## Dissipative structures

*28 Jan 1997 13:34*

Ilya Prigogine (NL) coined the phrase, as a name
for the patterns which self-organize in far-from-equilibrium
dissipative systems. *He* thinks they're unbelievably important, and
says so at great length in his books. Some of us physicists believe him; some
are skeptical; I am leaning towards skepticism.

But to explain. Dissipation inspires the wrath of the moralist and the envy of most others; for the physicist, however, it is merely faintly depressing. We call something dissipative if it looses energy to waste-heat. (Technically: if volume in the phase space is not conserved.) The famous Second Law of Thermodynamics amounts to saying that, if something is isolated from the rest of the world, it will dissipate all the free energy it has. Equivalently, it maximizes its entropy. Thermal equilibrium is the state of maximum entropy.

If something is (in a well-defined sense) near thermal equilibrium, one can show that its behavior is governed by linear differential equations (hence the name "linear thermodynamics" for the appropriate body of theory), and that left to itself it will approach equilibrium exponentially (hence the somewhat more common name "irreversible thermodynamics"). Here we are guided, not by the entropy, but by "entropy production," the rate of increase in entropy. Since, once we reach equilibrium, the entropy cannot increase (by definition), the entropy production at equilibrium is zero, and the entropy production is always decreasing (the "principle of minimum entropy production").

In general, however, things are not well-isolated from the rest of the world. If energy arrives from the outside as quickly as it is dissipated, even bodies in the linear regime can be kept away from equilibrium. (Hence various Creationist arguments about the Second Law are worthless: neither living things nor the Earth are well-isolated from the rest of the universe, as may be observed every day at sunrise.) Thus dissipation, and why dissipative systems are not necessarily dull as dish-water. So you can have structures in dissipative systems, and there's no reason not to call them "dissipative structures", though it's not obvious that there are many interesting generalizations about them.

"Far-from-equilibrium" means that your system is so far from its thermal equilibrium that the linear laws I mentioned a moment ago no longer apply; non-linear terms become important. The only general rule about the solution to non-linear differential equations is that there are no general rules; hence the interest in the subject. (Cf. Chaos and non-linear dynamics.) This is not good news, of course, if what you want to do is extend thermodynamics to the far-from-equilibrium case. But, one might suppose, matters are not totally hopeless; we aren't talking about just any arbitrary system of equations, but the particular ones important in thermodynamics; perhaps there is some general principle (like those of maximum entropy, or minimum entropy production) which can guide us to solutions. What Prigogine claims to have done is to have found, if not another extremum principle, then at least an inequality (a "universal evolution criterion"), and to have used it to work out the theory of dissipative structures, according to which patterns are supposed to form when the uniform, uninteresting "thermodynamic branch" of the system becomes unstable. The math for all this is analogous to that of equilibrium phase transitions with "broken symmetry", where, again, a uniform state becomes unstable, forcing the system into a patterned, coherent one to minimize free energy. Even without Prigogine's claims that this theory is Very Significant to biology and social science, even without the philosophical and cultural importance he claims for it, this would be very interesting, and the big question is whether he's right, i.e., whether and to what the theory applies, whether, so to speak, there are Dissipative Structures and not just dissipative structures.

"Of course he's right," one is tempted to say. "Everyone acknolwedges he's an expert on thermodynamics; he was part of the Brussels School which basically invented irreversible thermodynamics; he won the Nobel Prize, for crying out loud!" But irreversible thermodynamics is very different, and that was a long time ago --- the forties and fifties and early sixties; that was what the Nobel was for. ("And besides the wench is dead.")

And then there is the matter of his scientific peers --- not the systems theorists and similar riff-raff, but the experts in thermodynamics and statistical mechanics and pattern formation. One of them (P. Hohenberg, co-author of the latest Review of Modern Physics book on the state of the art on pattern formation) was willing to be quoted by Scientific American (May 1995, "From Complexity to Perplexity") to the effect that "I don't know of a single phenomenon his theory has explained."

This is extreme, but it becomes more plausible the more one looks into the
actual experimental literature. For instance, chemical oscillations and waves
are supposed to be particularly good Dissipative Structures; Prigogine and his
collaborators have devoted hundreds if not thousands of pages to their
analysis, with a special devotion to the Belousov-Zhabotisnky
reagent, which is *the* classic chemical oscillator. Unfortunately,
as Arthur Winfree points out (When Time Breaks Down, Princeton UP,
1987, pp. 189--90), "the Belousov-Zhabotinsky reagent ... is perfectly stable
in its uniform quiescence," but can be distrubed into oscillation and
wave-formation. This is precisely what *cannot* be true, if the theory
of Dissipative Structures is to apply, and Winfree accordingly judges that
"the first step [in understanding these phenomena], which no theorist would
have anticipated, is to set aside the mathematical literature" produced by a
"ponderous industry of theoretical elaboration". --- Needless to say,
Winfree is not opposed to theory or mathematics, and his superb The
Geometry of Biological Time (Springer-Verlag, 1980) is full of both.

Somewhat more diplomatic is Philip W. Anderson, one of the Old Turks of the Santa Fe Institute, and himself a Nobelist. I refer in particular to the very interesting paper he co-authored with Daniel L. Stein, "Broken Symmetry, Emergent Properties, Disspiative Structures, Life: Are They Related", in F. Eugene Yates (ed.), Self-Organizing Systems: The Emergence of Order (NY: Plenum Press, 1987), p. 445--457. The editor's abstract is as follows:

The authors compare symmetry-breaking in thermodynamic equilibrium systems (leading to phase change) and in systems far from equilibrium (leading to dissipative structures). They conclude thgat the only similarity between the two is their ability to lead to the emergent property of spatial variation from a homogeneous background. There is a well-developed theory for the equilbirium case involving the order parameter concept, which leads to a strong correlation of the order parameter over macroscopic distances in the broken symmetry phase (as exists, for example, in a ferromagnetic domain). This correlation endows the structure with a self-scaled stability, rigidity, autonomy or permanence. In contrast, the authors assert that there is no developed thoery of dissipative structures (despite claims to the contrary) and that perhaps there are no stable dissipative structures at all! Symmetry-breaking effects such as vortices and convection cells in fluids --- effects that result from dynamic instability bifurcations --- are considered to be unstable and transitory, rather than stable dissipative structures.Some quotes from the paper itself:Thus, the authors do not believe that speculation about dissipative structures and their broken symmetries can, at present, be relevant to questions of the origin and persistence of life.

"Is there a theory ofdissipative structurescomparable to that of equilibrium structures, explaining the existence of new, stable properties and entities in such systems?"Contrary to statements in a number of books and articles in this field, we believe that there is

no such theory,and it even may be that there are no such structures as they are implied to exist by Prigogine, Haken, and their collaborators. What does exist in this field is rather different from Prigogine's speculations and is the subject of intense experimental and theoretical investigation at this time.... [p. 447]Prigogine and his school have made a series of attempts to build an analogy between these [dissipative far-from-equilibrium systems which form patterns] and the Landau free energy and its dependence on the order parameter, which leads to the important properties of equilibrium broken symmetry systems. The attempt is to generalize the principle of maximum entropy production, which holds near equilibrium in steady-state dissipative systems, and to find some kind of dissipation function whose extremum determines the state. As far as we can see, in the few cases in which this idea can be given concrete meaning, it is simply incorrect. In any case, it is clearly out of context in relation to the observed chaotic behvaior of real dissipative systems. [pp.454--455]

Anderson and Stein cite two of their own papers (P. W. Anderson, "Can
broken symmetry occur in driven systems?" in G. Nicolis, G. Dewel and
P. Turner (eds.), Order and Fluctuations in Equilbirium and
Non-Equilibirum Statistical Mechanics, pp. 289-297; and D. L. Stein,
"Dissipative structures, broken symmetry, and the theory of equilibrium phase
transitions," J. Chem. Phys. **72**:2869-2874) for
the technical details of their critique; I haven't read 'em yet. Their joint
paper is reproduced in Anderson's Basic Notions of Condensed Matter
Physics, sans illustrations. Prigogine may be observed waxing
philosophical in Order Out of Chaos, obscure in From Being
to Becoming, and textbookish in Self-Organization in
Non-Equilibrium Systems.