Actually, it's the other way around. Nonequilibrium processes are responsible for self-organization. Systems in thermodynamic equilibrium, by contrast, have no way to change their internal organization.

Dear ADM Ider, I'm sorry but i general if a system is at equilibrium, it is not self-organized. Ilya Prigogine, among others, has pointed out the relationship between self-organization, dissipative structure and systems far from the equilibrium. Actually, I believe (but I may be wrong, :)) that you're not confusing equilibrium (a notion that have a strict definition in the thermodynamic framework) and steady state.

Thank you Olivier. So it is not legitimate to talk about self-organization when the system is out of equilibrium! I would like to know please if the coherent structures viewed in some classical flows such as Taylor-Couette flow or Rayleigh-Bénard convection have a permanent character or they can change their state with respect to time (transition). Thank you!

Actually, it's the other way around. Nonequilibrium processes are responsible for self-organization. Systems in thermodynamic equilibrium, by contrast, have no way to change their internal organization.

@ADM Ider,

D Grier is right,;). Like W Jackson, I think that the better way is to search for signature of self-organized systems.

I can recommend you the wiki page dedicated to self-organization:

http://en.wikipedia.org/wiki/Self-organization

http://csis.pace.edu/~marchese/CS396x/Computing/Ashby.pdf

Thank you for your suggestions and clarifications!

A beautiful example of self organization is us. Check hypercycles of Prof. Manfred Eigen - the evolution in the prebiotic structures assembling into RNA and proteins etc. It's really beautiful!

It was the original contribution of non-equilibroum thermodynamics (first by Onsager, and later on by Prigogine) to prove that systems in equilibrium do not know of self-organization. The most basic physical example of an equilibrium system are crystals. (Let's leave aside, for the time being the case of semi-crystals).

Working from biology, Kauffman's contribution was showing how self-organizations occurs for free ("order-for-free") and, hence, equilibrium is just an abstraction non-existing in nature (living systems).

A long part of research herewith has been in trying to explain what causes or how does self-organization emerge.

One final source here (for the sake of brevity) is Gould. Punctuated equilibria are the very source for the richness and diversity in nature - by and large more important than the stasis.

pp.20-21 of Sunagawa, I. (2005). Growth, morphology and perfection. Cambridge University Press, Cambridge.   I would be interested to know, both because of your response in general and your accurate, important note that certain organization within some systems which we might refer to as SOC processes are better (or at least more often) referred to as self-assembly rather than self-organization what you think of the utility of the concept of SOC itself. One the one hand, as you note, we find organization (even self-organization) at equilibrium states, while complex systems and self-organization is generally in part defined by being off or far from equilibrium. On the other, we typically don't find any identifiable "points" (in the sense of an actual well-defined or known point in the phase space of some system) for phenomena supposedly exhibiting SOC. Instead, those some systems  which might be said to exhibit SOC tend to be 1) generally stable or semistable apart from phase transitions occuring within intervals or arbitrarily small departures from some attractor which result not in chaotic behavior but in reconfiguration or 2) systems which are constantly in flux and generally far from thermodynamic equilibrium, and thus there seems little sense in defing them by critical points in their phase space. I would be interested to hear you thoughts. Thanks!   Andrew

Dear Dr. De Decker:

Could you give specific examples or organization processes that occur "at equilibrium"? I ask for two reasons. First, the example you give of solids seems like it may refer to our inability to provide formal rather than qualitative answerts to questions such as why groups of e.g., gold or copper atoms form the structures they do rather than form into fluid or semi-fluid arrays. While a fascinating question, the problems posed by many-body problems in QM and QFT aren't seem to me to be rather qualitatively different than those posed by complex systems in general. One is complicated because the level of analysis required in condensed matter physics or to examine excitation modes in quantum physics reaches and often surpasses the so-called classical limit. The other is complicated far beyond/above the microscale.

Second, while certain SAMs can have critical states or reach criticality "at equilibrium", including crystalline structures, once we move away from the classical limit it is off-equilibrium and away from equilibrium states (even if small perterbations) that enable the organization of monolayers: "At the equilibrium when no external stiimulus is applied, there is no possibility flux flow among molecues" & "It is obvious from [the equation for "the deviation of order parameter from equilibrium state"] that the deviation from equilibrium state comes from a step external compression simulation and that the thermal diffusion effect which is expressed as (Ai-A)/τi tends to bring molecules back to a new equilibrium state Seq|A≡A"

p.98 & 104 of Iwamoto, M., & Chen-Xu, W. (2001). The Physical Properties of Organic Monolayers. World Scientific.

I'm not sure how this differs from the way that the paradigm model system for self-organized criticality (sandpiles, which granular physics and empirical demonstration has shown is not actually characterized by SOC) and actual systems governed by SOC frequently retain (even complex) structures as final configuration states that are the result of SOC and are at equilibrium.

Personally, I'm not sure that the SOC model has much merit. It seems to me to belong to the period of "chaos theory" when catastrophe theory was in vogue. It is not sufficiently well-defined as distinct from self-organization yet empirically useful (again, the prototypical model of SOC turned out not to have the properties the founders of SOC stated), Phenomena said to exhibit SOC seem to me to almost never have properties unique enough to systems displaying SOC yet different enough from other properties to warrant the nomenclature, they often are not actually defined by the impetus for the concept in the first place as they tend to lack any known critical point/threshhold rather than some interval, and finally surveys of phenomena supposedly exhibiting SOC that are similar in how they are characterized by SOC are usually so in ways for which their exist better terms.

However, even relatively "simple" self-organized/complex systems tend to be off-equilibrium while all biological systems and many complex systems are far from thermodynamic equilibrium. Furthermore, the kind of orginazational structures we find at the molecular and even atomic level tend towards non-equilibrium with or without self-organization/assembly: "Atomic vibrations in a crystal, that is, displacements from equilibrium positions, have a frequency of the order of 1013 per second...Atoms in other unit cells will also exhibit this random deviation from their equilibrium positions, different for each such atom in the various unit cells throughout the crystal."

p. 82 of Glusker, J.P. & Trueblood, K.N. (2010) Crystal Structure Analysis: A Primer (3rd Ed.) (IUCr Texts on Crystallography Vol. 14). Oxford University Press

""It is only when a driving force causes a system to depart from its equilibrium condition that a nucleus of a crystal is formed and growth begins...In dealing with systems of metals, alloys, ceramics, and silicates, as well as growing single crystals, phase diagrams are the prerequisite in understanding the basic phase relations...In contrast to equilibrium thermodynamics, the main concerns of this book are to show how the phase transitions involved in growth and dissolution proceed at an atomic level, and to explain how the morphology, perfection, and homogeneity of single crystals and the textures of polycrystalline aggregates are determined through these processes."

Dear Abdallah, It is simply impossible for any self-organized system . whther natural or artificial, human or in nature, to be in an equlibroum state.

Firstly, because any system in equilibrium is an isolated system. In other words, it is a "controlled" experiment. And secondly, because, truly or ultimately there are no closed or isolated systems in nature. 

Dear Abdallah

Would you clarify what you mean by equilibrium in your question? It seems that the answers have focused on a static equilibrium, e.g thermodynamic equilibrium. But there is also dynamic equilibrium as exhibited in stable limit cycles,e.g temperature in your fridge or in some predator-prey populations (hare and lynx being the most well known). Organisms although they are far from thermodynamic equilibrium, are considered by some to be in a dynamic equilibrium maintaining their position by taking in energy etc and using it. So to avoid confusion it would be wise to explain what you mean by equilibrium in your question.

From an economic standpoint, equilibrium can be achieved through economics or a catallaxy. Economics denotes an "organizing system driven by a common goal" while catallaxy denotes "an emergent system arrived at despite diverse interests of its components" In the case of economics, it can be said that order is dictated and not spontaneous. In the case of catallaxy, "it is legitimate to consider that a given system at equilibrium is self-organized". An analogous description involves the notion of "basins of attraction" where equilibrium is not a point but a set of possible stability points, despite differing starting points (i.e. the marble dropped into a bowl metaphor)

@David G Grier · 60.25 · 852.41 · New York University

Actually, it's the other way around. Nonequilibrium processes are responsible for self-organization. Systems in thermodynamic equilibrium, by contrast, have no way to change their internal organization.

To provide an example, street networks or streets are self-organized in nature, although they are built by human beings:

Jiang B., Zhao S., and Yin J. (2008), Self-organized natural roads for predicting traffic flow: a sensitivity study, Journal of Statistical Mechanics: Theory and Experiment, July, P07008, Preprint: http://arxiv.org/abs/0804.1630

In the same vein, cities are self-organized as well:

Jiang B. (2015a), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77.