Determinism in the case of bifurcations and symmetry breaking

In the discussion about free will

http://trv-science-ru.livejournal.com/111384.html

cordek has said that mathematical equations might not have an unique solution. I have asked at the everything list about bifurcations and symmetry breaking. My messages are below. I should say that there was no good answer.

http://groups.google.com/group/everything-list/brt/788c5771ca89a292

23.03.2012 20:08:

In physics there are bifurcations and symmetry breaking. What happens then if I solve some transient problem for a system where a bifurcation or symmetry breaking happens. How the choice will be made?

25.03.2012 11:43:

Let us take Benard cells for example. It is a good idea. I guess that in this case the incompressible Navier-Stokes equations with the Boussinesq approximation for free convection should suffer.

If I understand correctly, bifurcation in this case arises when we increase the temperature difference between two plates. That is, if we consider the stationary Navier-Stokes equations on the top of thermal gradient Del T in the system, there is a critical Del T after that we have several solutions.

To be back to my question. One could construct a system of equations from the stationary Navier-Stokes equations + Del T(time). In this case we have a problem that at some time when we reach a critical Del T, the system of equation has suddenly several solution and the question would be which one will be chosen.

On the other hand, one could use the transient Navier-Stokes equations directly and it seems that in this case the problem of bifurcation will not arise as such. Well, in this case there are numerical problems.

My question would be if physical laws allow for the first situation when at some point during transient solution a mathematical model has several solutions. If yes, then I do not understand how physics chose the one of possible solutions.

25.03.2012 12:02:

I have found Logistic Map

http://mathworld.wolfram.com/LogisticMap.html

Here the system has very different outcomes depending from initial conditions (now I understand your use of statistics). Yet, each trajectory is deterministic.

Hence, this was not my question. Sorry for being unclear. Bifurcations in Logistic Map is a result of uncertainty in initial conditions. I was thinking more in terms of Transient Equation of Everything. Does it allow for multiple solutions at some times or not?

In this case, Wolfram seems to support determinism:

http://www.stephenwolfram.com/publications/recent/ultimateknowledge/

“It looks probabilistic because there is a lot of complicated stuff going on that we’re not seeing–notably in the very structure and connectivity of space and time.”

“But really it’s all completely deterministic.”

«That somehow knowing the laws of the universe would tell us how humans would act–and give us a way to compute and predict human behavior.»

«Of course, to many people this always seemed implausible–because we feel that we have some form of free will.»

«And now, with computational irreducibility, we can see how this can still be consistent with deterministic underlying laws.»

27.03.2012 20:33:

“Nature computes itself by evolving in time”

Let me put then my question this way. When Nature computes itself does the next state is uniquely determined the previous steps? Or, to make the next state, does Nature play a dice?

I would appreciate if you explain how bifurcations and symmetry breaking could happen along the way that you have described.


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