mechanical degree of freedom. Therefore, heat energy is transmitted at the microscopic level and not directly related to the macroscopic work.

1.1.5. Macroparameters: Physical Work as a Pure Mechanical Way to Change Macroparameters

Work looks like it is in opposition to heat: It is a way to change the internal energy of a macrosystem, the method of transmitting of energy in a process, when the transfer process is directly related to the change of macroscopic parameters.

The concept of work in thermodynamics comes from mechanics. In mechanics, the elementary work is the product of force on the small displacement:

(1.1)

It should be noted that the elementary work, even in mechanics, is not, generally speaking, the exact differential of any function of the displacement l, and, therefore, at the designation of its elementary value employed δ A, and the sign of the elementary change is not used 1. and 2.. It is also important because the work is not a measure or function of the state, but is only a measure of processes, a quantitative measure of the energy exchange in a process. Work is a function of the process not the state.

In a simple example of the thermodynamic case for an ideal gas, work is equal to the product

(1.2)

It should be emphasized that in thermodynamics, work is also not an exact differential of any function of the state, but work is a function of the process 2. and 3.. The formal property of this underscores the fact that the work is a process, there is a means of energy transfer, and it is not a function of state. On the other hand, work is a quantitative measure of energy transfer into the system through the action on it of some generalized forces from other systems.

1.1.6. Macroparameters: The Energy Conservation Law

The first law of thermodynamics imposes the quantitative relationship for the transformations between the macroscopic and microscopic forms of energy (in a wide sense between the qualitatively different degrees of freedom of physical motion) to another. Formally, this is the postulation of the existence of an additive value—the internal energy of the system.

The change in the internal energy of a system is equal to the sum of the heat into the system and the work done on the system, which is formally expressed as:

(1.3)

where d E is change in the internal energy of the system, δ Q is the amount of thermal energy supply to the system to heat the microscopic degrees of freedom, and δ A is the work done on the system or the amount of energy that the system gained by the nonthermal macroscopic degrees of freedom.

The first law of thermodynamics strictly delineates the possibility of different kinds of energy in relation to the processes in which the system participates. These processes are the microscopic and macroscopic forms of energy transfer: heat and work. Actually, this is a distinction in the microscopic and macroscopic aspect, as heat and work do, in this sense, belong to different levels of this two-leveled hierarchy.

The first law of thermodynamics does not discriminate between the macrodegrees and microdegrees of freedom, or the interaction between systems. This interaction depends on the hierarchical affiliation, which, as it turns out, is related to reversibility or irreversibility of the interaction process of energy exchange. It should be emphasized that this question arises only in thermodynamics. In mechanics, its emergence does not manifest itself so clearly. It is the second law of thermodynamics that raises the question of the status of energy as a measure of reversibility/equilibrity. The first law discriminates between the ways of energy exchange, in terms of thermal and nonthermal, and, naturally, states that the overall energy in their forms is conserved.

However, even if the macroscopic parameters remain constant, changes may occur at the microscopic level. This leads to the fact that for the same macrostate, the system can have multiple sets (numbers) of microstates that can be different in the sense of stability. This last fact leads to the second law of thermodynamics.

1.1.7. Macroparameters: Free Energy—Macroscopic Measure of Nonequilibrium

We can say that the thermodynamic study of the interaction of qualitatively different macroscopic degrees of freedom is an investigation of the redistribution of energy among the various structural and energetic macrostates. These macrostates represent the degree of freedom in the system, during its interaction with the environment or another system. This sense of imbalance (in the sense of equilibrium) in all degrees of freedom of the system, regardless of the inflow of external imbalance, or an existing imbalance in the system, manifests itself according to the second law of thermodynamics, as a more or less equilibrated state. More specifically, macroscopic forms of energy (related to microscopic degrees of freedom and macrostates) are divided into thermal and nonthermal. This division is a characteristic feature, the basis for thermodynamics, and its main laws define the relationship between all forms of energy, in accordance with this division.

In line with this interpretation of micro and macro forms of energy, the internal energy of a system can be qualitatively divided into the relationship between the possibilities of its transformation into the macroscopically ordered form of energy—work (particularly into mechanical work). This part of the internal energy that can be converted into any type of work—mechanical, chemical, electrical—can be defined as free energy. Another part of the internal energy, which cannot be converted into macroscopic work (as was already mentioned), is referred to as the bounded energy and is usually associated in thermodynamics with the energy of the thermal motion of particles that make up the thermodynamic system.

1.1.8. Macroparameters: Universal Fatality of the Processes—The Second Law of Thermodynamics and the Hierarchy of Energy

The second law of thermodynamics reveals the properties of reversibility/stability or irreversibility/instability of a process of interaction of one or another degree of freedom or that of another way of energy exchange. It reveals the reaction of the system, describes a macroscopic interaction as a way to change the nonequilibrity, and highlights the special status of the thermal degree of freedom as the most equilibrated (stable one), thereby selecting the thermal energy, both qualitatively and quantitatively. The second law underscores the crucial irreversibility of the thermodynamics of all processes of energy conversion and directs this irreversibility to the thermal degree of freedom as the most sustainable energy form. In terms of the relationships between the microscopic and macroscopic states of the system, the second law, to some extent, subordinates the status of macrostates to only a certain set of microstates.

It is the second law of thermodynamics that from a formal point of view allows us to introduce a macroscopic function: entropy S. The feature of this function is related to the spectrum of microstates. It is postulated that this function cannot decrease with time for a closed system.

Pure thermodynamical, or phenomenological, entropy is introduced by the ratio of elementary change in the heat transfer into the system, δ Q, to the absolute temperature T at which this increment happened:

(1.4)

where S is entropy. However, this introduction implies a reversible process of heat transfer.

For an irreversible process,

(1.5)

This means that there is an irreversible process of so-called dissipation of free energy, when there is some gain of entropy that did not come from heat, but which is also converted into the energy of the thermal motion of the particles. Therefore, the effective d S was greater than that in the case of the reversible process.

This additional increase in d S reflects the fact that other types of energy are transformed so that the energy of thermal motion increases. It is, therefore, why entropy is a parameter that characterizes the relationship to qualitatively define different degrees of freedom and their energy with the enormous reservoir of an energy absorber, which is their thermal degree of freedom.

1.1.9. Macroparameters: Helmholtz Free Energy

In an isothermal reversible process, when the system temperature does not change, taking into account Eq. (1.4), the work done on the system (1.3) can be represented as:

(1.6)

We can then define for systems at constant temperature and volume,

(1.7)

where F= E− TS. The value F is referred to as the Helmholtz free energy [3]. It should be emphasized, again, that our interest in free energy is associated with its role in providing energetical support for life processes.

1.1.10. Macroparameters: Enthalpy

For real systems and processes, it must constantly be borne in mind that these systems have volume and are under, sometimes constant, atmospheric pressure—i.e., the redistribution of energy is constantly followed by some mechanical work. Partly because of this, another function of the state, enthalpy H, is widely used. Taking into account the change in internal energy E and the change in volume V and pressure P, one can write enthalpy as:

(1.8)

In general, an elementary change in enthalpy d H, when under changing volume V and pressure P, can be expressed as:

(1.9)

The introduction of enthalpy can take into account the part of energy that can be converted into mechanical macroscopic work.

In thermodynamics, it is common to introduce another state function, Gibbs free energy G, which is defined as:

(1.10)

where H is enthalpy, S is entropy, and T is absolute temperature, which takes into account the real state of the macroscopic system under constant temperature and pressure P. Gibbs free energy is useful in the description of chemical processes, and when under experimental conditions the pressure is usually constant.

If energy exchange occurs at a constant temperature, the change in Gibbs free energy is expressed as:

(1.11)

It should be noted that for a certain process, the overall difference Δ is

(1.12)

Then the system is in equilibrium. If for the ongoing process

(1.13)

this means that there is a gain, a surplus of free energy in the process, and it cannot proceed spontaneously. If for the ongoing process

(1.14)

this is the criterion for spontaneous processes. The system then has the ability to perform work.

This does not mean that the total quantity of Δ G can be converted into work, Δ G—a measure of the maximum possible work that can be obtained from the system. It is the second law of thermodynamics that states that not all the amount of Δ G can be transformed into work, but only a certain part. The total value of Δ G can be converted only in the case of a reversible process. If all of Δ G can be converted into work, we could revert to this kind of Δ G term in its original form. So, based on the second law of thermodynamics, Δ G can be only partially converted into mechanical macroscopic work.

1.1.11. Link from Macro- to Microparameters: Physical Entropy

As noted previously, when considering thermodynamic systems, i.e., the systems consisting of a large number of particles, we must take into account that there are few levels of monitoring of the physical system and so two kinds of quantities characterizing the system. One of the monitoring levels is at the macrolevel which characterizes the system macroscopically. This is represented by the values of volume, pressure, internal energy, and so on. The second level is the microlevel of observations with microparameters—coordinates and momenta of the particles, and so on. It is clear that because of the identity and indistinguishability of the microparticles comprising the system, any macroscopic state that is represented by a large number of microstates is ambiguous. Therefore, a simple question is logical in this sense: how many microstates are represented by a macrolevel state, i.e., by a given state within this macroscopic system? This number could be treated as the degree of degeneration of the macroscopic level, with the given values of the macroparameters, which can be designated as W. Because this number is very large, a logarithmic measure is used. According to Boltzmann (see, for instance, Refs. 4. and 5.), this introduces the value

(1.15)

where k is the Boltzmann constant, W is degeneration of the macroscopic state, or the number of microstates consistent with the given macrostate, the number of microstates that represented this given macroscopic state. The value of S is usually called the physical (Boltzmann) entropy. Entropy, therefore, acts as a quantitative measure of the uncertainty governing