Acceleration of Universe As Acceleration of Probable Correlation 8: Phase Transitions

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

Theories of phase transitions often involve either changes in
correlations or differences between different types of correlations,
but a fascinating mystery so to speak is involved in this.

Recall that correlation coefficients used in mainstream mathematical
probability-statistics are themselves averaged (means, aggregated)
rather than localized at each point of space(-time) or other domain in
the pointwise type of localization.

Only Probable Influence/Causation (PI) uses a pointwise localized
Probable Correlation:

1) P(X<-->Y)(x,y) = P(X < = x, Y < = y) + P(X > x, Y > y)

for X, Y continuous random variables for simplicity (slight changes for
discrete random variables). But the first right hand side term is F(x,
y), the joint bivariate cumulative distribution function (cdf) of X and
Y at (x,y), and the second right hand side term is the joint bivariate
(engineering) reliability R(x, y). So (1) can be written:

2) P(X<-->Y)(x,y) = F(x,y) + R(x,y)

The word "reliability" comes from the fact that for various engineering
materials or objects or processes, the longer they survive the better
(for example, a battery). They are more "reliable" in an intuitive
sense.

A remarkable thing happens in phase transition theory using the
Mainstream (often linear) correlation, namely, it makes mistakes at
singularities. This is why the Ehrenfest classification scheme (see
"Phase transition" in Wikipedia) was mostly replaced by the new
non-mean-field-theory (non-averaged) classification of first-order and
second-order phase transitions. However, the researchers had no idea
what a pointwise Probable Correlation is (it doesn't arise in the
Mainstream of mathematical probability-statistics), so they changed
criteria physically.

In changing criteria physically from the Ehrenfest/Landau
classification scheme of lst and 2nd order phase transitions to the new
one which uses the presence or absence of latent heat for example as
the criterion, some useful insights of the Ehrenfest classification
were lost, in particular the first versus second derivative of free
energy discontinuity difference. It is true that the role of
thermodynamic fluctuations was neglected, but now there is some neglect
of first versus second derivative differences.

Although the acceleration of the universe via Probable Influence (PI)
does not require assumptions regarding phase transitions formally, it
is interesting that the mixed second order partial derivative Dxy which
is key to PI acceleration is of higher order in derivatives, somewhat
similar to the Ehrenfest/Landau classification criteria for second
order phase transitions. The latter are not mixed partials, but I
explained mathematically why mixed partials are needed here.

The Ginzburg-Landau theory which followed the Landau theory did
consider the thermodynamic fluctuations in the superconducting phase
via the coherence length which has inside its square root in the
denominator the phenomenological parameter alpha or rather the absolute
value of alpha where alpha is the coefficient of the "field" or complex
order parameter w or rather /w/^2 where w describes how deep the system
is into the superconducting phase and alpha/w/^2 is the second term of
the expansion of the free energy whose first term is the free energy in
the normal phase. Also a penetration depth lambda is involved, which
denominator is w_o or wo, the equilibrium value of the order parameter
without an EM field. Lambda describes how deeply an external magnetic
field can penetrate the superconductor. The Ginzburg-Laundau parameter
kappa is the ratio of lambda and the coherence length, and first order
vs second order phase transitions (respectively type I vs type II
superconductors) are discriminated by whether kappa < vs > sqrt(2)/2
respectively.

See hep-ph/0404288 for applications to Higgs and branes.

Osher Doctorow

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