Has Statistical Mechanics Produced Any Results on Self-Organization?

[nonequilibrium]

So I'm not asking if thermodynamics has, but specifically statistical mechanics, because it seems so unlikely that by probabilistic and mechanical reasoning, the phenomenon of self-organization would arise -- and if it has happened, I'd love to find out. I'm not looking for specific papers, as I probably won't understand them (2nd year undergrad), but references are always welcome. So to be clear: I'm asking about the creation of self-organization, not "merely" the description of it.

Thank you.

 

[Andy Resnick]

There is quite a bit of material discussing phase transitions in terms of the order parameter. This has allowed meaningful analysis of the transition from a disordered (short coherence length) state to an ordered (long range coherence) state.

The order parameter was used to develop the renormalization group, which makes calculations of the phase transition quantitative.

 

[schip666!]

There's a bit of work inspired by statmech and information theory that allows for analysis and characterization of seemingly "random" systems, e.g., chaotic dynamics. One aspect of "chaos theory" is self-organization. Here is a founding paper for what one of the authors calls "Computational Mechanics":
http://tuvalu.santafe.edu/~cmg/papers/ISC.pdf
Look for later papers which reference this one for more details.

Self-organization is a bit of a slippery thing. As mentioned by the previous poster, phase transitions are one area that can be examined. This initially lead folks to look at avalanches in sand or rice piles, which then lead to a big brouhaha over "power-law" relationships -- which indicate that there is some kind of non-random organization in a system. Per Bak over-stated his case, but at least made a start with this:
http://en.wikipedia.org/wiki/Bak–Tang–Wiesenfeld_sandpile

Cellular Automata can be used as small models of statmech systems as well, so maybe have a look at that literature. The Wolfram "New Kind of Science" book is a good, if exhaustive, review, as long as one takes the claims of discovery with a few grains of something.

 

[Llewlyn]

because it seems so unlikely that by probabilistic and mechanical reasoning, the phenomenon of self-organization would arise -- and if it has happened, I'd love to find out.

You don't need a statmech description because it often suffices the macroscopical trend. Just to say an example, go to the homepage of Michael Cross and you'll see a demonstration of the Swift-Hohenberg equation, a PDE you may think as an averaged statmech model.

http://www.cmp.caltech.edu/~mcc/Patterns/SwiftHohenberg.html

This is a system which exhibits self-organization. Using the applet you may watch the entire dynamic starting from an not-organized state. Does it satisfy you? Or were you looking for general theoretical argument about self-organization?

So to be clear: I'm asking about the creation of self-organization, not "merely" the description of it.

What do you mean? Are you not interested in equilibrium stat mech models?
The most familiar example of self-organization to physicists is a ferromagnet at equilibrium. The 2D Ising model is the simplest model which captures its phase transition -- you likely encountered during your statmech course. Using the math you have shown that it undergoes a phase transition. It may seems a "merely description" but you can appreciate the entire dynamic simulating the system with any monte carlo technique.

A last example worth to point out is the Langton's Ant.
http://en.wikipedia.org/wiki/Langton's_ant
This is not a physical system but rather an abstract one described by just a couple of (stochastic) rules. It is nice because it shows complex emergent behaviour despite its simplicity.

I have to say I haven't understood the title "Has Statistical Mechanics Produced Any Results on Self-Organization?". Self-organization and statistical mechanics aren't necessarily correlated. The former is a property exhibited by many dynamical system, whether stochastic or not, the latter the study of systems made by many components. By your post you seemed more impressed by self-organization itself rather than statmech but perhaps I have misunderstood you. If you were looking for the role played by having many components in a system which self-organize you should look any book about renormalization group.

Ll.

 

[nonequilibrium]

Thank you all three for your time.

Some of the terminology (order parameter, coherence length, renormalization group, Ising model [heard of it a bit, but haven't studied it]) went a bit over my head (when I said I was a 2nd year undergrad, I meant I'll be that this year which starts this september), and I looked it up a bit on wikipedia, but my original question was actually a bit more general, which I could probably have been more explicit on (--the computer simulations were very interesting though). So what I was basically wondering about is the possibility of statistical mechanics explaining the phenomenon of self-organization. First of all: can a system of idealized molecules governed by classical dynamics exort macroscopic self-organization? Are there accepted views on this? And if it's not possible, is it generally believed that quantum mechanics is essential and sufficient to derive/predict such phenomena on the microscale?

So Llewlyn, for example: is it possible to derive such PDEs from the microscopic laws as we know them? (and then mainly "in principle")

 

[Llewlyn]

can a system of idealized molecules governed by classical dynamics exort macroscopic self-organization? Are there accepted views on this?

What do you mean by self-organization? Is the gas at equilibrium?

Put aside self-organization for a while and let try to change the question.
Think about a classical gas, it's indeed a thermodynamic system. Since thermodynamic holds, it has an irreversible dynamic which eventually relaxes to an equilibrium state. But a gas is nothing more than a set of interacting particles, so it obeys also at F=ma rule. From this it stems that the whole thermodynamic is nothing but an "emergent behavior" obtainable from only mechanical reasoning. So, how does thermodynamic stem from mechanic?

That problem seemed *impossible* because mechanical systems have reversible dynamic which tends to return close to its initial state (see Poincaré recurrence theorem). In fact, the first historical answer was neglecting the existence of atoms and so splitting the thermodynamic systems from the mechanical ones.

Nowadays it is accepted by scientists that a system which obeys the laws of mechanics exhibits thermodynamic behavior, and statistical mechanics (kinetic theory) has taught us how it is possible. Unfortunately the approach is not free of fundamental problems. In order to apply statistical hypothesis one must require some properties on the dynamic (ergodicity, phase mixing) which are hard to sketch mathematically. There are results (Sinai theorem, KAM theorem, FPU simulation) which suggest that statistical mechanics is well applied on physical systems, but a complete proof is still missing.

And if it's not possible, is it generally believed that quantum mechanics is essential and sufficient to derive/predict such phenomena on the microscale?

I think no one (here) would question that we are made by nothing more than quantum particles. So the problem "Who will win next political election?" is somehow a quantum physics problem. But despite the fact that politicians are nothing more than a set of interacting quantum particles, Schroedinger's equation is hardly a tool to get insight of the problem.

So Llewlyn, for example: is it possible to derive such PDEs from the microscopic laws as we know them? (and then mainly "in principle")

"Yes".
There are stochastic models that starting from a microscopic view, derive a PDE to describe the averaged macroscopic behavior. A famous one is Einstein's treatment of Brownian's motion, in which the diffusion equation arises. However, the "microscopic laws" used are often far from the really fundamental physics laws (hence they are called "mesoscopic ones").

Ll.

 

[nonequilibrium]

Hello Llewlyn, thanks for your time.

"Put aside self-organization for a while and let try to change the question." But the main part of my question is self-organization :P Let me explain (whilst reacting on what you said).

Indeed, I'm aware of the explanation Statistical Mechanics offers on the phenomenological arrow of time and that's exactly why I'm very impressed by Statistical Mechanics and I'd love to work on some of its problems in the future. But on the other hand, self-organization is a whole other thing to be explained. The arrow of time is indeed an understandable thing to follow out of the symmetric fundamental laws once you've thought about it enough, but a tendency to form complex structures whilst striving to maximum entropy does not seem understandable (at the first sight...). If I have a theoretical gas with everything in the corner at t_0, I'll expect it to get uniform over time, but I won't expect the formation of complex regions of self-organization as it spreads out (okay, so with self-organization I mean things like the convection currents that suddenly start at a certain temperature difference etc) no matter what the force field. I'm not trying to claim it's impossible: I'd very much enjoy the sight of it being explained out of the microscopic world just like the arrow of time, but I'm asking if it is possible with the fundamental laws we know today. Okay first of all I haven't had any QM yet (although I know a lot of historical information and principles of it by reading books), so if I wanted to limit it to classical mechanics: the arrow of time can be explained with classical mechanics microstates, and now I was wondering if self-organization can also theoretically arise in worlds governed completely by classical mechanics. This seems doubtful(?) If it can't: would it arise out of a theoretical world construed only with all the fundamental laws we know to date (thus incl. QM)? Or does self-organization -if you want to explain it from the microscopic world- require an adaptation/addition concerning the fundamental laws?

There are stochastic models that starting from a microscopic view, derive a PDE to describe the averaged macroscopic behavior. A famous one is Einstein's treatment of Brownian's motion, in which the diffusion equation arises. However, the "microscopic laws" used are often far from the really fundamental physics laws (hence they are called "mesoscopic ones").

I'm confused: how can the microscopic laws be different from the fundamental physical laws? Wouldn't this put a "yes" on my above question of possibly needing new/changed fundamental laws for self-organization? I interpret your sentence as: we're using experimentally verified PDE's on the microscopic level which we can't trace back to the fundamental laws known to date.

 

[Llewlyn]

Self-organization does not require any adaptation/addition of the fundamental laws. It's surprising that only few features in the dynamic are needed to self-organize a system, but there's nothing against thermodynamics (or any microscopical laws). Remember that we are speaking about out-of-the-equilibrium system.

I'm confused: how can the microscopic laws be different from the fundamental physical laws?

Usually statmech models are "toy abstract models" built with simple rules "close" to fundamental laws of physics. Take Ising for example; it's just an hamiltonian with an interaction terms close (but not equal) to the one of interacting magnetic dipoles.

Ll.