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Non-equilibrium thermodynamics

 
Wikipedia: Non-equilibrium thermodynamics
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Non-equilibrium thermodynamics is a branch of thermodynamics concerned with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium because they are not isolated from their environment and are therefore continuously sharing matter and energy with other systems. This sharing of matter and energy includes being driven by external energy sources as well as dissipating energy. The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by equilibrium thermodynamics. Of particular importance for non-equilibrium thermodynamics are the concepts of local thermodynamic equilibrium[1][2][3][4][5] (see also Keizer (1987)[6], time rate of dissipation of energy (Section II of Rayleigh (1873)[7]), and of time rate of entropy production (Onsager 1931)[8]. Systems driven out of equilibrium do often exhibit patterned behavior such as fluctuations or phase transitions. Non-equilibrium thermodynamics, as contrasted with equilibrium thermodynamics, is most successful in the study of steady states, where there are nonzero forces, flows and entropy production, but no time variation. A favorite example of a non-equilibrium system is the Belousov-Zhabotinsky chemical oscillator. The non-equilibrium thermodynamics studies that conform to the requirement of local thermodynamic equilibrium are said by Jou, Casas-Varquez, Lebon (1993)[9] to be in the field of 'classical non-equilibrium thermodynamics', but those authors investigate processes that violate the local thermodynamic equilibrium hypothesis, processes which can be studied by more recent fast measurements; such studies are said to lie the field of 'extended non-equilibrium thermodynamics'. This present article is focused on classical non-equilibrium thermodynamics.

Contents

Basic concepts

There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary Couette flow, a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. Laser action is also a non-equilibrium thermodynamic process. Here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion). Damping of acoustical perturbations or shock waves are non-stationary non-equilibrium processes. Driven complex fluids, turbulent systems and glasses are other examples of non-equilibrium systems.

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 boundaries conditions impose to 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 Ei. Each extensive quantity has a conjugate intensive variable Ii (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

I_i = \partial{S}/\partial{E_i}.

We then define the extended Massieu function as follows:

\ k_b M = S - \sum_i( I_i E_i),

where \ k_b is Boltzmann's constant, whence

\ k_b \, dM = \sum_i (E_i \, dI_i).

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.

It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not.

Local thermodynamic equilibrium

The scope of classical non-equilibrium thermodynamics does not cover all physical processes.

Classical non-equilibrium thermodynamics of matter in laboratory conditions

According to Wildt (1972)[10] (see also Essex (1984a)[11] (1984b)[12] (1984c)[13]), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to matter under laboratory conditions with temperatures well below the those of stars. At laboratory temperatures, in laboratory quantities of matter, thermal radiation is weak and can be practically nearly ignored. For example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that are not necessarily within the scope of laboratory conditions. Thus current versions of classical non-equilibrium thermodynamics might be called classical non-equilibrium thermodynamics of matter in laboratory conditions. The exact boundaries of its scope may not be fully known, but one wants some confidence in what one is doing, and so one says that a sufficient condition for the validity of many studies in classical non-equilibrium thermodynamics of matter is that they deal with what is known as local thermodynamic equilibrium[1][2][3][4][5] (see also Keizer (1987)[6]. In classical thermodynamics, macroscopic variables such as temperature and entropy are strictly speaking defined only in systems at thermodynamic equilibrium.

Local thermodynamic equilibrium of matter means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' of small (infinitesimal) size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation. These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective. When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in local thermodynamic equilibrium.

One can think here of two 'relaxation times' separated by order of magnitude[14]. The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. The shorter is of the order of magnitude of times taken for a single 'cell' to reach local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning.[14] For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies.

Milne's 1928 definition of local thermodynamic equilibrium in terms of radiative equilibrium

Milne (1928)[15], thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the thermal radiation of the matter in the 'cell'. He forgot Planck's (1914)[16] (see Robitaille, (2003)[17] (2008)[18]) careful insistence on the presence and radiative equilibrium of a black body for the production of the Kirchhoff-Planck so-called "universal" thermal radiative spectrum. Milne (1928)[15] defined 'local thermodynamic equilibrium' by requiring that the 'cell' radiate as if it were a black body in radiative equilibrium in a cavity at the temperature of the matter of the 'cell'. In the absence of a black body in radiative equilibrium with the 'cell' and in the absence of a specification of the walls of the cavity, this definition is devoid of physical meaning. Milne (1928))[15] wrongly assumed that even in the absence of these requirements the cell would necessarily radiate as if it were a black body in thermodynamic equilibrium.

Flows and forces

The fundamental relation of classical equilibrium thermodynamics

dS=\frac{1}{T}dU+\frac{p}{T}dV+\sum_{i=1}^s\frac{\mu_i}{T}dN_i

expresses the change in entropy dS of a system as a function of the intensive quantities temperature T, pressure p and ith chemical potential μi and of the differentials of the extensive quantities energy U, volume V and ith particle number Ni.

Following Onsager (1931,I)[8], let us extend our considerations to thermodynamically non-equilibrium systems. As a basis, we need locally defined versions of the extensive macroscopic quantities U, V and Ni and of the intensive macroscopic quantities T, p and μi.

For classical non-equilibrium studies, we will consider some new locally defined intensive macroscopic variables. We can, under suitable conditions, derive these new variables by locally defining the gradients and flux densities of the basic locally defined macroscopic quantities.

Such locally defined gradients of intensive macroscopic variables are called 'thermodynamic forces'. They 'drive' flux densities, perhaps misleadingly often called 'fluxes', which are dual to the forces. These quantities are defined in the article on Onsager reciprocal relations.

Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (Ji) may be coupled. The article on Onsager reciprocal relations considers the stable near-steady thermodynamically non-equilibrium regime, which has dynamics linear in the forces and flux densities.

In stationary conditions, such forces and associated flux densities are by definition time invariant, as also are the system's locally defined entropy and rate of entropy production. Notably, according to Ilya Prigogine and others , when an open system is in conditions that allow it to reach a stable stationary thermodynamically non-equilibrium state, it organizes itself so as to minimize total entropy production defined locally. This is considered further below.

One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of non-stationary local quantities; these integrals are macroscopic fluxes and production rates. In general the dynamics of these integrals are not adequately described by linear equations, though in special cases they can be so described.

The Onsager relations

Main article: Onsager reciprocal relations

Following Section III of Rayleigh (1873)[7], Onsager (1931, I)[8] 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 L:

J_i = \sum_{j} L_{ij} \nabla I_j

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

Microscopic reversibility of the dynamics implies that the matrix Lij is symmetrical (Onsagers reciprocal relations).

Thermodynamic Extremum Principles for Energy Dissipation and Entropy Production

Jou, Casas-Vazquez, Lebon (1993)[9] note that classical non-equilibrium thermodynamics "has seen an extraordinary expansion since the second world war", and they refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine. Mahulikar and Herwig (2004)[19] wrote “The major revolution in the latter half of the previous century is the understanding with an expanded view of thermodynamics that the spontaneous production of order from disorder is the expected consequence of basic laws.” Martyushev and Seleznev (2006)[20] note the importance of entropy in the evolution of natural dynamical structures: "Great contribution has been done in this respect by two scientists, namely Clausius, ... , and Prigogine." Prigogine in his 1977 Nobel Lecture[21] said: "... non-equilibrium may be a source of order. Irreversible processes may lead to a new type of dynamic states of matter which I have called “dissipative structures”." Glansdorff and Prigogine (1971)[2] wrote on page xx: "Such 'symmetry breaking instabilities' are of special interest as they lead to a spontaneous 'self-organization' of the system both from the point of view of its space order and its function."

Principles of Maximum Entropy Production and Minimum Energy Dissipation

Onsager (1931, I)[8] 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." Careful note needs to be taken of the opposite signs of the rate of entropy production and of the dissipation function, appearing in the left-hand side of Onsager's equation (5.13) on Onsager's page 423[8].

Although largely unnoticed at the time Ziegler approached the idea early with his work in the mechanics of plastics in 1961[22], and later in his book on thermomechanics revised in 1983[23], and in various papers (e.g., Ziegler (1987)[24],). 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[23] 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”.

Chandrasekhar (1961)[25] 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."

Glansdorff and Prigogine (1971)[2] 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)[7] that was used also by Onsager (1931, I[8], 1931, II[26]). On page 79 of their book[2] Glansdorff and Prigogine (1971) concluded that at a stable steady state, the dissipation function was minimum.

Sawada (1981)[27], postulating a principle of largest amount of entropy increment per unit time, cites work in fluid mechanics by Malkus and Veronis (1958)[28] as having "proven a principle of maximum heat current, which in turn is a maximum entropy production for a given boundary condition". According to Tuck (2008)[29], "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975[30], 2001[31]). Initially Paltridge (1975)[30] used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978)[32], and in Paltridge (1979)[33]), he used the now current terminology "maximum entropy production" to describe the same thing. This point is clarified in the review by Ozawa, Ohmura, Lorenz, Pujol (2003)[34]. Paltridge (1978)[32] cited Busse's (1967)[35] fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) [36] discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. Again investigating planetary atmospheric dynamics, Shutts (1981)[37] used an approach to the definition of entropy production, different from Paltridge's, to investigate a more abstract way to check the principle of maximum entropy production, and found a good fit.

Independently, beginning in 1988, Swenson postulated a “principle of maximum entropy production” which by 1989 he had had formulated and stated in a universal form (applying to all scales, and ranges) as what he called the Law of Maximum Entropy Production (MEP or LMEP) which he and colleagues elaborated in papers over the next decade and a half and applied particularly to evolution e.g.,[38][39][40][41][42][43] . Swenson postulated that systems select the paths out of available paths that maximize the rate of their entropy production (“MEP” or “LMEP”). The demonstration of the principle was done with simple physical systems where it can be shown that if a system has multiple paths by which to bring disequilibria to equilibrium it will pick the paths out of available paths that do it at the fastest rate given the constraints. He then used this to derive a ‘universal ordering principle’ noting that if systems maximize the entropy at the fastest rate given the constraints and because ordered states produce entropy faster than disordered ones then ordered states will be selected whenever it is possible to do so.

C. Nicolis (1999)[44] concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production; in the present writer's opinion, there are few as expert in the theory of entropy production as Nicolis. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008)[45] is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's 1931[8] origination of this subject. Grandy (2008)[45] in section 4.3 on page 55 is careful to distinguish between the idea that entropy is related to order (which he considers to be a "mischaracterization" that needs "debunking"), and the idea of E.T. Jaynes[46] that entropy is related to the experimental reproducibility of process (which Grandy regards as correct).

Using the information theoretical formalism of Jaynes, Dewar (2003[47], 2005[48]) claimed to have derived a principle of maximum entropy production applying to nonlinear stationary states. With a background of previous literature, a number of authors[49] cited Dewar’s purported proof. Grinstein and Linsker (2007)[50] showed that Dewar’s derivation was invalid due to an unnoticed physical assumption of linearity in the purported proof; they concluded that "the question of the existence of possible extremal principles (and in particular, of MaxEP) that might apply to far-from-equilibrium regimes (having non-linear constitutive relations) has not been settled by [[47],[48]]". Lucia[51] offers a proof of a theorem for the eventual steady state of a constrained open system: "The principle of maximum irreversible entropy: The irreversible entropy reaches its maximum at the stability."

Prigogine’s Theorem of Minimum Entropy Production

In 1945 Prigogine [52] (see also Prigogine (1947)[53]) proposed a “Theorem of Minimum Entropy Production” which applies only to the linear regime near a stationary but possibly thermodynamically non-equilibrium state, and which some have mistakenly taken to be contradictory to MEP or LMEP in its universal form. They are not contradictory but answer different questions. Prigogine’s theorem can be visualized by imagining a system with several thermodynamic forces which are all allowed to equilibrate or adjust in a time-dependent way except that at least one is maintained so that the system cannot get to equilibrium. The result is that that the entropy production will start at some rate and because the forces are progressively depleted the rate of the entropy production will steadily (monotonically) go down until the system gets as close to equilibrium as it can and then stay in that state as long as the one force is maintained. So the entropy production starts at one place and goes down to a minimum where it stays. What the principle does not address is which paths out of available paths the system will take to do this and this is what MEP or LMEP in its universal form answers . So in the linear range where in the experimental model demonstrating MEP/LMEP[40] the system will select the paths out of available paths that will deplete the forces (maximize the entropy) at the fastest rate given the constraints thus validating LMEP/MEP while the rate of entropy production will still go monotonically down to satisfy Prigogine’s principle too. Outside of this range however while LMEP/MEP holds Prigogine’s theorem, as he himself emphasized, does not (for more discussion on this see the Wikipedia article on the Law of Maximum Entropy Production. Martyushev and Seleznev (2006)[20] refer to a "popular opinion" that the principle of maximum entropy production does not apply in the linear regime, claiming in their footnote 7 that it was held by Ozawa, Ohmura, Lorenz and Pujol (2003)[34]; the nearest that the present writer can find in that review[34] to support that claim of Martyushev and Seleznev (2006)[20] is the apparently innocent couple of sentences on page 18: "The maximum entropy production principle, then, acts as a guiding principle for choosing the most probable state among all other possible states allowed by the nonlinear system. The MEP principle is therefore fundamental to nonlinear systems and should not be confused with Prigogine's one for linear systems".

See also

References

  1. ^ a b Gyarmati, I. (1970). Non-equilibrium Thermodynamics. Field Theory and Variational Principles, Springer, Berlin.
  2. ^ a b c d e Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability, and Fluctuations, Wiley-Interscience, London, 1971, ISBN 0471302805.
  3. ^ a b Balescu, R. (1975). Equilibrium and Non-equilibrium Statistical Mechanics, John Wiley & Sons, New York, ISBN 0471046000.
  4. ^ a b Mihalas, D., Weibel-Mihalas, B. (1984). Foundations of Radiation Hydrodynamics, Oxford University Press, New York, ISBN 0195034376.
  5. ^ a b Schloegl, F. (1989). Probability and Heat: Fundamentals of Thermostatistics, Freidr. Vieweg & Sohn, Brausnchweig, ISBN 3528063432.
  6. ^ a b Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, ISBN 0387965017.
  7. ^ a b c Strutt, J.W., Lord Rayleigh (1873). Some general theorems relating to vibrations, Proceedings of the London Mathematical Society 4: 357-368.
  8. ^ a b c d e f g Onsager, L. (1931). Reciprocal relations in irreversible processes, I, Physical Review 37:405-426.
  9. ^ a b Jou, D., Casas-Vazquez, J., Lebon, G. (1993). Extended Irreversible Thermodynamics, Springer, Berlin, ISBN 3540558748, ISBN 0387558748.
  10. ^ Wildt, R. (1972). Thermodynamics of the gray atmosphere. IV. Entropy transfer and production, Astrophysical Journal 174: 69-77.
  11. ^ Essex, C. (1984a). Radiation and the irreversible thermodynamics of climate, Journal of the Atmospheric Sciences 41(12): 1985-1991.
  12. ^ Essex, C. (1984b). Minimum entropy production in the steady state and radiative transfer, Astrophysical Journal 285: 279-293.
  13. ^ Essex, C. (1984c). Radiation and the violation of bilinearity in the irreversible thermodynamics of irreversible processes, Planetary and Space Science 32: 1035-1043.]
  14. ^ a b Zubarev D. N.,(1974). Nonequilibrium Statistical Thermodynamics, translated from the Russian by P.J. Shepherd, New York, Consultants Bureau. ISBN 030610895X; ISBN 9780306108952.
  15. ^ a b c Milne, E.A. (1928). The effect of collisions on monochromatic radiative equilibrium, Monthly Notices of the Royal Astronomical Society 88: 493-502
  16. ^ Planck, M. (1906/1914). The Theory of Heat Radiation, second edition, translated by M. Masius, P. Blakiston's Son & Co., Philadelphia.
  17. ^ Robitaille, P.-M. L. (2003). On the validity of Kirchhoff's law of thermal emission, IEEE Transactions on Plasma Science 31(6): 1263-1267.
  18. ^ Robitaille, P.-M. L. (2008). Blackbody radiation and the carbon particle, Progress in Physics 2008(3): 36-55.
  19. ^ Mahulikar, S.P, & Herwig, H. (2004), Conceptual investigation of the entropy principle for identification of directives for creation, existence and total destruction of order, Physica Scripta, Vol. 70, 212-221, p.212
  20. ^ a b c Martyushev, L.M., Seleznev, V.D. (2006). Maximum entropy production principle in physics, chemistry and biology, Physics Reports 426: 1-45.
  21. ^ Prigogine, I. (1977). Time, Structure and Fluctuations, Nobel Lecture.
  22. ^ Ziegler, H. (1961). Zwei Extremalprinzipien der irreversiblen Thermodynamik, Ingenieur-Archiv 30:410-416.
  23. ^ a b Ziegler, H. (1983). An Introduction to Thermomechanics, North-Holland series in applied mathematics and mechanics, v. 21, 2nd rev. ed. North-Holland Publishing, New York.
  24. ^ Ziegler, H., Wehrli, C. (1987). On a principle of maximal rate of entropy production, J. Non-Equilib. Thermodyn. 12: 229-243.
  25. ^ Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford.
  26. ^ Onsager, L. (1931). Reciprocal relations in irreversible processes. II, Physical Review 38: 2265-2279
  27. ^ Sawada, Y. (1981). A thermodynamic variational principle in nonlinear non-equilibrium phenomena, Progress of Theoretical Physics 66: 68-76.
  28. ^ Malkus, W.V.R., Veronis, G. (1958). Finite amplitude cellular convection, Journal of Fluid Mechanics 4(3): 225-260.
  29. ^ Tuck, Adrian F. (2008) Atmospheric Turbulence: a molecular dynamics perspective, Oxford University Press. ISBN 9780199236534. See page 33.
  30. ^ a b Paltridge, G.W. (1975). Global dynamics and climate - a system of minimum entropy exchange, Quarterly Journal of the Royal Meteorological Society 101:475-484.
  31. ^ Paltridge G.W.(2001). A physical basis for a maximum of thermodynamic dissipation of the climate system, Quarterly Journal of the Royal Meteorological Society 127:305-313.[1]
  32. ^ a b Paltridge, G.W. (1978). The steady-state format of global climate, Quarterly Journal of the Royal Meteorological Society 104: 927-945.
  33. ^ Paltridge, G.W. (1979). Climate and thermodynamic systems of maximum dissipation, Nature 279:630-631.[2]
  34. ^ a b c Ozawa, H., Ohmura, A., Lorenz, R.D., Pujol, T. (2003). The Second Law of Thermodynamics and the Global Climate System: A Review of the Maximum Entropy Production Principle, Reviews of Geophysics, 41, 4: 1-24.
  35. ^ Busse, F.H.(1967). The stability of finite amplitude cellular convection and its relation to an extremum principle, Journal of Fluid Mechanics 30(4): 625-649.
  36. ^ Nicolis, G., Nicolis, C. (1980). On the entropy balance of the earth-atmosphere system, Quarterly Journal of the Royal Meteorological Society 125:1859-1878.
  37. ^ Shutts, G.J. (1981). Maximum entropy production states in quasi-geostrophic dynamical models, Quarterly Journal of the Royal Meteorological Society 107: 503-520.
  38. ^ Swenson, R. (1989d). Gauss-in-a-box: Nailing down the first principles of action. Perceiving Acting Workshop Review (Technical Report of the Center for the Ecological Study of Perception and Action, CESPA, University of Connecticut) 5, 60-63.
  39. ^ Swenson, R. (1991). End-directed physics and evolutionary ordering: Obviating the problem of the population of one. In The Cybernetics of Complex Systems: Self-Organization, Evolution, and Social Change, F. Geyer (ed.), 41-60. Salinas, CA: Intersystems Publications
  40. ^ a b Swenson, R. and Turvey, M.T. (1991). Thermodynamic reasons for perception-action cycles. Ecological Psychology, 3(4), 317-348.
  41. ^ Swenson, R. (1992). Order, evolution, and natural law: Fundamental relations in complex system theory. In Cybernetics and Applied Systems, C. Negoita (ed.), 125-148. New York: Marcel Dekker Inc
  42. ^ Swenson, R. (1997). Autocatakinetics, evolution, and the law of maximum entropy production: A principled foundation toward the study of human ecology. Advances in Human Ecology, 6, 1-46.
  43. ^ Swenson, R. (1998a). Thermodynamics, evolution, and behavior. In The Handbook of Comparative Psychology, G. Greenberg and M. Haraway (eds.), Garland Publishing, New York.
  44. ^ Nicolis, C. (1999). Entropy production and dynamical complexity in a low-order atmospheric model, Quarterly Journal of the Royal Meteorological Society 125:1859-1878.
  45. ^ a b Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press. ISBN 9780199546176.
  46. ^ Jaynes, E.T. (1965). Gibbs vs Boltzmann Entropies, American Journal of Physics 33: 391-398.
  47. ^ a b Dewar, R. (2003). Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states. J. Phys. A: Math. Gen. 36, 63.
  48. ^ a b Dewar R C (2005). J. Phys. A: Math. Gen. 38 L371
  49. ^ Kleidon, A. & R.D. Lorenz (2005). Nonequilibrium Thermodynamics and the Production of Entropy. Life, Earth, and Beyond, Springer-Verlag, Berlin/Heidelberg. ISBN 3450224955
  50. ^ Grinstein, G. & R. Linsker (2007). Comments on a derivation and application of the 'maximum entropy production' principle, Journal of Physics A: Mathematical and Theoretical 40: 9717-9720.
  51. ^ Lucia, U. (2007). Irreversible entropy variation and the problem of the trend to equilibrium, Physica A 376: 289-292.
  52. ^ Prigogine, I. (1945). Moderation et transformations irreversibles des systemes ouverts, Bulletin de la Classe des Sciences., Academie Royale de Belgique 31: 600-606.
  53. ^ Prigogine, I. (1947). Etude thermodynamique des Phenomenes Irreversibles, Desoer, Liege.

Books

  1. Ziegler, Hans (1977): An introduction to Thermomechanics. North Holland, Amsterdam. ISBN 0444110801. Second edition (1983) ISBN 0444865039.
  2. Kleidon, A., Lorenz, R.D., editors (2005). Non-equilibrium Thermodynamics and the Production of Entropy, Springer, Berlin. ISBN 3540224955.
  3. Prigogine, I. (1955/1961/1967). Introduction to Thermodynamics of Irreversible Processes. 3rd edition, Wiley Interscience, New York.
  4. Zubarev D. N. (1974): Nonequilibrium Statistical Thermodynamics. New York, Consultants Bureau. ISBN 030610895X; ISBN 9780306108952.
  5. Keizer, J. (1987). Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, ISBN 0387965017.
  6. Zubarev D. N., Morozov V., Ropke G. (1996): Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory. John Wiley & Sons. ISBN 3055017080.
  7. Zubarev D. N., Morozov V., Ropke G. (1997): Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes. John Wiley & Sons. ISBN 3527400842.
  8. Tuck, Adrian F. (2008). Atmospheric turbulence : a molecular dynamics perspective. Oxford University Press. ISBN 9780199236534.
  9. Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems. Oxford University Press. ISBN 9780199546176.
  10. Kondepudi, D., Prigogine, I. (1998). Modern Thermodynamics: From Heat Engines to Dissipative Structures. John Wiley & Sons, Chichester. ISBN 0471973939.

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