Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics.Ziegler, H., (1983). An Introduction to Thermomechanics, North-Holland, Amsterdam, According to Kondepudi (2008),Kondepudi, D. (2008)., Introduction to Modern Thermodynamics, Wiley, Chichester UK, , page 172. and to Grandy (2008),Grandy, W.T., Jr (2008). Entropy and the Time Evolution of Macroscopic Systems, Oxford University Press, Oxford, . there is no general rule that provides an extremum principle that governs the evolution of a far-from-equilibrium system to a steady state. According to Glansdorff and Prigogine (1971, page 16), irreversible processes usually are not governed by global extremal principles because description of their evolution requires differential equations which are not self-adjoint, but local extremal principles can be used for local solutions. Lebon Jou and Casas-Vásquez (2008)Lebon, G., Jou, J., Casas-Vásquez (2008). Understanding Non-equilibrium Thermodynamics. Foundations, Applications, Frontiers, Springer, Berlin, . state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997)Šilhavý, M. (1997). The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, , page 209. offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)." It follows that any general extremal principle for a non-equilibrium problem will need to refer in some detail to the constraints that are specific for the structure of the system considered in the problem.

Fluctuations, entropy, 'thermodynamics forces', and reproducible dynamical structure Apparent 'fluctuations', which appear to arise when initial conditions are inexactly specified, are the drivers of the formation of non-equilibrium dynamical structures. There is no special force of nature involved in the generation of such fluctuations. Exact specification of initial conditions would require statements of the positions and velocities of all particles in the system, obviously not a remotely practical possibility for a macroscopic system. This is the nature of thermodynamic fluctuations. They cannot be predicted in particular by the scientist, but they are determined by the laws of nature and they are the singular causes of the natural development of dynamical structure.

It is pointed outGrandy, W.T., Jr (2004). Time evolution in macroscopic systems. I: Equations of motion. Found. Phys. 34: 1-20. See .Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. II: The entropy. Found. Phys. 34: 21-57. See .Grandy, W.T., Jr (2004). Time evolution in macroscopic systems. III: Selected applications. Found. Phys. 34: 771-813. See .Grandy 2004 see also . 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 it can metaphorically be thought of as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.

As indicated by the quotation marks of Onsager (1931), such a metaphorical but not categorically mechanical force, the thermal "force", , 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write . Actually this thermal "thermodynamic force" is a manifestation of the degree of inexact specification of the microscopic initial conditions for the system, expressed in the thermodynamic variable known as temperature, . Temperature is only one example, and all the thermodynamic macroscopic variables constitute inexact specifications of the initial conditions, and have their respective "thermodynamic forces". These inexactitudes of specification are the source of the apparent fluctuations that drive the generation of dynamical structure, of the very precise but still less than perfect reproducibility of non-equilibrium experiments, and of the place of entropy in thermodynamics. If one did not know of such inexactitude of specification, one might find the origin of the fluctuations mysterious. What is meant here by "inexactitude of specification" is not that the mean values of the macroscopic variables are inexactly specified, but that the use of macroscopic variables to describe processes that actually occur by the motions and interactions of microscopic objects such as molecules is necessarily lacking in the molecular detail of the processes, and is thus inexact. There are many microscopic states compatible with a single macroscopic state, but only the latter is specified, and that is specified exactly for the purposes of the theory.

It is reproducibility in repeated observations that identifies dynamical structure in a system. E.T. JaynesJaynes, E.T. (1985). Macroscopic prediction, in Complex Systems - Operational Approaches in Neurobiology, edited by H. Haken, Springer-Verlag, Berlin, pp. 254-269 . explains how this reproducibility is why entropy is so important in this topic: entropy is a measure of experimental reproducibility. The entropy tells how many times one would have to repeat the experiment in order to expect to see a departure from the usual reproducible result. When the process goes on in a system with less than a 'practically infinite' number (much much less than Avogadro's or Loschmidt's numbers) of molecules, the thermodynamic reproducibility fades, and fluctuations become easier to see.Wang, G.M., Sevick, E.M., Mittag, E., Searles, D.J., Evans, D.J. (2002) Experimental demonstration of violations of the Second Law of Thermodynamics for small systems and short time scales, Physical Review Letters 89: 050601-1 - 050601-4.

According to this view of Jaynes, it is a common and confusing abuse of language, that one often sees reproducibility of dynamical structure called "order".Dewar, R.C. (2005). Maximum entropy production and non-equilibrium statistical mechanics, pp. 41-55 in Non-equilibrium Thermodynamics and the Production of Entropy, edited by A. Kleidon, R.D. Lorenz, Springer, Berlin. . Dewar writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,Jaynes, E.T. (1963). pp. 181-218 in Brandeis Summer Institute 1962, Statistical Physics, edited by K.W. Ford, Benjamin, New York. 1965, 1988,Jaynes, E.T. (1988). The evolution of Carnot's Principle, pp. 267-282 in Maximum-entropy and Bayesian methods in science and engineering, edited by G.J. Erickson, C.R. Smith, Kluwer, Dordrecht, volume 1 . 1989Jaynes, E.T. (1989). Clearing up mysteries, the original goal, pp. 1-27 in Maximum entropy and Bayesian methods, Kluwer, Dordrecht.)." Grandy (2008) in section 4.3 on page 55 clarifies the distinction between the idea that entropy is related to order (which he considers to be an "unfortunate" "mischaracterization" that needs "debunking"), and the aforementioned idea of Jaynes that entropy is a measure of experimental reproducibility of process (which Grandy regards as correct). According to this view, even the admirable book of Glansdorff and Prigogine (1971) is guilty of this unfortunate abuse of language.

Local thermodynamic equilibrium Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),Glansdorff, P., Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, London. in general, these principles apply only to systems that can be described by thermodynamical variables, in which dissipative processes dominate by excluding large deviations from statistical equilibrium. The thermodynamical variables are defined subject to the kinematical requirement of local thermodynamic equilibrium. This means that collisions between molecules are so frequent that chemical and radiative processes do not disrupt the local Maxwell-Boltzmann distribution of molecular velocities.

Linear and non-linear processes Dissipative structures can depend on the presence of non-linearity in their dynamical régimes. Autocatalytic reactions provide examples of non-linear dynamics, and may lead to the natural evolution of self-organized dissipative structures.

Continuous and discontinuous motions of fluids Much of the theory of classical non-equilibrium thermodynamics is concerned with the spatially continuous motion of fluids, but fluids can also move with spatial discontinuities. Helmholtz (1868)Helmholtz, H. (1868). On discontinuous movements of fluids, Philosophical Magazine series 4, vol. 36: 337-346, translated by F. Guthrie from Monatsbericht der koeniglich preussischen Akademie der Wissenschaften zu Berlin April 1868, page 215 et seq. wrote about how in a flowing fluid, there can arise a zero fluid pressure, which sees the fluid broken asunder. This arises from the momentum of the fluid flow, showing a different kind of dynamical structure from that of the conduction of heat or electricity. Thus for example: water from a nozzle can form a shower of droplets (Rayleigh 1878, and in section 357 et seq. of Rayleigh (1896/1926)Strutt, J.W. (Baron Rayleigh) (1896/1926). Section 357 et seq. The Theory of Sound, Macmillan, London, reprinted by Dover, New York, 1945.); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975Thom, R. (1975). Structural Stability and Morphogenesis: An outline of a general theory of models, translated from the French by D.H. Fowler, W.A. Benjamin, Reading Ma, ). Helmholtz pointed out that the sounds of organ pipes must arise from such discontinuity of flow, occasioned by the passage of air past a sharp-edged obstacle; otherwise the oscillatory character of the sound wave would be damped away to nothing. The definition of the rate of entropy production of such a flow is not covered by the usual theory of classical non-equilibrium thermodynamics. There are many other commonly observed discontinuities of fluid flow that also lie beyond the scope of the classical theory of non-equilibrium thermodynamics, such as: bubbles in boiling liquids and in effervescent drinks; also protected towers of deep tropical convection (Riehl, Malkus 1958), also called penetrative convection (Lindzen 1977Lindzen, R.S. (1977). Some aspects of convection in meteorology, pp. 128-141 in Problems of Stellar Convection, volume 71 of Lecture Notes in Physics, Springer, Berlin, .).

Historical development W. Thomson, Baron Kelvin

William Thomson, later Baron Kelvin, (1852 a,Thomson, William (1852 a). "On a Universal Tendency in Nature to the Dissipation of Mechanical Energy" Proceedings of the Royal Society of Edinburgh for April 19, 1852 [This version from Mathematical and Physical Papers, vol. i, art. 59, pp. 511.] 1852 b) wrote

"II. When heat is created by any unreversible process (such as friction), there is a dissipation of mechanical energy, and a full restoration of it to its primitive condition is impossible.

III. When heat is diffused by conduction, there is a dissipation of mechanical energy, and perfect restoration is impossible.

IV. When radiant heat or light is absorbed, otherwise than in vegetation, or in a chemical reaction, there is a dissipation of mechanical energy, and perfect restoration is impossible."