Judith
Fully agree with the the diagnostic of inadequacy of reductionism when we face highly non linear, possibly non ergodic out of equilibrium systems.
I partly disagree with the statement of Barabasi that establishes a (fundamental ?) difference between non linear dynamics what many call now “chaos theory” and “network theory”.
Even if I already partly did it on some previous threads, I would like to illustrate the reductionism inadequacy for those of readers who are not necessarily familiar with the technical, mathematical apparatus of chaos.
- The Feigenbaum constant.
It has been proven that the transition to chaos by period doubling is governed by the Feigenbaum constant.
This constant is independent of the precise (reduced) mechanism of the process being considered – actually one can mathematically prove that it only depends on the presence of a quadratic extremum which may appear in many very different processes.
This is an example of “order” in “chaos” which couldn’t be found by any reductionist method.
So there is an infinity of very different and chaotic non linear systems but they will all transit to chaos with the same Feigenbaum constant if they do so by period doubling.
- The topology of attractors
The attractors are invariant subsets of the phase space (space of dynamical states) where the chaotic systems live. Their dimension is vastly inferior to the dimension of the phase space itself. Their topology e.g dimension, form and boundaries (fractal or not) depends on the coupling between the dynamical variables.
A reductionist approach typically neglects couplings (or considers that the system is linear) and is therefore fully unable to describe or even discover the attractors and their topology. In practice it means that a reductionist approach of a chaotic system would let the system live in states which are in reality forbidden because they are not on the attractor.
- Coupled map lattices
This paper finds a major and beautiful result for a specific case of spatio temporal chaos (http://amath.colorado.edu/faculty/juanga/Papers/PhysicaD.pdf).
Technically it could be called indiferently “network theory” or “chaos theory” what shows that the Barabasi’s distinction is largely artificial.
The paper considers a lattice of coupled oscillators and shows that the transition to coherent behaviour depends on the product of a parameter depending only on the uncoupled dynamics of each oscillator (this is what the reductionist approach would exclusively consider) AND a parameter depending on the network structure.
This kind of systems cannot be studied by neglecting the whole or considering that all of the “whole” is contained in the properties of its elements.
In other words the reductionism utterly fails on this system and can’t explain an apparition of synchronisation.
This paper needs a good level of mathematical training and I would like to point out for the too fast readers that the authors show that the ergodicity hypothesis results in an incoherent state of the system.
The same type of problem is studied in http://matisse.ucsd.edu/~hwa/pub/ks2d.pdf.
This rather technical paper deals with spatio-temporal chaos via interaction between small scales and large scales. It is precisely because of this interaction that turbulence is still not understood and it is arguably one of the top most difficult unsolved problems in physics.
It is this scale interaction/coupling which clearly shows that reductionism doesn’t work on this kind of complex chaotic systems.
Also, among others, Chief Hydrologist is right when he keeps repeating that this kind of complexity exhibits qualitatively very different behaviours for some critical values of couplings.
Some could compare that to phase changes what is a rather fitting analogy. Somebody on the thread mentionned “damping” stating that “damping” was some magics which prevents chaotic systems to be chaotic.
This is obviously very uninformed because “damping” (or energy dissipation) is precisely a necessary condition to have chaos.
So “damping” has nothing to do with the problem of predictability.
On a more anecdotical scale Webhub wrote :
I can actually believe that all that “chaos” map to random functions which leads to a straightforward transfer function.
There is a math research area called Random Matrix Theory, whereby large matrices with arbitrary elements reduce to eigenvalue solutions that have a distribution corresponding to semi-circle PDF. The issue is explaining why the solutions are predictable independent of the input. Terence Tau is currently working on this subject
First it is Terry Tao. Btw he wrote a marvelous paper about why Navier Stokes is so difficult. A must read.
Second believing that “chaos” equals “randomness” shows a, mostly subconscious, postulate of ergodicity. One cannot stress enough that ergodicity is not a given!
And if a system is not ergodic (there are enough examples which show that many are not) then there is no invariant pdf in the phase space.
It is not because we are used to popular examples of chaotic systems which are ergodic (f.ex Lorenz chaos, rigid spheres e.g statistical thermodynamics etc) that ALL chaotic systems are necessarily ergodic.