Re: The least action principle as a consequence of Maxent
- From: "Juan R." <juanrgonzaleza@xxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 5 Oct 2007 10:57:25 +0000 (UTC)
On Sep 28, 12:49 pm, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:
Juan R. schrieb:
If you are interested, you would contact them [2] for details on
recent Brushles-Austin theory.
Brushles Austin has no useful hits in google. It is also not
mentioned in
Sorry by typo; it is Brussels-Austin.
The Brussels-Austin group is the group was leaded by Prigogine. We
informally call the theory the "Brussels-Austin theory". Still a
Google search will return results. The best way is to follow directly
the authors' literature [2].
Please give more details. What is it about?
The idea fundamental is that Poincare resonances split models into two
large classes: integrable and non-integrable. See [1, 2, 3] and
previous message for details on that follows.
Non-integrable systems need of an extension of dynamics and that
extension deal with phenomena like irreversibility, dissipation, and
so called non-deterministic chaos.
Examples of non-integrable class (called LPS [*] in the theory) are
thermodynamics systems, measurement apparatus in quantum mechanics,
instable particles (fields)...
Regarding collapse, this theory *dinamically* differentiates simple
quantum systems (following Schrödinger equation) from measurement
apparatus (Schrödinger does not apply). The difference finds in the
spectra.
Spectral decomposition for an usual quantum system (e.g. atom):
H = SUM_j |Phi_j> E_j <Phi_j|
Spectral decomposition for a measurement apparatus:
L = SUM_v SUM_alpha |F_alpha^(v)>> Z_alpha^(v) <<TILDE F_alpha^(v)|
Equation of motion for an usual quantum system (e.g. atom):
{PARTIAL |PHI> / PARTIAL t} = H |PHI>
Equation of motion for a measurement apparatus:
{PARTIAL |RHO_B>> / PARTIAL t} = THETA_B |RHO_B>>
whith
THETA _B = { LAMBDA_B L {LAMBDA_B}^-1 }
This dynamical distinction is a fundamental point you do not find in
other theories dealing with quantum measurement and collapse. For
example, decoherence approaches _a la_ Zurek, or the histories
formalism of Gell-Mann and Hartle are based in _ad hoc_ splitting into
quantum system and measurement apparatus. Therefore, the development
of realistic models may be guided by experience instead pure theory.
Penrose also tries to dinamically explain collapse. Penrose speculates
that quantum gravity effects in 'large' systems collapse wave
functions. He even offers an estimation of size and find a large
enough system would be of order of average molecules. This is right
since we know that average molecules are not completely quantum [**]
but in the border between quantum and classical worlds.
Prigogine and me agree in several fundamental points but both disagree
on the role of Poincare resonances. I do not think Poincare resonances
are fundamental but product in the large limit (LPS). I think a nice
research program would be so find the roots for LPS because that would
solve some weak points on the Brussels-Austin approach.
As stated in a previous message, it seems that a Brussels-Austin like
equation
{PARTIAL RHO_B / PARTIAL t} = THETA_B RHO_B
can be derived like a special case from a Keizer-like one
{PARTIAL RHO / PARTIAL t} = S + D + f
One can formally obtain the equation in the large limit (LPS), with a
Theta superoperator verifying (6.7) in [1]:
THETA _B = { LAMBDA_B L {LAMBDA_B}^-1 } --> L_0
Therefore there exists a clear link between both theories. But this is
still under active research.
A reference?
In [2] you can see list of recent published references including best-
seller books like _The End of Certainty, Time, Chaos and the New Laws
of Nature_ for general audiences.
In that book Prigogine presents the novel theory in a broad context
(cosmology, evolutionary biology, nonlinear chemistry, geometry and
general relativity) and discusses philosophical implications.
A concise self-contained presentation of the extended quantum
formulation including appendices to specific topics (wave collapse,
Friedrichs model, instable particles, quantum Lorentz gas, anharmonic
lattices...) and further references on detailed models is [1].
Extension of classical mechanics is presented on [3].
Arnold Neumaier
Due to flaming in sci.physics.relativity my email is disabled.
Therefore, I am not acessible now.
[*] Large Poincare Systems.
[**] Its electronic structure is, of course, purely quantum but the
nuclear framework is not, any quantum chemist know.
[1] Adv. Chem. Phys. 1997, 99, 1. Petrosky, Tomio; Prigogine, Ilya.
[2] http://order.ph.utexas.edu/Prigogine.htm
[3] Cha. Sol. & Fract. 1996, 7(4), 441-497. Petrosky, Tomio;
Prigogine, Ilya.
.
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