User:Jlancaster/Draft Evolutionary Thermodynamics

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Evolutionary thermodynamics is a branch of thermodynamics that describes the radial expansion of dynamically structured energetic systems, by component energetic structures self-organizing to form successively emerging energetic systems, to be the basis of evolution and the root of life. This branch of science addresses the fundamental relationship of energy and evolution in natural systems over time. A fundamental Theory (or Law) of Evolutionary Thermodynamics states that characteristic dynamic processes responsible for the evolution of subatomic particles into atoms, and then atoms into molecules, are the same characteristic dynamic processes responsible for the inevitable progression of molecules forming energetic systems that are deemed "living systems."

A restatement of the Theory in 1989 [1], postulated that energy forms itself into structured energetic systems, where an energetic system has an observed boundary of dimension related to the spatial extent of subsystems having interactive relationship to the system.

This science approaches the study of energetic systems at multiple scales, from sub-atomic structure to living organisms, to societies of organisms and to cosmological scales. This developing branch of thermodynamics relates to non-equilibrium thermodynamics and to self-organization theories, as well as to many aspects of cosmology and astrophysics.

Energetic systems evolving in nature are not in thermodynamic equilibrium. They change over time, being dependent on flows of energy from and to other systems in their environment and on internal dynamic processes and chemical reactions. Evolutionary thermodynamics is concerned with developing understanding of root principles that govern the dynamics of energetic systems, including formation, emergence, growth, stability, longevity, association, bonding and other aspects of evolving systems and their components.[1]

The thermodynamic study of evolving systems requires more general concepts than are dealt with by equilibrium thermodynamics. One fundamental difference between equilibrium thermodynamics and evolutionary thermodynamics lies in the behaviour of inhomogeneous systems, which require for their study knowledge of rates of reaction which are not considered in equilibrium thermodynamics of homogeneous systems. This is discussed below. Another fundamental difference is the difficulty in defining entropy in macroscopic terms for systems not in thermodynamic equilibrium.[2][3]

Overview[edit]

Evolutionary thermodynamics joins non-equilibrium thermodynamics with ecological energetic analysis to reveal a strong analogy between chemical, biological, social and ecosystem evolution. An early summary of the field of study was presented by Justin Lancaster in 1989 in his paper "Theory of Radially Evolving Energy."

Definitions

Energetic Structure: An organizational process, O.sub.r.sub.i, for which there exists an organizational radius, r.sub.i. An energetic structure may be an energetic system.

Energetic System: Characterized by an organizational radius, r.sub.j=m and is an assemblage of energetic structures within an observed boundary, which structures are characterized by organizational radii r.sub.i<n.

Organizational Radius: The scale radius, r.sub.i, corresponding to the spherical volume, V.sub.i, that is defined by the total energy of the system, E.sub.i, per constant energy density, u.sub.s = 1 J/cm^3, such that


We then define the extended Massieu function as follows:

where is Boltzmann's constant, whence

(EQ.14)

Observed Boundary: The minimum spatial boundary that will circumscribe all the components of the energetic system, as defined by inter-relationships between the energetic subsystems comprising the system and by energy and material responsive to those subsystems, as determined by an observer.

Dynamic Coordination: A process whereby kinetic energies become stored through harmonization, or non-interference.

Subsystem: A subset of energetic structures within an energetic system that share a common functional relationship to the system, which relationship differs from relationship of other subsets to the system.

The Restatement: "The evolution of energy proceeds radially from and into energetic structure. Energetic structure disorganizes into energy, and the unbound energy diffuses radially, moving from a volume of higher energetic density to lower. Energy self-organizes into energetic structure, and the bound energy aggregates radially. Radial evolution pertains to the growth of all energetic systems, including abiotic and biotic structures."

The following principles describe the emergence of energetic systems:

I. General. Radial evolution is increasing the volume and energy contained by the observed boundaries of successively emerging energetic systems. The increasing energy in successive systems can be defined by the product of a unit energy density and an increasing organizational radius, r.sub.i, from subatomic through biospheric scales. Energetic structure of lesser organizational radii, O.sub.r.sub.i-1, self-organize into emergent energetic systems of greater organizational radii, O.sub.r.sub.i, through successive self-encompassing steps, the organizational process at any level i being a function of the series of preceding functions for each level of component

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle \ k_{b}M=S-\ O_{r}_{i}=f_{i}(f_{(}i-1))(....,O_{r}_{(}i-2))....,O_{r}_{(}i-1)).(EQ.15)}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle \ O_{r}=f_{i}(f_{(}i-1))(....,O_{r}_{(}i-2))....,O_{r}_{(}i-1))} (EQ.15)




I(a). The change in organization of an energetic system, dO.sub.r.sub.i, with respect to time, is a function Ψ of the total energy of that system, E.sub.r.sub.i, the change in total energy with time, dE.sub.r.sub.i/dt, the organizational process of the system, O.sub.r.sub.i, and time t, which we can write

dO.sub.r.sub.i/dt = Ψ(E.sub.r.sub.i, dE.sub.r.sub.i/dt, O.sub.r.sub.i, t) (EQ.16)

I(b). The change, with respect to time, in energy contained in an evolving series of emerging structures can be modeled as the product of a radial evolutionary force, F.sub.e, and a radial evolutionary velocity, v.sub.e,

dE/dt = F.sub.e * v.sub.e (EQ.17)

The force is the product of a pressure and surface area at radius i. The pressure is the energy density, u.sub.s, so that the force is equivalent to the change in energy with increasing scale radius

F.sub.e = u.sub.s * 4 * pi * (r.sub.i)^2 = dE.sub.r.sub.i/d.sub.r (EQ.18)

The evolutionary velocity is the rate at which the organizational scale radius is increasing, which rate is greater than zero

v.sub.e = d.sub.r.sub.i/dt > 0 (EQ.19)

This rate is nonconstant; it will be described by a growth function or set of such functions. From Eq. (17), with substitution from Eqs. (18) and (19), we can model the change in energy in terms of the change in organizational radius with respect to time, obtaining the differential of Eq. (14) above:

dE/dt = u.sub.s * 4 * pi * (r.sub.i)^2 * dr.sub.i/dt (EQ. 20)

I(c). Energetic systems at each level incorporate energy by absorbing energetic structures that are characterized by smaller organizational radii, but not structures characterized by greater organizational radii.

I(d). As the evolution of energy proceeds to greater organizational radii the component structures increase in diversity (few types of subatomic particles, many types of organisms) and decrease in number (many subatomic particles to few organisms).

I(e). Subsystems (SS) of energetic systems with organizational radius r.sub.i will correspond to strongly analogous subsystems in systems of higher level radii r.sub.i+n.

The set of the former, {SS.sub.1,..., SS.sub.y}.sub.r.sub.i, will become a subset of the latter, {SS.sub.1,..., SS.sub.y,..., SS.sub.z}.sub.r.sub.i+n. Owing to emergent relationship between environment and system at successively greater organizational radii, which allows new subsystem opportunities, not all higher level subsystems will be traceable to subsystems found within component systems.

II. Component coordination. The total potential energy in an energetic system is greater than the sum of the potential energies of the individual components by the amount that the sum of the kinetic energies of the components exceeds the kinetic energy of the system, i.e., by the amount that is stored through dynamic coordination, which can be written

E.sub.(i pot) - SUM.sub.(i-1) [G*m.sub.α * m.sub.β / R.sub.αβ ] = 1/2 * SUM.sub.(n-1)(i-1 [m.sub.n * (v.aub.n)^2 - E.sub.(i kin) ] (EQ. 21)

where m.sub.α and m.sub.β are the masses of component pairs at r.sub.(i-1), G is the gravity constant, R.sub.αβ is the distance between component pairs prior to organization, and v.sub.n is the velocity of a component at r.sub.(i-1).

II(a). System functions, or techniques, that enhance the incorporation of energy previously external to the system, will provide a positive energy feedback to the extent that the incoming energy can be used to enhance those functions. Similarly, innovative techniques that yield more efficient work create a positive feedback by making conserved energy available for greater work. Such techniques, as well as those that reduce destructive interference and minimize degenerative transformations, will enhance the stability and evolutionary competitiveness of the encompassing system.

II(b). Stability in an energetic system is the ability of the dynamic structure to withstand perturbations. Fluctuations in the energy flow through the system boundaries, as well as fluctuations caused by component mutation, innovation or degradation, can perturb the system. A positive feedback, between energy storage, technique and increasing energy absorption, controls the susceptibility of the system to perturbation. Positive feedbacks in the energy flow that support a fluctuation will drive the system to a new dynamic configuration based on the fluctuation. This configuration will be relatively stable until another branch point is reached by another such fluctuation.

II(c). Periodic transformations between potential and kinetic energy create oscillatory responses within the system that are a product of growth. Oscillations also result when a system succeeds in maintaining its identity against fluctuations, giving rise to the homeostatic notion of "elastic limit". II(d). Dynamic coordination, while enhancing the growth and stability of an emerging energetic system, reduces the degrees of freedom of component energetic structures.

II(e). An energetic system has a lifetime, which begins upon organization of the system, and ends upon disorganization. The system exists so long as its organization can control the energy flow within its boundaries. As the system ages, its subsystems lose the elasticity that is necessary to survive the fluctuations that are present. Eventually, the dynamic coordination is interrupted when a subsystem fails to co-operate, and the system disorganizes.


III. Biospheric utilergy storage hypothesis (BUSH). Bioenergetic systems evolve both by expanding their observed boundaries and by increasing their useful energy density, or "utilergy". The evolving biosphere converts an increasing fraction of the solar influx to chemical potential or structure, so that the total utilergy contained within the observed boundary of the biosphere is an increasing function. The change in total biosphere energy over time is greater than zero

dE.sub.bio/dt > 0 (EQ. 22)

The second derivative was positive following the onset of life, but became negative as photosynthetic life reached nutrient and surface limits. More recently, the development of animal populations has accelerated the increase of biospheric energy density, so that currently

d^2(E.sub.bio)/dt^2 >0 (EQ. 23)

This hypothesis predicts that the total energy reflected or re-emitted from the Earth's surface is a decreasing fraction of the total incoming solar energy. Solving Eq. (14) for the scale radius as a function of energy, and Eq. (18) for the scale radius as a function of force, we can write an expression relating a biospheric growth force, F.sub.bio , and the total biospheric energy, E.sub.bio

( F.sub.bio/(3*lamda))^(1/2) = ( E.sub.bio/lamda )^(1/3) (EQ. 24)

Where λ = u.sub.s * (4/3) * π (gs^(-2) cm^(-1)). Making a further substitution, Λ = 3λ^(1/3), which is numerically equal to 4.836, we can write the growth force as a function of biospheric energy

F.sub.bio = Λ * (E.sub.bio)^(2/3) (EQ. 25)

III(a). The energy density of the biosphere is increasing in three ways: (i) the residence time of the solar throughput is increasing by lengthened pathway and structural storage; (ii) the amount of solar energy being channeled through the biosphere is increasing; (iii) terrestrial materials, including increasingly heavier elements, are being incorporated into the biosphere.

III(b). Organismic societies, and symbiotic assemblages of organisms in ecosystems, are energetic systems that have succeeded lower level biological systems. As energetic systems, they are attracted to energy in order to increase their energy density.




Evolutionary thermodynamics is an emerging science, but has a heritage reaching back more than 100 years.

Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,[4] Onsager 1931,[5] also[6][7]), time rate of entropy production (Onsager 1931),[5] thermodynamic fields,[8][9][10] dissipative structure,[11] and non-linear dynamical structure.[7]


'Quasi-zero-dimensional' non-equilibrium thermodynamics[edit]

It is assumed that the local entropy density is the same function of the other local intensive variables as in equilibrium; this is called the local thermodynamic equilibrium assumption[6][7][11][12][13][14][15][16] (see also Keizer (1987)[17]).


The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of extensive quantities. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential Helmholtz free energy (A = U - TS), a Legendre transformation of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the Gibbs free energy (G = U + PV - TS), where the system's properties are determined both by the temperature and by the pressure. Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundary conditions impose on them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc.), often called thermodynamic forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the entropy in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential. By definition, the entropy (S) is a function of the collection of extensive quantities . Each extensive quantity has a conjugate intensive variable (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

We then define the extended Massieu function as follows:

where is Boltzmann's constant, whence

The independent variables are the intensities.

Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system.

Stationary states, fluctuations, and stability[edit]

In thermodynamics, one is often interested in a stationary state of a process. If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy. The reproducible response of the system is then to increase the entropy back to its maximum by irreversible processes: the fluctuation cannot be reproduced with a significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323).[18] The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system.

If the stationary state is unstable, then a fluctuation can lead the system to depart from the unstable stationary state. This can be accompanied by increased export of entropy.


Entropy in evolving systems[edit]

It is pointed out[19][20][21][22] by W.T. Grandy Jr that entropy, though it may be defined for a non-equilibrium system, is when strictly considered, only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.

Flows and forces[edit]

The fundamental relation of classical equilibrium thermodynamics [23]

expresses the change in entropy of a system as a function of the intensive quantities temperature , pressure and chemical potential and of the differentials of the extensive quantities energy , volume and particle number .

Following Onsager (1931,I),[5] and considering non-equilibrium systems, one can locally define versions of the extensive macroscopic quantities , and and of the intensive macroscopic quantities , and .

According to Ilya Prigogine and others, when an open system is in conditions that allow it to reach a stable, stationary, thermodynamic non-equilibrium state, it organizes itself so as to minimize total entropy production defined locally.

The Onsager relations[edit]

Following Section III of Rayleigh (1873),[4] Onsager (1931, I)[5] showed that in the regime where both the flows are small and the thermodynamic forces vary slowly, there will be a linear relation between them, parametrized by a matrix of coefficients conventionally denoted :

The second law of thermodynamics requires that the matrix be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix is symmetric. This fact is called the Onsager reciprocal relations.

Principles for energy dissipation and entropy production[edit]

Analyzing the Rayleigh-Bénard convection cell phenomenon, Chandrasekhar (1961)[24] wrote "Instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." With a temperature gradient greater than the minimum, viscosity can dissipate kinetic energy as fast as it is released by convection due to buoyancy, and a steady state with convection is stable. The steady state with convection is often a pattern of macroscopically visible hexagonal cells with convection up or down in the middle or at the 'walls' of each cell, depending on the temperature dependence of the quantities; in the atmosphere under various conditions it seems that either is possible. (Some details are discussed by Lebon, Jou, and Casas-Vásquez (2008)[3] on pages 143-158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going.

Glansdorff and Prigogine (1971)[11] on page xv wrote "Dissipative structures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions." They were referring to the dissipation function of Rayleigh (1873)[4] that was used also by Onsager (1931, I,[5] 1931, II[25]). On pages 78–80 of their book[11] Glansdorff and Prigogine (1971) consider the stability of laminar flow that was pioneered by Helmholtz; they concluded that at a stable, steady state of sufficiently slow laminar flow, the dissipation function was minimum.

These advances have led to proposals for various extremal principles for the "self-organized" régimes that are possible for systems governed by classical linear and non-linear non-equilibrium thermodynamical laws, with stable stationary regimes being particularly investigated. Convection introduces effects of momentum which appear as non-linearity in the dynamical equations. In the more restricted case of no convective motion, Prigogine wrote of "dissipative structures". Šilhavý (1997)[26] offers the opinion that "... the extremum principles of [equilibrium] thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)."

Prigogine’s proposed theorem of minimum entropy production[edit]

In 1945 Prigogine [27] (see also Prigogine (1947)[28]) proposed a “Theorem of Minimum Entropy Production” in the linear regime near a stationary, non-equilibrium state.

Principle of maximum entropy production and minimum energy dissipation[edit]

Onsager (1931, I)[5] wrote: "Thus the vector field J of the heat flow is described by the condition that the rate of increase of entropy, less the dissipation function, be a maximum." [5]

Although largely unnoticed at the time, Ziegler proposed an idea early with his work in the mechanics of plastics in 1961,[29] and later in his book on thermomechanics revised in 1983,[9] and in various papers (e.g., Ziegler (1987),[30]). Ziegler never stated his principle as a universal law but he may have intuited this. He demonstrated his principle using vector space geometry based on an “orthogonality condition” which only worked in systems where the velocities were defined as a single vector or tensor, and thus, as he wrote[9] at p. 347, was “impossible to test by means of macroscopic mechanical models”, and was, as he pointed out, invalid in “compound systems where several elementary processes take place simultaneously”.




See also[edit]

References[edit]

  1. ^ Fowler, R., Guggenheim, E.A. (1939). Statistical Thermodynamics, Cambridge University Press, Cambridge UK, page vii.
  2. ^ Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press. ISBN 9780199546176.
  3. ^ a b Lebon, G., Jou, D., Casas-Vázquez, J. (2008). Understanding Non-equilibrium Thermodynamics: Foundations, Applications, Frontiers, Springer-Verlag, Berlin, e-ISBN 9783540742524.
  4. ^ a b c Strutt, J. W. (1871). "Some General Theorems relating to Vibrations". Proceedings of the London Mathematical Society. s1–4: =. doi:10.1112/plms/s1-4.1.357.  Unknown parameter |unused_data= ignored (help)
  5. ^ a b c d e f g Onsager, L. (1931). "Reciprocal relations in irreversible processes, I". Physical Review. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405. 
  6. ^ a b Gyarmati, I. (1970). Non-equilibrium Thermodynamics. Field Theory and Variational Principles, translated by E. Gyarmati and W.F. Heinz, Springer, Berlin.
  7. ^ a b c Lavenda, B.H. (1978). Thermodynamics of Irreversible Processes, Macmillan, London, ISBN 0-333-21616-4.
  8. ^ Gyarmati, I. (1967/1970) Non-equilibrium Thermodynamics. Field Theory and Variational Principles, translated by E. Gyarmati and W.F. Heinz, Springer, New York, pages 4-14.
  9. ^ a b c Ziegler, H., (1983). An Introduction to Thermomechanics, North-Holland, Amsterdam, ISBN 0444865039. Cite error: Invalid <ref> tag; name "Ziegler_1983" defined multiple times with different content (see the help page).
  10. ^ Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, Wiley-Interscience, New York, ISBN 0-471-04600-0, Section 3.2, pages 64-72.
  11. ^ a b c d Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-Interscience, London, 1971, ISBN 0471302805.
  12. ^ Jou, D., Casas-Vázquez, J., Lebon, G. (1993). Extended Irreversible Thermodynamics, Springer, Berlin, ISBN 3540558748, ISBN 0387558748.
  13. ^ De Groot, S.R., Mazur, P. (1962). Non-equilibrium Thermodynamics, North-Holland, Amsterdam.
  14. ^ Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, John Wiley & Sons, New York, ISBN 0471046000.
  15. ^ Mihalas, D., Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York, ISBN 0195034376.
  16. ^ Schloegl, F. (1989). Probability and Heat: Fundamentals of Thermostatistics, Freidr. Vieweg & Sohn, Brausnchweig, ISBN 3528063432.
  17. ^ Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, ISBN 0387965017.
  18. ^ Kondepudi, D., Prigogine, I, (1998). Modern Thermodynamics. From Heat Engines to Dissipative Structures, Wiley, Chichester, 1998, ISBN 0471973947.
  19. ^ Grandy, Jr. (2004). "Time Evolution in Macroscopic Systems. I. Equations of Motion". Foundations of Physics. 34: 1. Bibcode:2004FoPh...34....1G. arXiv:cond-mat/0303290Freely accessible. doi:10.1023/B:FOOP.0000012007.06843.ed. 
  20. ^ Grandy, Jr. (2004). "Time Evolution in Macroscopic Systems. II. The Entropy". Foundations of Physics. 34: 21. Bibcode:2004FoPh...34...21G. arXiv:cond-mat/0303291Freely accessible. doi:10.1023/B:FOOP.0000012008.36856.c1. 
  21. ^ Grandy Jr., W. T. (2004). "Time Evolution in Macroscopic Systems. III: Selected Applications". Foundations of Physics. 34 (5): 771. Bibcode:2004FoPh...34..771G. doi:10.1023/B:FOOP.0000022187.45866.81. 
  22. ^ Grandy 2004 see also [1].
  23. ^ W. Greiner, L. Neise, and H. Stöcker (1997), Thermodynamics and Statistical Mechanics (Classical Theoretical Physics) ,Springer-Verlag, New York, P85, 91, 101,108,116, ISBN 0387942998.
  24. ^ Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford.
  25. ^ Onsager, L. (1931). "Reciprocal relations in irreversible processes. II". Physical Review. 38 (12): 2265–2279. Bibcode:1931PhRv...38.2265O. doi:10.1103/PhysRev.38.2265. 
  26. ^ Šilhavý, M. (1997). The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, ISBN 3-540-58378-5, page 209.
  27. ^ Prigogine, I. (1945). "Modération et transformations irreversibles des systemes ouverts". Bulletin de la Classe des Sciences., Academie Royale de Belgique. 31: 600–606. 
  28. ^ Prigogine, I. (1947). Étude thermodynamique des Phenomènes Irreversibles, Desoer, Liege.
  29. ^ Ziegler, H. (1961). "Zwei Extremalprinzipien der irreversiblen Thermodynamik". Ingenieur-Archiv. 30 (6): 410–416. doi:10.1007/BF00531783. 
  30. ^ Ziegler, H.; Wehrli, C. (1987). "On a principle of maximal rate of entropy production". J. Non-Equilib. Thermodyn. 12 (3): 229–243. Bibcode:1987JNET...12..229Z. doi:10.1515/jnet.1987.12.3.229. 

Further reading[edit]

  • Kleidon, A., Lorenz, R.D., editors (2005). Non-equilibrium Thermodynamics and the Production of Entropy, Springer, Berlin. ISBN 3540224955.
  • Prigogine, I. (1955/1961/1967). Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York.
  • Zubarev D. N. (1974): Nonequilibrium Statistical Thermodynamics. New York, Consultants Bureau. ISBN 030610895X; ISBN 9780306108952.
  • Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, ISBN 0387965017.
  • Zubarev D. N., Morozov V., Ropke G. (1996): Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory. John Wiley & Sons. ISBN 3055017080.
  • Zubarev D. N., Morozov V., Ropke G. (1997): Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes. John Wiley & Sons. ISBN 3527400842.
  • Tuck, Adrian F. (2008). Atmospheric turbulence : a molecular dynamics perspective. Oxford University Press. ISBN 9780199236534.
  • Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press. ISBN 9780199546176.
  • Kondepudi, D., Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures. John Wiley & Sons, Chichester. ISBN 0471973939.

External links[edit]


Category:Evolutionary thermodynamics Category:Fundamental physics concepts Category:Branches of thermodynamics