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Differential equations of macrothermodynamics

HomepageINSTITUTE of Physico-Chemical Problems of EvolutionBiological Evolution and AgingBook Review: G.P. Gladyshev, Thermodynamic Theory of the Evolution of Life FormsDifferential equations of macrothermodynamics

The systems and the processes

As a rule, modern thermodynamics studies simple or complex systems with similar processes taking place in a single fixed time scale. (Usually, the processes are localized in one or several hierarchies.) However, the processes in real systems usually imply complicated transformations involving structures of various hierarchies and relating to different time scales.

Let us consider some differential equations and characteristic functions that are important for macrothermodynamics - thermodynamics of complex hierarchic natural systems.
We use the commonly known equation combining the first and the second laws of thermodynamics for complex closed systems where physical-chemical processes take place:

. (1)

Here T denotes temperature, S the entropy, U the internal energy, p pressure, V volume, Xk any generalized force except pressure, xk any generalized coordinate except volume, m k chemical potential, mk the mass of the k-th substance, which can be replaced by the number of moles. The equality sign relates to the case of reversible changes, the inequality describes the irreversible ones. The work performed by the system is negative. (In some literature it is considered to be positive; for instance, see the monographs by K.Denbigh and V.Sychev).

Suppose that in the system under study not only chemical reactions take place but also transformations between the structure elements of other j-th hierarchic levels (i-th partial evolution). Moreover, let the reagents of each i- th evolution condense into the particles of the i-th evolution phase - a structure of higher level of substance organization. In their turn, the latter are reagents of the next, (i+1) -th, partial evolution. (The order of the partial evolution, i and of the structure hierarchy, j corresponds to the hierarchy order of the reagents.) Transformation of similar reagents of some hierarchic level (j) into similar reagents of the next hierarchic levels (j+1, j+2,...) can be represented as follows:


Here n, m, r >>1, and to each j-th structure hierarchy the i-th process of phase-transition type is put into correspondence. Each partial evolution (i, i+1, i+2, ...) takes place in its own time scale t (ti, ti+1, ti+2, ...), which is comparable with the life-span of the corresponding system or its subsystem (j, j+1, j+2, ...) of the given structure level (for instance, organells, cells, organisms). At times t the natural subsystems (j) are partly kinetically quasi-closed, i.e., non-stationary. Since a hierarchic system is kinetically closed only with respect to separate components of the flow coming out, it is open as a whole, and therefore, its volume and composition vary. At the same time, in the subsystems of the j-th, (j+1), (j+2) levels local quasi-equilibrium are achieved at small times t i, t i+1, t i+2, ... . These quasi-equilibrium are drifting ones, they are achieved at every moment during the evolution of the corresponding system. (Different-size arrows in the equalities (2) show that in the process of thermodynamic self-organization - self-assembly we consider namely quasi-equilibrium drifting to the right; the large upper arrow shows the direction where the equilibrium is shifted.) Therefore, each partial evolution is a process of self-assembly (i) for the substance of variable composition in an open non-stationary system (j). At any moment of the i-th partial evolution, the subsystem j (for instance, the subsystem of supramolecular structures) can be characterized by the value of Gibbs function of the system formation,. One can also use the variation of the specific value of other state function, see Chapter II. The valuerelates to the process (i) of self-assembly (condensation) of the elements of j-th structure level, which are initially in the standard state: for instance, the ideal gas or ideal solution. As a result of such self-assembly, the products of partial evolution i are formed, their composition varying in time. These products are the reagents of the partial evolution (i+1). One of the partial evolution with the participation of different reagents of the k-th type can be presented as


where t i,k do not differ much, and, therefore, the total process of fast relaxation takes place in a single time scale.

We stress that the schemes (2) and (3) describe a real evolution process consisting of partial evolution. These partial evolution are characterized by different mean times of fast relaxation t i, t i+1, ... and go on in the corresponding time scales ti, ti+1, ... . To a certain approximation, these schemes reflect the hierarchy of structures (j) and the corresponding hierarchy of processes (i) in Nature. The evolution of natural biostructures can be characterized as avalanche-condensation structure formation. From the viewpoint of its mechanism, it is opposite to avalanche or branched processes, like electron avalanches accompanying the electric discharge in gases or branched chemical reactions.

Taking into account the schemes (2) and (3), one can write the generalized equation of the first and the second laws in the form:

, (4)

where the subscript i relating, as before, to the partial evolution and k to the component of the i-th evolution. The upper index j indicates that we consider the behaviour of the j-th system.

Relation (4) introduces the potentials of the components of various partial evolution. It is the most general equation in the thermodynamics of hierarchic structures, for it accounts for any kind of interaction (work), transformations of the substance in the system (variation of its amount in different subsystems), and also the entropy variation in the course of all partial evolution. Note that this relation can be used, to a certain approximation, within any given time scale.

It follows from the relation (4) that the full differentials of the characteristic functions of corresponding characteristic (independent) variables given below can be written for the case of complex system as

, (5)

, (6)

, (7)

, (8)

where the indices j are omitted for convenience and the superscript * means that we study the behaviour of complex systems. The functions H* and G*, which do not correspond with their analogues for simple systems, H and G, can be written as H*=H-Xx, G*=G-Xx. The evolution potentials in the equations (5)-(8) are defined as

, (9)

where all corresponding variables of the type m are chosen constant, with the exception for those used as the differentiation variables. Sincein the equation (9) relates to a complex system, it can be denoted by.

For a complex hierarchic, say, ecological, system, contributions of different terms in (4)-(9) are determined by the choice of the time scale and the parameters i,k. For instance, for a real system where higher evolution take place involving rather large structures (organisms and so on), the contribution of the terms TidSi, pidVi and some others, is extremely small. However, the valuesandcan be rather large in this case. Indeed, in the case of small particles the energy of thermal motion kBT>>En. (En is the energy of natural magnetic, electric, and gravity fields acting on the particle.) Still, for sufficiently large structures the inverse is valid: En >>kBT. Hence, the relative contribution of the corresponding terms is large. While passing from lower evolution to higher ones, one can also observe variations of the relative contributions of the terms characterizing surface, photochemical, magnetic-hydrodynamic, deformational, and other kinds of work.

While studying real processes with fixed time scales, one can simplify considerably the equation (4) by neglecting the small terms. Certainly, the thermodynamics (thermostatic) of systems is not interested whether the transition from one state into another one is equilibrium or not. Still it should be noted that there are many real chemical, biological, and other processes studied experimentally that go practically in equilibrium regimes. (Such processes are studied by the thermodynamics of equilibrium processes.) When fast chemical reactions take place in the case of a stationary flow (the situation is close to the one of the Vant -Hoff box), the actual useful work of the system (Wx)T,p,... approaches the maximal work of the process, i.e., the relation -(Wx)T,p,...@ - (Wrev)T,p,...= - D G* holds true. This relation is also satisfied with high accuracy in many natural chemical and biological systems where stationary processes take place. As an example, recall that for the processes in fuel cells the rational efficiency h R = -(Wx)T,p/-D G* is close to unity. (In the industrial hydrogen-oxygen fuel cells, the value of h R can reach 0,9 at low currents.) For electrochemical reactions carried out in laboratory conditions, one can easily achieve h R = 0,99. Several equilibrium (quasi-equilibrium) chemical transformations in living organisms are described in literature. Thus, we come to the conclusion that many natural processes can be characterized by low irreversibility: to a certain good approximation they can be considered as quasi-static (quasi-equilibrium) ones and investigated by means of the classical thermodynamics of drifting equilibrium. As to the natural chemical and biological systems, in the majority of cases they can be characterized with high accuracy by the corresponding functions of the states (see the Chapter I). Besides, from the principle of structure stabilization and the statements following from macrothermodynamics one can conclude that the rational efficiencies of spontaneous evolutionary processes tend to the unity in the course of the evolution (h R 1). This conclusion is in good agreement with the general statements of modern ecology.

Certainly, the results presented do not neglect the existence of dissipative structures and the effects of dynamic self-organization in natural systems (I.Prigogine), for instance, when such systems undergo "revolutionary changes" due to the influence of external factors, and their state is far from the equilibrium one.

General references
  1. Alberty R.A. (1987). Physical chemistry. 7th Ed. New York, etc.: Wiley, 934 p.
  2. Bazarov I.P. (1983). Termodinamika. (Thermodynamics). 3rd Ed. Moscow: Vysshaya shkola, 344 p.
  3. Gladyshev G.P. (1988). Termodinamika i Makrokinetika Prirodnykh Ierarkhicheskih Processov (Thermodynamics and Macrokinetics of Natural Hierarchic Processes). Moscow: Nauka, 287 p.
  4. Gladyshev G.P. (1995). Termodinamika of the Ierarkhicheskih Sistem (Thermodynamics of the Hierarchic Systems). In: Khimicheskaya Enciklopedia (Chemical Encyclopedia). M.: v.4, p.535.
  5. Gladyshev G.P., et.al. (1993). Macrothermodynamics of Biological Systems and Evolution. // J. Biol. Syst., v. 1, N 2, p. 115.
  6. Gladyshev G.P. (1995). Thermodynamic Trends of Biological Evolution. // Izvestia RAN. Seria biol., 1995, N 1, p. .5 . (// Biology Bulletin, v. 22, N 1, p.1. ISSN 1062-3590; N 1, 1995).
  7. Gladyshev G.P. (1996). Thermodynamic Trends of Biological Evolution. Model and Reality. // Izvestia RAN. Seria biol., N 4. (// Biology Bulletin ISSN 1062-3590, N 4, p. 315).
  8. Gibbs J.W. (1928). The Collected Works of J. Willard Gibbs. Thermodynamics. V.1. New York: Longmans, Green and Co. (Russian transl.: Moscow: Mir, 1950, 444 p.).
  9. Guggenheim E.A. (1977). Thermodynamics: An Advanced Treatment for Chemists and Physicists. 6th Ed. Amsterdam etc.: North Holland, 390 p.
  10. Denbigh K.G. (1971). The Principles of Chemical Equilibrium. 3rd Ed Cambridge: Cambridge Univ. Press, 491 p.
  11. Kubo R. (1968). Thermodynamics. Amsterdam: North-Holland Publ. Co. (Russian transl.: Moscow: Mir, 1970, 304 p.).
  12. Ovsyannikov A.A. (1995). Samoorganizatsiya. (Self-organization). In: Khimicheskaya Enciklopedia (Chemical Encyclopedia). M.: v.4, p. 574.
  13. Relaksatsiya. (Relaxation) (1995). In: Khimicheskaya Enciklopedia (Chemical Encyclopedia). M.: v.4, p. 462.
  14. Sychev V.V. (1986). Slozhnye termodinamicheskie sistemy (Complex thermodynamic systems). M.: Energoatomizdat, 208 p. (Translated into English: M.: Mir, 1985).
  15. Haywood R.W. (1980). Equilibrium Thermodynamics. Chichester, New York, Brisbane, Toronto: A Wiley Interscience Publ. (Russian transl.: Moscow: Mir, 1983, 491 p.).

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