A founding study of how irreversible processes behave far from equilibrium
What the Work Is
Étude thermodynamique des phénomènes irréversibles is a scientific study of irreversible processes, first published in 1947. It is the work in which Ilya Prigogine set out his early thermodynamic treatment of systems that are not in equilibrium, and it cites the earlier reciprocal relations of Lars Onsager (1931) as a foundation. Prigogine was a Belgian physical chemist of Russian origin whose work on dissipative structures, complex systems, and irreversibility would later earn him the 1977 Nobel Prize in Chemistry.
What the Book Is About
The book addresses the thermodynamics of irreversible phenomena — processes that run in one direction and cannot simply be run backwards, such as the conduction of heat or the diffusion of matter. Its central contribution is a proposed “Theorem of Minimum Entropy Production,” which Prigogine put forward in 1945 and developed in this 1947 work. The theorem applies only to the purely diffusive linear regime, with negligible inertial terms, near a stationary thermodynamically non-equilibrium state. In that restricted setting, Prigogine’s proposal is that the rate of entropy production is locally minimum at every point.
Key Ideas and Themes
The organizing idea of the book is that irreversible processes can be described through thermodynamic “forces” and the flows they drive. In this framework, a thermal “force” such as the one driving the conduction of heat is written in terms of the temperature gradient, and it is understood not as a mechanical force but as a metaphor — a way of representing how a macroscopic variable, here temperature, stands in for the inexactly specified microscopic state of the system. Temperature is only one example: all the macroscopic thermodynamic variables constitute inexact specifications of initial conditions, and each has its respective “thermodynamic force.” These inexactitudes are the source of the apparent fluctuations that drive the generation of dynamical structure.
The book’s proposed theorem belongs to the linear regime — the case of sufficiently slow transfer in which flux and generalized force are linearly related and inertial terms can be neglected. Prigogine’s claim is that in this regime the rate of entropy production is locally minimum at every point. The proof he offered has been open to serious criticism. A critical and unsupportive discussion of the proposal was later given by Grandy, and it has been shown by Barbera that the total whole-body entropy production cannot be minimum, though that paper did not consider the pointwise minimum proposal. A closely related proposal holds that the pointwise rate of entropy production should have its maximum value minimized at the steady state — compatible with, but not identical to, Prigogine’s. N. W. Tschoegl proposed a proof, perhaps more physically motivated than Prigogine’s, that would if valid support the conclusion of Helmholtz and of Prigogine that under these restricted conditions entropy production is at a pointwise minimum.
The book sits within a broader search for extremal principles in non-equilibrium thermodynamics — principles that would predict the likely steady states and dynamical structures a physical system might show, following their successful use elsewhere in physics. Prigogine’s minimum-entropy-production theorem is one such proposal, and it is specifically a theorem for very slow, purely diffusive transfer. It is distinct from the maximum-entropy-production principle that Ziman proposed for the thermodynamics of irreversible processes, under which, of all current distributions for which intrinsic and extrinsic entropy production are equal for a given set of forces, the steady state distribution makes entropy production a maximum. Ziman noted the difference between his principle and “Prigogine’s theorem,” which states, crudely speaking, that if not all the forces acting on a system are fixed, the free forces will take such values as to make the entropy production a minimum. Prigogine was present when Ziman’s paper was read and is reported to have given notice that he doubted the validity of part of Ziman’s thermodynamic interpretation.
The book’s scope is bounded by its linear, near-equilibrium assumptions. Faster transfer, in which non-linearity appears and heat flow can develop into phases of convective circulation, falls outside it: in such cases the time rate of entropy production has been shown to be a non-monotonic function of time during the approach to steady state, so the local rate of entropy production is not an adequate variable for predicting the time course of far-from-equilibrium processes, and the principle of minimum entropy production does not apply. Covering those cases would require at least one further non-equilibrium state variable, the so-called second entropy — a step toward generalizing beyond the classical second law, which refers only to states of thermodynamic equilibrium.
Significance and Influence
The 1947 study is an early landmark in the development of non-equilibrium thermodynamics, the field that expanded extraordinarily after the Second World War and that produced Nobel Prizes for both Onsager and Prigogine. Its proposed theorem became a reference point for later work: it is cited in surveys of extremal principles, and it drew both extensions and criticism. Gyarmati’s systematic presentation of the field cites numerous papers and books authored or co-authored by Prigogine, and Prigogine’s own later collaboration with Glansdorff, published in 1971, carried the subject further — that later book is itself noted for treating dissipative structures, formed and maintained through the exchange of energy and matter in non-equilibrium conditions, and for concluding that at a stable steady state of sufficiently slow laminar flow the dissipation function was minimum.
At the same time, the prospects for general extremal principles of this kind have remained clouded. Glansdorff and Prigogine themselves observed that irreversible processes usually are not governed by global extremal principles, because their description requires differential equations that are not self-adjoint, though local extremal principles can be used for local solutions. Other writers, including Lebon, Jou and Casas-Vásquez, and Šilhavý, have likewise felt that prospects for general global extremal principles are clouded; Šilhavý has offered 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 has been shown that heat convection does not obey extremal principles for entropy production and that chemical reactions do not obey extremal principles for the secondary differential of entropy production, so the development of a general extremal principle seems infeasible. The lasting importance of Prigogine’s 1947 study lies in having framed the question sharply — proposing a definite extremal principle for a well-defined regime, and thereby setting terms that later thermodynamics would test, extend, and dispute.
Onsager’s own 1931 formulation of the subject included the statement that “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,” and careful note must be taken of the opposite signs of the rate of entropy production and of the dissipation function as they appear in the left-hand side of equation (5.13) on page 423 of that work. The dissipation function itself was introduced by Rayleigh in 1873, and again in Sections 81 and 345 of his 1896/1926 treatise, for the description of dissipative processes involving viscosity; it was conceived from a mechanical viewpoint and did not refer in its definition to temperature, so that it needed to be generalized before it could serve in non-equilibrium thermodynamics. More general versions of the function have since been used by many investigators of dissipative processes and dynamical structures.
Rayleigh’s study of jets of water from a nozzle noted that when a jet is in a state of conditionally stable dynamical structure, the mode of fluctuation most likely to grow to its full extent and lead to another such state is the one with the fastest growth rate; a jet can settle into a conditionally stable state but is likely to suffer fluctuation so as to pass to another, less unstable, conditionally stable state. He used like reasoning in a study of Bénard convection, and these physically lucid considerations seem to contain the heart of the distinction between the principles of minimum and maximum rates of dissipation of energy and entropy production.
Earlier still, William Thomson, later Baron Kelvin, wrote in 1852 that when heat is created by any unreversible process such as friction there is a dissipation of mechanical energy and full restoration to the primitive condition is impossible; that when heat is diffused by conduction there is a dissipation of mechanical energy and perfect restoration is impossible; and that when radiant heat or light is absorbed, otherwise than in vegetation or in a chemical reaction, there is again a dissipation of mechanical energy and perfect restoration is impossible. In 1854 Thomson wrote about the relation between two previously known non-equilibrium effects: in the Peltier effect an electric current driven by an external electric field across a bimetallic junction carries heat across the junction when the temperature gradient is constrained to zero, while in the Seebeck effect a flow of heat driven by a temperature gradient across such a junction causes an electromotive force when the electric current is constrained to zero, so that thermal and electric effects are said to be coupled. Thomson proposed a theoretical argument, partly based on Carnot and Clausius and in those days partly speculative, that the coupling constants of the two effects would be found experimentally to be equal; experiment later confirmed this, and the idea was later among those that led Onsager to his results.
Helmholtz in 1869 stated his minimum dissipation theorem, subject to a certain kind of boundary condition, as a principle of least viscous dissipation of kinetic energy: for a steady flow in a viscous liquid, with the speeds of flow on the boundaries given steady, in the limit of small speeds the currents distribute themselves so that the dissipation of kinetic energy by friction is minimum. In 1878, again citing Carnot and Clausius, Helmholtz wrote about electric current in an electrolyte solution with a concentration gradient, showing a non-equilibrium coupling between electric effects and concentration-driven diffusion, and found a reciprocal relation that was another of the ideas noted by Onsager. Korteweg in 1883 gave a proof that in any simply connected region, when the velocities along the boundaries are given, there exists, as far as the squares and products of the velocities may be neglected, only one solution of the equations for the steady motion of an incompressible viscous fluid, and that this solution is always stable; he attributed the first part to Helmholtz, and because of the restriction to cases in which the squares and products of the velocities can be neglected, these motions are below the threshold for turbulence.
Onsager made great theoretical progress in 1931 and again in 1953, and Casimir extended the theory in 1945. Gyarmati’s 1967/1970 systematic presentation extends Onsager’s principle of least dissipation of energy to a more symmetric form known as Gyarmati’s principle, and cites eleven papers or books authored or co-authored by Prigogine; in Section III 5 it gives a precis of the subtleties of Casimir’s work, explaining that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, while Casimir derived anti-symmetric relations concerning variables which are odd functions of those velocities. Hans Ziegler extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.
Ziman’s 1956 account commented that the maximum-entropy-production principle he proposed was a known general principle, discovered by Onsager, but was “not quoted in any of the books on the subject.” Ziegler, though largely unnoticed at the time, proposed an idea early in his work on the mechanics of plastics in 1961 and later in his book on thermomechanics revised in 1983 and in various papers; he never stated his principle as a universal law but may have intuited it, demonstrating it 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 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.”
Local Equilibrium and the Scope of the Principles
According to Glansdorff and Prigogine (1971, page 15), the various principles proposed by diverse authors over more than a century apply in general 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, meaning that collisions between molecules are so frequent that chemical and radiative processes do not disrupt the local Maxwell-Boltzmann distribution of molecular velocities. Grandy has pointed out 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, and the approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.
Reproducibility, Entropy, and the Language of Order
It is reproducibility in repeated observations that identifies dynamical structure in a system. E. T. Jaynes explains how this reproducibility is why entropy is so important in this topic: entropy is a measure of experimental reproducibility, telling 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 of molecules — much less than Avogadro’s or Loschmidt’s numbers — the thermodynamic reproducibility fades and fluctuations become easier to see. 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 writes that “Jaynes considered reproducibility — rather than disorder — to be the key idea behind the second law of thermodynamics (Jaynes 1963, 1965, 1988, 1989).” Grandy (2008), in section 4.3 on page 55, clarifies the distinction between the idea that entropy is related to order, which he considers an “unfortunate” “mischaracterization” that needs “debunking,” and the 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.
Non-linearity, Dissipative Structures, and Self-Organization
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. Prigogine in his 1977 Nobel Lecture 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) 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.” On page xv they wrote: “Dissipative structures have a quite different status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions,” referring to the dissipation function of Rayleigh that was used also by Onsager. Martyushev and Seleznev (2006) 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.”
Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961) 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; that steady state 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, and in the atmosphere under various conditions either seems possible, with some details discussed by Lebon, Jou, and Casas-Vásquez (2008) on pages 143–158. With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going. Convection introduces effects of momentum which appear as non-linearity in the dynamical equations, and in the more restricted case of no convective motion Prigogine wrote of “dissipative structures.”
Discontinuities of Fluid Motion
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) 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. Water from a nozzle can form a shower of droplets, and waves on the surface of the sea break discontinuously when they reach the shore. 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, since 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, and many other commonly observed discontinuities of fluid flow also lie beyond its scope, such as bubbles in boiling liquids and in effervescent drinks, and protected towers of deep tropical convection, also called penetrative convection.
Atmospheric and Planetary Applications
The physics of the earth’s atmosphere includes dramatic events like lightning and the effects of volcanic eruptions, with discontinuities of motion such as those noted by Helmholtz (1868), and turbulence is prominent in atmospheric convection; other discontinuities include the formation of raindrops, hailstones, and snowflakes, so that the usual theory of classical non-equilibrium thermodynamics will need some extension to cover atmospheric physics. According to Tuck (2008), “On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975, 2001).” Initially Paltridge (1975) used the terminology “minimum entropy exchange,” but afterwards, for example in Paltridge (1978) and Paltridge (1979), he used the now current terminology “maximum entropy production” to describe the same thing, a point clarified in the review by Ozawa, Ohmura, Lorenz, and Pujol (2003). Paltridge (1978) cited Busse’s (1967) fluid mechanical work concerning an extremum principle, and Nicolis and Nicolis (1980) discuss Paltridge’s work, commenting that the behaviour of the entropy production is far from simple and universal; this seems natural in the context of the requirement of some classical theory of non-equilibrium thermodynamics that the threshold of turbulence not be crossed. Paltridge himself nowadays tends to prefer to think in terms of the dissipation function rather than in terms of rate of entropy production, and the logic of his earlier work is open to serious criticism.
Sawada (1981), also in relation to the Earth’s atmospheric energy transport process, postulated a principle of largest amount of entropy increment per unit time and cited work in fluid mechanics by Malkus and Veronis (1958) as having “proven a principle of maximum heat current, which in turn is a maximum entropy production for a given boundary condition,” but this inference is not logically valid. Shutts (1981), investigating planetary atmospheric dynamics, 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 reported a good fit.
Further Critical Assessments
C. Nicolis (1999) 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, comments that this is to some extent disappointing, and points to the difficulty of finding a thermodynamically consistent form of entropy production. Grandy (2008) devotes Chapter 12 to an extensive and very cautious discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy, finding difficulty in defining the “rate of internal entropy production” in many cases, and finding 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 origination of the subject. According to Kondepudi (2008) and to Grandy (2008), there is no general rule that provides an extremum principle governing the evolution of a far-from-equilibrium system to a steady state. Lebon, Jou and Casas-Vásquez (2008) state that “In non-equilibrium … it is generally not possible to construct thermodynamic potentials depending on the whole set of variables.” 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. Jou, Casas-Vazquez, and Lebon (1993) note that classical non-equilibrium thermodynamics “has seen an extraordinary expansion since the second world war,” and refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine.
