The common extremalities in biology and physics maximum energy dissipation principle in chemistry biology physics and evolution

Therefore, the apparent micro- and macrodifferentiation dominateswhen considering the hierarchy of forms of energy in thermodynamics.1.1.4 Macroparameters: Heat as a Nonmechanical Method

The Common Extremalities

in Biology and Physics

Maximum Energy Dissipation Principle

in Chemistry, Biology, Physics and

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The science of living nature is known as biology Biology, in the modern sense ofthe 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 asense, 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 thisimmense universe of dust, stars, planets and vacuum? Is it merely down to randomchance? Or, if life is not accidental, what does this signify? Biology does notexplain the transition from inorganic objects to organic life perhaps because thereasons are too broad to be understood in purely biological terms Moreover, theconcept of evolution has infiltrated and now permeates physics, that other ancientvision of mankind and nature A complex question arises: to which laws does lifeowe its existence? Essentially, the answer lies partially within the realm ofphysics a science which is fundamentally concerned with non-living nature andpartially 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 theinvolvement 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 theappearance of primitive or increasingly sophisticated beings in nature at intermedi-ate stages Therefore the question of the origin of life can be reformulated in vari-ous ways: To what extent do the laws of inorganic nature and of physics derivefrom, produce and require the emergence of biological processes? Is it possible todeduce biological laws from physical and chemical laws? How do we define therelationships between physical and biological processes? According to which laware physical processes transformed into biological processes? To what degree arebiological regularities governed by physical regularities? Success in answeringthese questions, even at an elementary level, might well enable the development of

a conceptual methodology that would generate biological laws based on physicallaws 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 amode of thinking that could be considered to allow some additional steps on thepath to better understanding the relationship between biotic and physical processes

However, one should note that any concept about nature, whether a simple tal picture or a complex formal mathematical scheme, is only one of many modelsrelating to matter Concepts such as these are produced within the social forms ofinformational mapping, cognition or information reflection Mathematical science(including the theory of models and the theory of systems) is itself merely oneform of information reflection, mapping and modelling It can be characterised by

men-a dissocimen-ation from the mmen-aterimen-al world (from supporting mmen-aterimen-al messengers men-andprocesses), creating an ideal, almost spiritual, models, and sometimes could bethought that nature itself moves according to these models

Nevertheless, mathematics, though eloquent in its description of nature, is ply 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 andhave 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

sim-an idealisation that could be considered to be rather two-dimensional “paper” formand recently appears to have taken on a distinctly electronic character

It is well known that the formal mathematical modeling has achieved the est success in explanation, description, and the forecasting of physical phenomena,

great-as well great-as in formal reconstruction of processes that take place within physical tems At the highest level, the description of physical systems and processes pro-ceeds from an extreme ideology to enable the formal mapping of physicalinteraction or dynamics This ideology is based on the least action principleemploying the variational method The methodology of this approach contains thefollowing stages:

sys-G There is a physical value called theaction, which has the dimensional representation ofthe product of energy by time

G 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

G From the principle using variational technique, one can obtain equations of movement of

a physical system (the Euler Lagrange equations)

G The trajectories, or the laws of movement of the system, can be obtained from theEuler 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 turnsout, physical evolution It is only at the final stage of the modeling process that theoutcome appears as a purely kinematical result—the movement trajectories Thelast 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 realsystem

From this point of view, the formal mathematical description in biology has nificant methodological difference, possibly a halfway policy Here one can ini-tially proceed from concepts and terms of a dynamic system (also of some formaldesign), and in the majority of classical cases, from a system of differential

sig-equations and hybrid systems for more complex models The solution of such asystem represents the law of movement or trajectory, providing information on thelocation of a real system at any moment of time in the multidimensional phasespace of parameters of a biological system.

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

ener-of the nature ener-of biological phenomena

This argument proceeds from the suggestion that it is the energy sense that caninitiate the level of formalism, similar to top-level variational formalism in physicaldescription, and consequently, it is ideology of a common and unified approach inbiology and physics

In connection with the above, it is important to look at the most common energylaws of biological phenomena (which, in fact, are the thermodynamic laws) inorder to mathematically formalize, with the purpose of development on the basis ofthese laws, a universal, informative, and formal scheme generalizing the laws ofbiology We expect that such ideas could result in a formalism, similar to varia-tional formalism in physics, and that it could be a basis for the ideological unifica-tion 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 qualitativelyirreversible 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 variationalapproach with reference to the processes with explicit dissipation, i.e., to relaxationprocesses in chemical and biological kinetics

In this connection, it is expedient to reflect on the energy sense of the phenomenarelated to these areas, i.e., the hidden dynamic reason of one or another biologicalprocesses and the form of their representation (mapping) in the correspondingformal 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 thecorresponding under-integral function, the Lagrange function, needs to be foundfrom the equations of motion, from a dynamic system or a system of differentialequations The solution of such a problem would enable us to analyze in an explicitform the energy properties of the phenomena initially presented within the para-meters of a dynamic system However, the reverse variational problem could besolved for a very limited range of cases, and there is little optimism about findingthe successful solution as far as biokinetics is concerned

Thus, it is possible to follow two different approaches in the formal cal and deterministic descriptions of these rather opposite groups of phenomena—biology and physics The first is related to physics, with an explicit energy senseoutgoing from energy properties of the physical phenomena, from the least actionprinciple, leading through the Euler Lagrange equations to the laws of motion or

mathemati-trajectories And the second, more widespread in biology, likely begins with a parison of a physical description, directly from so-called dynamic models, of thesystems of differential or other kinds of equations, and it finally results in the samestage—the laws of motion, or trajectories.

com-We expected that the mutual penetration of both approaches could to a greatextent promote mutual development as well as the technical and ideological enrich-ment of physics and biology

We shall emphasize that the undertaken consideration concerns rather classicalmodels—the models presented by systems of differential equations; however, evensuch a phenomenological consideration is difficult to implement consecutively withinthe frameworks of these two broad and opposing phenomena—the biological and thephysical

1 Extreme Energy Dissipation

1.1.1 Thermodynamics—A Science That Connects Physics and BiologyThe general laws connecting biology and physics are particularly related to energytransformations, since thermodynamics is the phenomenological science that describesthe energetical macroscopic characteristics of systems Thermodynamics, which dire-ctly relates to biology, is known as biological thermodynamics It covers subjects con-nected to the interconversions of different forms of energy, ranging from those in thesimplest chemical reactions and ending with energy complex trophic changes of thebiomass of different species The energy and structure conversions in these complexchanges eventually end and, can be saying in a different way, transfer to another qual-ity in the large number of social processes

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

One can also say that biotic organisms are complex, phase-separated, chemicalreactions that contain very specific molecular forms of informational supportprocesses It can be said that these reactions, in the process of evolution, haveallowed organisms to acquire not only mechanical but also the development ofmore complex high-adaptive degrees of freedom—informational On some stages

of the evolution, these complex reactions significantly enhanced the role of modynamic regulatory feedback loops, regulating for instance the heat balance

ther-in the process of cellular respiration or mather-intather-inther-ing the temperature of the bodyand so on

However, thermodynamic systems operate with some characteristics that reflectthe hierarchy of the physical quantities in the process of energy transformation.Biological thermodynamics, in turn, mirrors the hierarchy of the complex biologi-cal world It is, therefore, useful to remind ourselves of the construction of the

The Common Extremalities in Biology and Physics DOI: 10.1016/B978-0-12-385187-1.00001-0

© 2012 Elsevier Inc All rights reserved.

hierarchal thermodynamic terms and the definition of these with respect to thecrucial differences in the organizational hierarchy—a central point in the differencebetween pure thermodynamic and biological phenomena.

1.1.2 Hierarchy of the Processes and Parameters in ThermodynamicsThermodynamics is known as a phenomenological science Thermodynamics repre-sents a classical and historical example of a macroscopic description of the energetictransformations in various macrosystems However, it is important to note that theunderstanding of macroscopic and particularly microscopic phenomena has steadilybeen changing with time

Thermodynamics, as we know, deals with the systems containing a large number

of particles (around 1010
1030) 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 scopic world, or the system as a whole Two classic examples of these variables are vol-ume and pressure

macro-2 Microscopic parameters—characterizing the properties of the particles that make up thesystem (mass of the particles, their velocities, momenta, and so on) Now, it seems obvi-ous that in any study of processes and systems, it is possible to set at least two fundamen-tally 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 inthermodynamics, 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 twolevels of variables are interrelated, although they have their own dynamism Theinconvenience of describing a one level (macro), which employs the microscopicdescription of the states of all components of a system of microparticles that carrythe microscopic parameters, leads to a statistical interpretation of these quantities,which connects them to the macroscopic parameters The fundamental relationshipsinvolved are closely related to thermodynamics—a form of statistical mechanics.Thermodynamic consideration deals only with the macroscopic parameters of thesystems, 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 forthe thermodynamic systems the concept of two types of processes is considered

In some sense, thermodynamics is the first hierarchical science within physics If inmechanics the reversible character of processes is the rule, and the irreversibility

in some way is an exception, in thermodynamics, perhaps, reversibility of processes isthe exception, and irreversibility is the rule Thermodynamics, therefore, requiresspecific fundamental law to take account of its macroscopic nature—the second law

of thermodynamics The apparent dominance of irreversible processes in the world is associated with the peculiarity of the dynamic nature of the relationship of

macro-microstates and macrostates of the thermodynamic system Reversible processes areunderstood as taking place in such a way that all the macroscopic parameters can bechanged in the opposite direction, without any other macroscopic changes, even out-side the system The irreversible processes occur so that they can run in the oppositedirection, just when connected with other macroscopic changes, such as the environ-ment Reversibility and irreversibility, which manifest themselves macroscopically,are closely linked with the microscopic characteristics of particles, i.e., their owndynamism Due to the dynamic nature of these macroparameters and the large range

of energy that characterizes (changes/transformations) the system, these values have acertain hierarchy

1.1.3 Macroparameters: Energy and the Forms of Its Exchange

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

inter-The internal energy of a system takes all the available energy into account, out regard to the hierarchy of interactions at the macrolevel or microlevel Thisenergy includes the energy of all microscopic particles, at all levels of the hierarchy

with-of the system, and includes the energy with-of all known interactions between them, aswell as the macroscopic part of energy (related to the system macroparameters, likepressure, volume) It should be emphasized that because of this broad concept ofinternal energy, it is impossible to establish its full value for any system, because itincludes 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 systembetween any of the states of the system

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

microparti-of a system in the process microparti-of work

Because nonequilibrium states are characteristic of macrosystems, the energy inthermodynamics acquires one other property The energy can also be considered as

a measure that characterizes the aspiration of processes and systems to reach theirequilibrium In other words, it can be considered as the measure of the relationshipbetween 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 flow by the motion of degrees of freedom (a measure of the nonequilibrium

over-structural state) Therefore, the apparent micro- and macrodifferentiation dominateswhen 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 alization of the mechanical motion of a large number of microparticles Actually, theenergy of this motion is characterized as thermal energy

materi-Paradoxically, heat is a macroscopic manifestation of microscopic changes, and atthe same time, it is a microscopic form of energy exchange, having a macroscopiceffect However, it should be more rigorously understood that heat is the microscopicform of energy transfer that is related to the change of macroscopic parameters, liketemperature, which has both a macroscopic and microscopic sense Therefore, heattransfer is only a microscopic form of change in internal energy

The temperature reflects the macroscopic manifestation of the intensity of themicroscopic motion Temperature is the molar heat of the kinetic energy per onemechanical degree of freedom Therefore, heat energy is transmitted at the micro-scopic 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 thetransfer 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:

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

to the product

It should be emphasized that in thermodynamics, work is also not an exact ferential of any function of the state, but work is a function of the process [2,3]

dif-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 formations between the macroscopic and microscopic forms of energy (in a widesense between the qualitatively different degrees of freedom of physical motion) toanother Formally, this is the postulation of the existence of an additive value—theinternal energy of the system

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

where dE is change in the internal energy of the system, δQ is the amount of mal energy supply to the system to heat the microscopic degrees of freedom, and

ther-δA 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 cesses 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 andwork do, in this sense, belong to different levels of this two-leveled hierarchy.The first law of thermodynamics does not discriminate between the macrodegreesand microdegrees of freedom, or the interaction between systems This interactiondepends on the hierarchical affiliation, which, as it turns out, is related to reversibility

pro-or irreversibility of the interaction process of energy exchange It should be sized that this question arises only in thermodynamics In mechanics, its emergencedoes not manifest itself so clearly It is the second law of thermodynamics that raisesthe question of the status of energy as a measure of reversibility/equilibrity The firstlaw discriminates between the ways of energy exchange, in terms of thermal and non-thermal, and, naturally, states that the overall energy in their forms is conserved.However, even if the macroscopic parameters remain constant, changes mayoccur 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 inthe sense of stability This last fact leads to the second law of thermodynamics

empha-1.1.7 Macroparameters: Free Energy—Macroscopic Measure

of Nonequilibrium

We can say that the thermodynamic study of the interaction of qualitatively ent macroscopic degrees of freedom is an investigation of the redistribution of

differ-energy among the various structural and energetic macrostates These macrostatesrepresent the degree of freedom in the system, during its interaction with the envi-ronment 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 ance, or an existing imbalance in the system, manifests itself according to the sec-ond law of thermodynamics, as a more or less equilibrated state More specifically,macroscopic forms of energy (related to microscopic degrees of freedom andmacrostates) are divided into thermal and nonthermal This division is a character-istic feature, the basis for thermodynamics, and its main laws define the relation-ship between all forms of energy, in accordance with this division

imbal-In line with this interpretation of micro and macro forms of energy, the internalenergy of a system can be qualitatively divided into the relationship between the pos-sibilities of its transformation into the macroscopically ordered form of energy—work(particularly into mechanical work) This part of the internal energy that can be con-verted into any type of work—mechanical, chemical, electrical—can be defined asfree energy Another part of the internal energy, which cannot be converted into mac-roscopic work (as was already mentioned), is referred to as the bounded energy and isusually associated in thermodynamics with the energy of the thermal motion of parti-cles 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 offreedom or that of another way of energy exchange It reveals the reaction of thesystem, 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 equil-ibrated (stable one), thereby selecting the thermal energy, both qualitatively andquantitatively The second law underscores the crucial irreversibility of the thermo-dynamics of all processes of energy conversion and directs this irreversibility tothe thermal degree of freedom as the most sustainable energy form In terms of therelationships between the microscopic and macroscopic states of the system, thesecond law, to some extent, subordinates the status of macrostates to only a certainset 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 isrelated to the spectrum of microstates It is postulated that this function cannotdecrease 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,δQ, to the absolute perature T at which this increment happened:

tem-dS5δQ

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

For an irreversible process,

1.1.9 Macroparameters: Helmholtz Free Energy

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

1.1.10 Macroparameters: Enthalpy

For real systems and processes, it must constantly be borne in mind that these tems 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 Vand pressure P, one can write enthalpy as:

In general, an elementary change in enthalpy dH, when under changing volume

V and pressure P, can be expressed as:

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

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

where H is enthalpy, S is entropy, and T is absolute temperature, which takesinto account the real state of the macroscopic system under constant tempera-ture and pressure P Gibbs free energy is useful in the description of chemicalprocesses, and when under experimental conditions the pressure is usuallyconstant

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

ΔG can be transformed into work, but only a certain part The total value of ΔGcan be converted only in the case of a reversible process If all ofΔG can be con-verted into work, we could revert to this kind ofΔG term in its original form So,based on the second law of thermodynamics, ΔG can be only partially convertedinto mechanical macroscopic work

1.1.11 Link from Macro- to Microparameters: Physical Entropy

As noted previously, when considering thermodynamic systems, i.e., the systemsconsisting 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 quantitiescharacterizing the system One of the monitoring levels is at the macrolevelwhich characterizes the system macroscopically This is represented by the values

of volume, pressure, internal energy, and so on The second level is the microlevel ofobservations with microparameters—coordinates and momenta of the particles,and so on It is clear that because of the identity and indistinguishability of the micro-particles comprising the system, any macroscopic state that is represented by a largenumber of microstates is ambiguous Therefore, a simple question is logical in thissense: how many microstates are represented by a macrolevel state, i.e., by a givenstate within this macroscopic system? This number could be treated as the degree ofdegeneration 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 mea-sure is used According to Boltzmann (see, for instance, Refs.[4,5]), this introducesthe value

where k is the Boltzmann constant, W is degeneration of the macroscopic state, orthe number of microstates consistent with the given macrostate, the number ofmicrostates that represented this given macroscopic state The value of S is usuallycalled the physical (Boltzmann) entropy Entropy, therefore, acts as a quantitativemeasure of the uncertainty governing which microstates are responsible for theobserved macroscopic state

It should be noted that there is an informational nuance of physical entropy: S is

a measure of the uncertainty about which microstate of the system is responsiblefor a macrostate with a given energy

The physical meaning of above definition of entropy lies in the fact that thenumber of macroscopic states of the system and the number of microscopic states

of the system are different Moreover, the number of microstates of the system is

so many orders of magnitude greater than the number of macrostates It thereforemakes sense to introduce a logarithmic measure of the representability of a macro-scopic number of microstates

It can also be stated that entropy is a measure of disorder in the system Indeed,the larger is the degeneration of the system’s energy state, the higher the microscopicdisorder and the greater the entropy of this state However, what might happen if theparticles are physically impossible to move—if they are not so indistinguishable, atleast in a spatial sense? Then each macroscopic state has a unique microstate Then

W5 1 and S 5 0 In this case, the entropy description of systems does not work well.When the system is highly personalized, the concept of entropy makes enough sense.Then the energy characteristics of the system seem to be more constructive Entropy

is, therefore, likely to “depersonalize/dehumanize” the world and make the systemfaceless

Final note: Entropy is good when the microlevel degrees of freedom (states) can

be easily counted That is, ideally, when it is just one microlevel degree of freedomand all the probabilities can be easily calculated

1.1.12 Microparameters: Statistical Interpretation of Free Energy

and Entropy

Statistical mechanics offers another possible interpretation of entropy, associateddirectly with the distribution of the probability of finding the system in the micro-state at the realization of a particular macrostate This leads to the following defini-tion of entropy:

N

i 51

where piis the probability of finding the system in microstate i

However, this microstate can be characterized by a certain energy: The ity of pi microstates are associated with this energy in a certain way This depen-dence was first obtained by Gibbs in 1901 and is called the Gibbs distribution (see,for example, Ref.[6]) or the canonical distribution:

It can be found that free energy is linked to the Gibbs distribution Substituting

in the formula for the distribution of entropy, we then obtain

 

where C is the normalization constant in the Gibbs distribution

Therefore, free energy can be expressed as a measure related to the deviation ofthe actual energy distribution from the most natural; in some sense, optimal forthese macroconditions There are a number of other definitions of entropy Themost well known is the Tsallis entropy [7], which is a generalization ofBoltzmann
Gibbs entropy However, all of these are based on the accountability

of probabilities of microstates in a two-leveled thermodynamic model: the state and the macrostate In actual fact, the hierarchy of biological systems is muchmore complex

micro-1.1.13 The Removal of Energetical Nonequilibrium and the Entropy

Production

The process of irreversible transformation of energy from unequilibrated forms, insome sense unstable material forms, to the more equilibrated, stable forms, is theprocess of increasing entropy It is called the energy dissipation or the entropyproduction

The full balance of the elementary changes of thermodynamic quantities tials) is given by the Gibbs equation, which for the case of small deviations fromequilibrium appears as (see, for example, Prigogine[8
12]):

in particular the free energy, indicating that at an increase in entropy in the systemfree energy dissipates

If the change in internal energy dE5 0, one can express the elementary entropy

in the system as the sum

The production of entropy P in a closed system can be defined as:

N

i 51

where X is generalized thermodynamic forces that initiate the irreversible processes

in the system and J is generalized thermodynamic flows that implement these cesses and address the imbalance

pro-One can say that the hierarchical sense of the second law is that small andreversible changes at the microscopic level produce irreversible changes at themacroscopic level

1.1.14 Dissipation in Chemical Transformations

In this study, we consider chemical kinetics as having been closely related to logical kinetics and which, in some sense, even generates biological kinetics Itwill, therefore, be a good example to consider entropy production, or energy dissi-pation, during chemical reactions, because it is such a dissipation that causes lifeprocesses In addition, it can be largely argued that chemical kinetics is a conve-nient example, because the microparameters coincide with the macroparameters,and at the level of biokinetic microparameters, the processes become even biologi-cal (specific to the kinetics of species densities, ecological kinetics)

bio-An example of the generalized thermodynamic forces, which lead to imbalances

in the case of chemical nonequilibrium, is the affinity of a chemical reaction,divided by absolute temperature T[12]:

Xl5 Al

Accordingly, the generalized thermodynamic fluxes Jl, eliminating this chemicalimbalance, are associated with the chemical reaction rates, and can be written interms of the so-called extent coordinates of the chemical reactionsξl:

where _ξl is the velocity of the lth chemical reaction andξlis the coordinate of thelth reaction, or extent from equilibrium

The extent coordinates from equilibriumξl, participated in the expression for thegeneralized thermodynamic flows (1.28), are related to the concentrations of anyinvolved in the lth reaction of substance xiin the stoichiometric ratioνlias:

where _ξ is the vector of rates and N2 Tis the inverted transposed matrix of

stoichio-metric coefficients Then the density of entropy production(1.26)in such a system

of l independent reactions can be written in a vector form as:

1.1.15 Dissipation of Nonequilibrium in Open Systems

Often one uses the following purely thermodynamic classification of namic systems as:

thermody-G isolated—isolated from the environment, which do not share any substance or energy;

G closed—exchange energy;

G open—exchange energy and matter

It is believed that systems are in steady state or equilibrium if they do notchange with time in the macroscopic state

The description of the dissipative processes in open thermodynamic systems hasoften had a tendency to interpret the entropy due to the fact that most of the pro-cesses do not necessarily occur at a constant temperature The rate of increase inentropy in an open system can be divided as follows[10
12]:

where dSextis the inflow of entropy from the outside and dSint is the entropy duction inside the system However, it should again be noted that it is difficult, interms of entropy, to describe the large number of unique processes of energy and astructure’s conversion that takes place, even in relatively simple chemical systems;this is even more difficult to do so in biochemical or ecological/biocenoticalkinetics

pro-In this sense, the energy
structural description has a large variety and potentialand can be more adequate than an entropic description Therefore, paying tribute tothe entropical description, the energy
structural description in terms of the irre-versible transformation of energy and its structural forms should be considered asfor the processes taking place in any isolated systems, as well as in the open sys-tems This should be done in view of the fact that entropy is only a measure of theambiguity/uncertainty of the macroscopic state with certain energy; it is a measure

of the representability of the macrostate by a set of microstates

Because the chemical processes of generalized flows can always be expressed interms of reaction rate (and, hence, a derivative of concentration), it makes sense todirectly address the kinetics of entropy production, or dissipation of energy in terms

of concentrations and their derivatives This would also facilitate a clearer tation of the mechanisms of chemical processes themselves, since these mechan-isms are usually expressed in terms of dynamical systems The close relationshipwith the mechanisms would allow one to consider the ways of energy conversion

interpre-in the formulation of the variational problem For the latter, it would be of no smallimportance compared with the mechanics, where the description is also made interms of coordinates and derivatives

1.1.16 Energy Dissipation or Entropy Production—The Energy Picture

Can Play a Role

So do some processes only become clear when the production of entropy is ered within the system itself? In terms of entropy, this aspect has the physical mean-ing of the design review processes It is known that the entropy representation isuseful in analyzing nonisothermal processes But entropy is statistically significantonly as a measure of the representability of the macrostate from a number of micro-states It is only a measure of the degeneration of the macrostate of the system, thelogarithm of the number of microstates responsible for a given energy state What isthe point of assuming such a case—the flow of degeneration of the state fromoutside? And from what material structural state is this degeneration imported?

consid-From the very interpretation of entropy, it can be seen that it is a somewhatsimplified description of the exchange of energy and structure This is particularlytrue in the case of biotic systems Real substances produce effects such as smell.Maybe biotic systems reveal their sources of food by the smell of entropy? Whatfunctional distinction in language is there between the entropy of a lysozyme andribonuclease?

Perhaps this might be, in this sense, a purely energetic description of a muchmore detailed fact For example, the spectral (and, therefore, energetical) descrip-tion of the proteins in the nuclear magnetic resonance (NMR) spectral region showshuge differences between the proteins Moreover, the energetical description isclosely related to the structure, and from the energy-structural representability per-spective, this has much more diversity than entropy Moreover, in cases of entropicdescription, the structure is not represented at all and, in some sense, the wholenature in terms of it (entropy) is just one face—the face (and maybe more appropri-ate, the shadow) of the degeneration of the macrostates

Therefore, it makes even more sense to consider the processes in isolated tems, as well as in open systems, as the irreversible transformation of energy andstructure Moreover, when one considers the dissipation of free energy, it is easier

sys-to observe this dissipation, for example, by the spectroscopic method If we recallthe interpretation of entropy as a shadow of energy, following F Wald (see, forexample, Ref [14]), the monitoring of the reality by its shadow, if not meager,may present the result

In some sense, entropy is only a consequence of the representability of a macrostate

by the microstates, the result of some “democracy” on the microlevel Therefore, thiswork seems more impressive by the diversity in nature of the faces of energy: a struc-tural, rather than an entropy-based description of the processes Entropy still fasci-nates, but this does not deny the huge role of energy Moreover, it can be associatedwith energy management, meaning the principle of least action To us, it seems logical

to concentrate on an energetical representation for the reasons noted above Let usalso note that the energetical character of our study stresses its phenomenologicalnature

1.1.17 Biological Hierarchy and Its Complexity

However, biological hierarchy is not as simple as in thermodynamics/statistics.Biological hierarchy is much more complicated The quantities and values dis-played in the macroscopic biological world are not just the averages of somemicroprocesses The constituent microprocesses cannot be thought of as those hav-ing just one level of organization The mean value is not so productive and more-over fails to represent microorganization at the microlevel It seems that entropy,

as a thermodynamic and statistical definition, cannot fail in a proper quantitativerepresentation of order or disorder at the microlevel The character of order/organi-zation at the molecular level of biological processes is huge/unique Its tremendousnature is beyond the imagination and can sometimes be seen as higher than that atthe macrolevel It is one of the reasons that make the applicability of entropy, in

any definition, uncertain [5] It is very difficult to reduce the microdescription tothe energetic description because different sorts of substances that are involved inmicroprocesses are organized in different ways One of the examples can be on thecellular level of microscopic processes.

Hence, significant progress has been done in the field of complex systems[15], one can also note that classical phenomenological, two-leveled thermody-namic-like, macroscopic description also has certain difficulties outside physics

It does not work sufficiently well in biology, economics, or sociology It works inchemistry, where, in fact, a macroscopic description coincides with elementarychemical kinetics (which is also a microscopic description) In more complicatedchemical systems, it has some limitations However, taking into account the factthat chemical kinetics is indeed formally linked to biological kinetic processes at

a molecular level (also evolutionary bioprocesses start at the chemical level), it

is a good reason to start the study of biological and physical descriptions with aconsideration of the chemical processes described Therefore, in our study, weconcentrate at the phenomenological, e.g., the energetical description, keepingalso in mind that it is also limited

As was mentioned, in contrast to the two-leveled thermodynamic-like model,biological systems are characterized by multilevel, nonlinear interconnections.Multilevel interpenetrating feedback results in a very complex system of regulation.Due to the links between different biological layers, new functionalities emerge asimportant life properties from this biological complexity These can be character-ized by self-organization, optimal adaptation, self-replication, and coevolution Ingeneral, biological systems are far from equilibrium due to the multiple controlloops needed to maintain the biological system in homeostasis

A global hierarchical structure of biological processes can be represented by thefollowing sequence: ecosystem—species interdependence; animal populations—competition and the food chain; individual organisms—physiological functioning;limbs and physiological systems—organism homeostasis; tissues—growth, mainte-nance, and repair; cells—growth, specialization, and death; organelles—cell homeo-stasis; and biological macromolecules—folding, molecular recognition and binding.The hierarchical structure and control in biological systems has developedduring a long period of evolution The complexity of biological systems is required

to create new functionalities, which can be characterized as self-organization, mal adaptation, self-replication, and coevolution Many regulatory processes have adynamic and cyclical nature, manifesting themselves over different characteristictime scales Regulation in any biological system cannot be adequately understood

opti-in the framework for any static two-leveled model (the simplest model from amathematical point of view) Behavioral control in a biological system should beconsidered in the framework of a dynamical system approach From a thermody-namic point of view, biological systems are too far from the equilibriumstate; therefore, only dynamical models can be used to investigate their complexbehavior Many attempts to describe the informational processes in biosystems,

on the basis of entropy-informational principles, have failed—possibly becausebiosystems are multileveled, autonomic, dissipative, and intelligent systems Here

we have adopted phenomenological methods for deriving nonlinear dynamicalmodels, which we will use in a study of regulatory processes in a multilayersystem.

Hierarchical regulation in multileveled biological systems has evolved to vide optimal adaptation and robustness At the same time, biological systems haveacquired the energetic-structural resources for adaptation and competition The pro-posed research will investigate these features of biological system behavior usingmathematical modeling with multilevel hierarchical feedbacks

pro-1.1.18 Some Conclusions

In light of above, thermodynamics is the simplest hierarchal model in physics Itand its quantities and parameters describe the energetical, main properties ofchange within physical systems on the macroscale Statistical mechanics/thermody-namics is concerned with dealing with a microscopic description of the processes

in thermodynamic systems, i.e., the description of a thermodynamic system on themicroscale It has its own terminology and complicated concepts, together withmethods to link microdescription with the macrodescription One of the most pow-erful concepts in this link is the concept of entropy The different ways to defineentropy are well developed for various physical systems within a mechanical vision

of their microstates However, there are also some difficulties in the universal nition of entropy that can be applied to systems with a complex, non
two-levelhierarchy, such as biological systems Consequently, the energetical laws and,indeed, energetical properties of systems with a complex hierarchy can be consid-ered as another optional framework in a phenomenological description that canplay a unification role in physical and biological complexity

defi-The part of thermodynamics known as biological thermodynamics describeswell the main phenomenological properties in energy transformation in the biologi-cal world Because the hierarchy of the biological world is tremendous, statisticalconcepts based on two-level physical models of macro
micro relations are noteffective in producing a description, as they are in statistical mechanics/thermodynamics

Can biological hierarchy be described in terms of entropy, in a fashion lar to a hierarchal description in thermodynamics—the two-leveled model ofhierarchy? If we are sure that the universal entropy definition can be defined,then (1) it is a universal description for both physics and biology, (2) it can beuniversal for different biological organizational levels (e.g., for cell and for soci-ety)—we might need to concentrate on this route However, if we are not surethat it is possible, we should not ignore the need to seek the development ofother approaches Another possible approach is the pure phenomenologicalapproach based on the relationship between kinetics and energetical transforma-tions We, therefore, will concentrate on phenomenology, on the parameters thatare related directly to the energetical properties of processes At this stage, wesee this way as being important to the development of a common interface forphysics and biology

simi-Certainly, the models of hierarchy of physical phenomena are not limited to thetwo-level classical physics model Some examples can be shown by quantum statistics,e.g., Bose
Einstein or Fermi statistics—hierarchical concepts based on symmetries inparticle physics or atom
nuclei
quarks
strings However, by comparing the hierar-chy in the fields of physics/thermodynamics and biology, it is easy to see that we aredealing with levels based on the laws of classical physics Molecules and macromole-cules in biology (biological thermodynamics and kinetics) are considered as classicalobjects So, by simply regarding the microlevel in thermodynamics, while comparingthe hierarchy with biology, we consider only classical thermodynamics/statistics-basedmechanics and physics In this sense, classical thermodynamics can be thought of asbeing a two-level hierarchical model Certainly, this description should be balanced inthe sense of the micro- and macroparameters involved.

1.2.1 Comparing Extreme Approaches

The extreme approaches can be said to represent the pinnacle of physics formalism.Extreme approaches in physics are mathematically based on the variational techni-ques Developed in recent decades, the unification and evolutionary physical theo-ries are the best proof of it and paradoxically the variational extreme formalism isnot just the formalizational pinnacle and the perfect technical basis, but also pene-trates deeply the nature of physical phenomena However, thermodynamics still hassome difficulties in developing such a consistent variational formulation, as havethe branches of mechanics or physics of fields

The classical expansion of the description in classical thermodynamics tually proceeds from the formulation of the laws of the equilibrium state and thenaddresses the simplest nonequilibrium state—the stationary one Later, classicalthermodynamics was lost in the different approaches to describe the increasingvariety/branches of nonequilibrium states However, obviously, there is a necessityfor the common ideological and technical description of the infinite variety of non-stationary states and processes considerably removed from the equilibrium andbeing stationary—e.g., the biological processes

concep-At the same time, it is obvious that nonstationary and nonequilibrium processesand states occur in nature even more frequently than stationary ones and more soequilibrium states One can say that the matter moves because it is in nonequilibrium.Therefore, one could try to alter the conceptual direction of the classical thermody-namic consideration and employ a methodological inversion, i.e., it is probable thatone needs to proceed in developing the thermodynamic description of the nonstatio-narity directly from the laws of inequilibrium and instationarity In this moment, itneeds to give the answer to the question: Do unstable or nonstationary states have acommon law?

Furthermore, one can say that the main characteristics of all nonstationary andnonequilibrium states are probably that they are striving to the equilibrium or

stationarity (in a dynamic case) What will occur if the fastest manner of striving ofthe nonequilibrium to the equilibrium could be applied, would it also be a desiredextreme approach that is routine for all physics?

How much would it be possible to connect a similar extreme approach to the leastaction principle in physics, on the basis of striving of instability to stability, initiallyfor thermodynamics of chemical transformations or biokinetics? It could be a primaryimplementation of the conceptual relationship between biology and physics In partic-ular, such an introduction of a common extreme principle to chemical and, especially,biochemical and biological kinetics could be a useful first step

In connection to the above, it would be interesting to compare the ideologicalaspects of the most well-known extreme approaches in thermodynamics to selectthe possible shape of the variational formalism closest to the least action principlefor biology

There are a number of well-known extreme principles that are employed in modynamics [16
23] However, recently more and more papers have appearedthat revise the Prigogine minimum entropy production principle [8
12]and refor-mulate in different ways the maximum entropy production (MEP) principle, ini-tially proposed by Ziegler as the maximum energy dissipation (MED)/MEPprinciple [24
28] A number of earlier and more recent studies, e.g., in globaldynamics and climate [29
35], photosynthesis [36], plasticity [37], heat transport[38
40], evolution and in relation to the least action principle[41
43,44], theoret-ical generalization studies [45,46]based on the work of Jaynes [47], support thisMED hypothesis In their interesting article, Martyushev and Seleznev [48]reviewed the Prigogine principle, illustrating its limitations and stating that thePrigogine principle is a result of the Ziegler principle, which also supports our con-clusions that the Ziegler principle is more generalized The brief but systematicconsideration can be found, for example, in Ref.[48]

ther-In the aspect of appearance, the principle of Prigogine [8
12] as the mostknown and Ziegler [24
28] as less known are possibly, to some (verbal) extent,the most opposite ones Prigogine considers the entropy production P in a closedsystem:

where P is the total production of entropy, dXP=dt is the change due to the change

in Xk (entropy production by means of generalized thermodynamic forces), and

dJP=dt is the change due to the change in Jk(entropy production by means of eralized thermodynamic flows)

gen-The above-mentioned work showed that the first term dXP=dt in the expression(1.35) satisfied the general inequality, that is, the generalization of the theorem ofminimum entropy production in a steady state (Prigogine[8
12]):

dXP

(the equality sign corresponds to the steady state) and demonstrated that thisinequality (1.36) had the same general character as did local thermodynamics.Prigogine called this inequality(1.36)the “universal evolutional criterion”[8
12]

At the same time, even from the structure of the decomposition of elementary ferential of entropy production, as in the above(1.35), as well as from the physicalcharacter of the structure of decomposition used by Prigogine, it should be noted thatthe value dXP=dt is the change in the elementary differential of the entropy productiondue to the change in generalized thermodynamic forces at fixed generalized thermody-namic flows In this aspect, the sense of the value dXP=dt; analyzed by Prigogine, is apossible change in the entropy production due to the change in generalized forces thatare the sources of an inequilibrium in the system and the cause of the occurrence ofgeneralized flows (naturally, considered at constant flows) Therefore, the physicalsense of the conclusion of Prigogine consists in the statement that the partial develop-ment of the processes of entropy production (or energy dissipation) in a system bymeans of generalized forces strives to reduce this destabilizing factor that formallyexpressed as the generalized thermodynamic forces X, and, consequently, the value

dif-dXP=dt is negative or equal to zero in a steady state

However, can the constraints for the dJP=dt differential of the entropy tion (occurring by means of generalized thermodynamic flows) or even for the total

produc-dP=dt differential be obtained for the general case?

The constraints to the total entropy production (or to dissipating energy) in themost explicit manner follows from the principle of Ziegler [24
28] Consideringthe rate of general entropy production, Ziegler suggests that the total entropy produc-tion in a system strives to a maximum and formulates it as the principle of the maxi-mum rate of energy dissipation or the principle of the maximum rate of entropyproduction [24
28] The Ziegler principle may seem to contradict the principle ofPrigogine; however, it is difficult to see the contradiction between these principles.The Prigogine principle states that the differential of entropy production due to theincrease of the generalized force’s contribution is either absent or reduces the generalentropy production differential The principle of Ziegler insists that the total entropyproduction strives to a maximum Hence, the part dP=dt strives to a possible maxi-mum value It means that the increase in the differential of entropy production is due

to the raise of the generalized flows dJP=dt; owing to the extremely possible ing in a system of the processes of enhancement of energy dissipation caused by theincrease of generalized thermodynamic flows Therefore, the total entropy productionstrives to a maximum, although the entropy production at the expense of the increasedcontribution of generalized thermodynamic forces can even decrease This reasoning

increas-is naturally more acceptable for the case when the external effect increas-is steady During

the increase of the external effect intensity, the contribution of the total entropy duction at the expense of the production of generalized forces can even be increased.There is a possible simple illustration for entropy production kinetics or freeenergy dissipation (the latter with the accuracy equal to some constant related tothe temperature for isothermal processes), which is qualitatively equivalent to aconsideration in terms of entropy production It can be qualitatively represented as

pro-in Figure 1.1 Let the rate of the external inflow of instability or the free energyinto a system be taken as constant

According to the principle of Ziegler, the rate of the energy dissipated, designated

as dEdissipated=dt; strives to reach the rate of the free energy inflow in the system

dEinflow=dt as rapidly as possible, i.e., it strives to equilibrate the rate of the instabilityinflow into the system, approximating in an asymptotic manner (Figure 1.1A).Figure 1.1B represents the kinetics of this asymptotic approximation of dissipated

Steady rate of free energy inflow in a system

Free energy dissipation rate in a system

Free energy inflow in a system

Free energy dissipated

Time (A)

(B)

(C)

Figure 1.1 The MEDprinciple: the inequilibriumstrives to the equilibrium in anextreme rapid manner Theundissipated free energystrives to the possibleminimum extremely rapidly in

a way when the area betweenfree energy inflow into asystem and dissipated freeenergy strives to a minimum.The area under the differencecurve (C) has the dimension ofthe product of energy by time,i.e., action

energy as functions are dependent on time to the value of free energy inflowing into

a system, accordingly

The difference between free energy coming into the system E(t)inflow and freeenergy dissipated E(t)dissipatedis shown inFigure 1.1C As the dissipation rate strives toachieve the inflow rate as quickly as possible, the area under the curve inFigure 1.1Cstrives to a minimum value, accordingly However, this area has certain physicaldimension—action Moreover, the extreme character of the reduction of this valuebetween some states according to the principle of Ziegler corresponds to the extremecharacter of behavior of “action” in mechanics, and in physics, generally This simplerelation based on dimensions illustrates the fact that the principle of Ziegler can bejust a special case of the most general principle for all physics—of the least actionprinciple In this case, it can probably cover the mechanics of the explicit dissipativesystems—thermodynamics and, in particular, chemical thermodynamics

Thus, it seems that the principle of Prigogine explains only the insufficiency ofthe processes of the entropy production by means of generalized forces and methodo-logically indicates a decisive role of processes of energy dissipation stipulated by thecontribution of generalized flows to the total increase of entropy production

Because the principle of Ziegler requires a maximum rate of dissipation and freeenergy, the quantity, which expresses a measure of nonequilibrium, in some sensethe energetical instability, plays the role of a potential in chemical thermodynamics,one could believe that the principle of Ziegler can conceptually correspond to the leastaction principle where free energy (Gibbs or Helmholtz’s) may be used as a part ofthe Lagrange function From the above reasoning, it is clear that the least action prin-ciple could be treated as a more general principle: It also includes the thermodynamicarea of physical processes With reference to thermodynamics, it maintains that in thefield of irreversible transformations of energy from its various material forms to a ther-mal one, the rate of these transformations strives to a possible maximum value.Given that the least action principle in physics is formally employed in the varia-tional form, one should find the appropriate variational formulation for the extremeapproach mentioned above It is also important for realizing the methodologicalrelation of thermodynamics and biokinetics to physics in general on the basis of theleast action principle A graphical illustration of the MED principle and differentways of free energy dissipating in an isolated system is shown inFigure 1.2 It illus-trates that free energy achieves its minimum in the way that minimizes the areaunder the dissipation curve—the physical action of the dissipative process This can

be a methodological basis for comparing these two principles

It was also mentioned above that there is a widely known criterion for scopic direction of processes in thermodynamics based on the requirement of striv-ing of Gibbs free energy to a minimum, see, for example, Prigogine[8
12], whichactually is the second law Therefore, it is reasonable to require this striving to beexerted as rapidly as the material variety of a system permits in a way when thearea between free energy inflow into a system and dissipated free energy strives to

micro-a minimum (due to the methodologicmicro-al cmicro-auses indicmicro-ated micro-above) Then within micro-anysufficiently small temporary interval [t1, t2], or within an open-end consideration,the free energy value will strive to a possible minimum In this form, the expression

may correspond to the variational formulation of the least action principle in whichfree energy could be used in the Lagrange function.

The extreme properties of thermodynamic energy transformation are usually mulated in the framework of a variational approach to keep it in line with all phys-ics[16
23]

for-1.2.2 How Is the Elementary Variational Problem Solved?

The variational method is based on the concept of a functional and its variation Thefunctional is the numerical function that maps the number of some function from acertain class; it is sometimes referred to as a function on a class of functions Theintegral

The elementary and fundamental problem of the variational calculation is ing of the extremes (minimum or maximum) of the functional of the form

in the way thatminimizes the areaunder the dissipationcurve—the physicalaction of thedissipative process

where x(t) is the unknown function on the time t The form of the function L,named the Lagrange function or the Lagrangian, is given explicitly It is necessary

to find such functions x(t), which give the extremes for the specified functional.There are so-called necessary conditions for the extremum of a functional Forthe function x(t) to give extremum to the integral J, the x(t) should satisfy the dif-ferential equations of Euler
Lagrange:

Such a function is referred to as extremal

One important case is when function L is not dependent on time, t,

L5 LðxðtÞ; _xðtÞÞ: Then Euler
Lagrange equation can be written (in one-dimensionalcase) as:

it is interpreted physically as energy

1.2.3 Other Necessary Conditions for Local Minimum

The Legendre condition is the necessary condition for a local minimum [49] forsimplicity in one-dimensional case:

@2L

The necessary condition for a global minimum is the Weierstrass condition(Gelfand and Fomin [49,50]) It states that for a strong minimum of the functional(1.38), the following E0—the Weierstrass function

E0ðt; x; _x; lÞ 5 Lðt; x; lÞ 2 Lðt; x; _xÞ 2 ðl 2 _xÞUL0_xðt; x; _xÞ ð1:44Þ

and the inequality

in all points of the extremal trajectory x(t) and for any number l are executed

1.2.4 Canonical Equations or Hamiltonian Formulation

As one can see from Eq (1.40), the Euler
Lagrange equations are the system ofthe second-order differential equations There is a method to reduce the order ofequations, and bring the system to a normal system of first-order equations, to aso-called canonical form In fact, it is another formulation of the extreme problem,so-called canonical [49] It is based on the Legendre transform, which introducesthe value pi, according to the transformation

These equations form the system of first-order 2N equations, which is equivalent

to the system of the Euler
Lagrange equations and is referred to as the canonicalsystem of equations or the Hamilton equations In mechanics, the variables pihave

a specific physical sense and represent momenta The expression (1.47) will be afirst integral, when the functional (1.38) will not change at the transformation

t05 t 1 a, where a is a constant, which means that Hamiltonian is not dependentexplicitly on t

Usually, the application of variational methods to mechanics is based on anassumption that the interval [t1, t2] is very small In thermodynamics, chemical andbiological kinetics, it is likely not possible to set this condition Then an interestingand important case should be considered when the terminal time in Eq (1.38) isnot specified and is totally free (so-called free-end or open-end problem or natural

boundary condition) Then instead of boundary conditions for terminal time t2, thetransversality conditions should be applied:

Taking a one-dimensional example, one can illustrate that applying a standardvariational approach to dissipative kinetics, based on Lagrange function as the dif-ference between the kinetic and potential parts, gives little chance to describe therelaxation trajectories

1.2.5 Conclusions

Thus, the principle of striving of the free energy dissipation rate to a maximum can

be interpreted in terms of physical dimensions as corresponding to the least actionprinciple The least action principle in such a sense is the form of the law according

to which the physical and chemical processes in a system are directed to theextremely fast elimination of the inequilibrium, as far as the structural variety ofthe system allows it, in the way that the area under free energy dissipation curvestrives to the possible minimum

At the same time, the possible suggestion on the dependence of free energy on therates of the dissipative processes violates the principle of local equilibrium in thermo-dynamics and chemical kinetics Therefore, the free energy cannot at all satisfactorilyplay the role of the Lagrange function and could be just a part of it It should berecalled that free energy is a thermodynamic potential and, consequently, is the mea-sure of the inequilibrium on which some dissipative flows are built, so it can only playthe role of a potential in the Lagrange function, while the kinetic part has to be added.Also in mechanics, the Lagrange function traditionally has two parts: the first is akinetic part and the second is a potential one Can free energy be such a potential?What form can the kinetic term, executing dissipation, take? In mechanics and ther-modynamics, the kinetic term depends on the rates of generalized flows It may beassumed that in the case of chemical kinetics the kinetic part depends on the rate, butonly on that of the chemical reactions

In many extreme approaches in thermodynamics, the kinetic part (dissipativefunction) is introduced in the thermodynamic Lagrangian However, this partdepends on the generalized flows Such a dependence somewhat masks the inter-pretation, whereas in transition to biological kinetics it has a rather too complexconstructiveness Therefore, it would be useful to search for the formulations wherethe potential and kinetic terms would depend directly on the concentrations, num-bers, or population densities in the case of biotic species, or in a phenomenologicalcase on the so-called extent from the equilibrium coordinate

1.3 Optimal-Control-Based Framework for Dissipative

Chemical Kinetics

1.3.1 Optimal Control and Mechanics

If we recall the Lagrange approach in classic mechanics, so each mechanical tem is compared with the Lagrange function, being the function of the coordinates

sys-q (coordinates in any curvilinear system of coordinates), velocities _q (velocities inany curvilinear system of coordinates), and time t:

con-Usually, applications may be limited by an explicitly time-independentLagrangian

The pure variational problem fromEqs (1.50) and (1.51)could be reformulated

as a dynamic optimal control Lagrange problem by the following designation:

In these terms, the corresponding functional will be

ð1:56ÞHere the control variables ui shall be considered as having no restrictions Inthis case, we obtain a classical dynamic optimal control problem A powerful tech-nique to solve this problem,Eqs (1.55) and (1.56), is based on the Pontryagin max-imum principle[52] Then the Hamiltonian is

qi1, qi(t2)5 qi2; therefore, the additional demand of the Pontryagin maximumprinciple of H5 Const corresponds, in fact, toEq (1.54)—the energy conserva-tion law

However, in the case of the optimal control formulation (Eqs (1.55) and(1.56)) of the classical mechanics problem (Eqs (1.50) and (1.51)), the interpre-tation of the Lagrange function L, Eq (1.53), seems to be cost-like, and it isdifficult to interpret the negatively defined and responsible for the interactionterm U(q) from the optimal control cost-explicit perspective Also, the costate(adjoint) variables pi (mechanical momenta) are difficult to interpret in the opti-mal control sense Conversely, one can see that if this term U(q) is positivelydefined, it could simply be interpreted from the optimal control in a cost-likemanner as the energetical penalty for being in an nonequilibrium state.Moreover, in this case, the character of motion is opposite to the harmonic-likestate and is just a relaxation to the equilibrium—this manner is characteristic ofall dissipative processes

1.3.2 Dynamic Optimal Control Formulation

Taking into account the relationship discussed in the foregoing between the variational(Eqs (1.50)
(1.54)) and optimal control approaches (formulated in Eqs (1.55)
(1.59)), let us consider the problem in terms of dissipative systems, since they are theclosest to biological and chemical kinetics and thermodynamics We can rewrite

In thermodynamic terms, the concept of a dissipative function was introducedtogether with the first thermodynamic principle of the least energy dissipation byOnsager[16,17]in the form

For chemical kinetics, the dissipation function (1.61) needs to be written interms of the extent coordinates (generalized displacements) of a set of independentchemical reactions[21,28]:

Taking into accountEqs (1.61)
(1.63), the Lagrangian forEq (1.60)for a closedsystem of chemical reactions, where state variables are the extents of independentchemical reactions, following our optimal control-based approach from Refs.[53,54],could be written in a general form and in terms of the pure variational technique as:

L L

TF

T F

1 1

“F ”—thermodynamic potentialΨ against decimal logarithm of time In the upper corner

of B, these functions are shown in linear scale

Applying the Legandre transform to the Lagrangian (1.64), we can build thethermodynamic Hamiltonian

Hðξi; piÞ 5X

N

i 51_ξiðpiÞpi2 Φð_ξiðpiÞÞ 2 ΨðξiÞ; ð1:70Þ

where pi are thermodynamic momenta Then the canonical system can be writtenas:

which could easily be transformed toEq (1.69)

One can see from the example above that the exponential relaxational kineticscould be considered in terms of a variational approach where dissipative functionand potential are positively defined In addition, we need to take into account thatthe relaxational kinetics of dissipative systems undermines the property that closedsystems relax to the equilibrium and stay there for a long and ideally indefinitetime This means that the Lagrange problem should be formulated as an open-endproblem and then transversality conditions should be employed

In summary, given the choice of the Lagrangian in the form of the sum of the itively defined dissipative functionΦ and thermodynamic potential Ψ, it is now easy

pos-to explain from the optimal control perspective In the optimal control sense, thecumulative dissipative penalty
dissipative/thermodynamical action (that consists of

as the integrand the sum of dissipative functionΦ and thermodynamic potential Ψ) is

to be minimized So, in earlier classical approaches, the Lagrange function is ratherthe sum of positively defined kinetic part Φ and negatively defined potential part Ψ.However, the optimal control-based approach supports the vision that a thermody-namical Lagrangian could be a sum of the positively defined kinetic part and thepositively defined potential part Defined in such a way, the Lagrangian seems less

limited and could solve some principal difficulties in the sense of conceptualagreement of high-level formalism in thermodynamics (including chemical kinetics)with the widely used Lagrange method in physics.

1.3.3 More General Case

Following the optimal control scenario ofEq (1.60a)with an open end, due to thefact that the dissipative (pure thermodynamical, chemical, or biochemical) pro-cesses are rather relaxational, and in a case when the dynamical constraints(Eq (1.60b))had a more complicated relationship, e.g.,

_ξi5 fiðξ1; ; ξN; u1; ; uNÞ; ξiðt0Þ 5 ξi0; ð1:74Þ

we can generalize the problem and later formulate it as a pure variational Let uschoose the minimization functional in the form of Eq (1.60a) with an open-endsubject to autonomous dynamical system (1.74), fixed initial time t0, unspecifiedfinal timeτ, and a fixed target state ξi5 0: