On 3/25/2012 2:46 PM, meekerdb wrote:
On 3/25/2012 2:43 AM, Evgenii Rudnyi wrote:
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.

And then one could use molecular dynamics directly - but this raises a different kind of numerical problem: how to put in the initial conditions for 1e28 molecules. But nature manages.

Brent

Dear Brent and Evgenii,

Nature computes itself by evolving in time. "The universe is not a program running somewhere else. It is a universal computer, and there is nothing outside of it." ~David Deutsch

A possible easy answer as to how Nature manages to "put in the initial conditions" is to consider that the actual evolution physical system of those 1e28 molecules, etc. is the actual computation of its behavior, as Stephen Wolfram has already pointed out. This makes sense once we cast aside the idea that computations are somehow objectively alienated from the physical world. When and if we consider that the evolution of each and every physical system is its own computation of itself and that computational universality is more or less just a mapping of the functions involved and not some crypto-substance dualism that completely separates the computations from the physical systems, then the difficulties of measures and so forth vanish. We no more need to invoke immaterial programs than we need to conjure immaterial spirits to explain these things and trying in vain to eliminate that which is so obviously real, our subjective consciousness, as at best an illusion, is equally a fools game. Dualism will work iff used correctly. In a sense, we might think of all of the functionally equivalent computations of the behavior of a system define transformations (endomorphism?) on a space whose fixed point is identified with the actual physical system itself. Dually we can say that all of the physical dynamics of a system define a logical algebra whose (Kleene)fixed point <http://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem> is the semantics of the computation. Abstract and concrete aspects "touch" in the actual objects themselves. IMHO, it is what Hegel and Marx tried to explain with their theories of alienation <http://answers.yahoo.com/question/index?qid=20110402044539AAlw9hI> that is the error. There is no actual dichotomy between mind and body or particular from Totality, there is only a problem of how to explain how bodies interact with bodies and how to minds interact with minds. We have most of the solutions to these problems already outlined before us in QM, GR and the work of Marchal, Turing, Barwise, Kleene, Wolfram, etc. What we actually struggle for is our individual understanding of these principles. When we are trying to built predictive models of physical phenomena to control aspects of them, we are not capable of creating simulations that are more faithful to the systems themselves and so have to use approximations and other devices to overcome this shortcoming.

Onward!

Stephen

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