Science > Physics > Causation/Causality, Memory, and Convolution 16: Power Laws vs sqrt(1 - v^2/c^2)
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
25 Feb 2006 11:40:02 AM |
| Object: |
Causation/Causality, Memory, and Convolution 16: Power Laws vs sqrt(1 - v^2/c^2) |
From Osher Doctorow
There always was something rather curious about the beta or gamma (or
1/beta, 1/gamma) factor sqrt(1 - v^2/c^2) in Special Relativity (SR),
and recently I realized what it is. It's not far from "power laws",
which are central to psychological perception via the Fechner or
Weber-Fechner laws! The Lorentz-Fitzgerald contraction or Lorentz
transformations seem at first glance more "fundamental", but in fact
the "power law" is very, very deep in perception.
The keyword "power law" brings up 139 papers in Front for the
Mathematics ArXiv, from 1992 to 2006. One of the most important is
Michel Planat's (Institut FEMTO-ST, Department LPMO, France), "on the
cyclotomic quantum algebra of time perception," quant-ph/0403020 v1 2
Mar 2004. I originally accessed the paper under "perception" if I
recall correctly, but it cross-references under several very important
keywords and relates to memory encoding, phase transitions, and the
quantum analog of Fechner's law.
Even Fechner's law is sometimes confused with Shannon
entropy/information. It says that the magnitude of subjective
sensation is positively proportional to log(stimulus intensity):
1) subjective sensation = k log(stimulus intensity)
but remember that this also turns out to be:
2) (stimulus intensity)^k = exp(subjective sensation)
and that is arguably where the law comes from.
I have pointed out in this thread that the so-called finite speed of
light is arguably the maximum humanly visible or
light-sensitive-"mechanical" apparatus-sensitive speed, and these
equations indicate part of the reason for that. Even a linear or
quadratic power law approximation to sqrt(1 - v^2/c^2) is quite close
under many scenarios.
Osher Doctorow
.
|
|
| User: "OsherD" |
|
| Title: Re: Causation/Causality, Memory, and Convolution 16: Power Laws vs sqrt(1 - v^2/c^2) |
25 Feb 2006 11:59:05 AM |
|
|
From Osher Doctorow
Another very rich research literature involving power integrands is
fractional differential equations and fractional integral equations,
which not only generalize many ordinary integral expressions but often
have considerable importance related to topics of this thread and
several of my previous threads.
Power law potentials are arguably the most important potentials in
physics including the Schrodinger equation, and likewise Coulomb and
Coulomb/like expressions/laws and power law probability-statistics
distributions.
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Causation/Causality, Memory, and Convolution 16: Power Laws vs sqrt(1 - v^2/c^2) |
25 Feb 2006 12:51:20 PM |
|
|
From Osher Doctorow
In fact, power law types are arguably the main types of
probability-statistics distributions. Gamma distributions (which
include chi square and exponential distributions) are products of power
law factors in x and exp(+/-x) (including negative exponents), so are
uniform distributions and normal/Gaussian distributions (exp(-kx^2)),
etc.
Expressions of the form sqrt(1 - v^2/c^2), or sqrt(1 - x^2), or their
inverses, are quite rare in probability-statistics distributions, and I
don't recall any major use of them offhand. Only the Cauchy
distribution, which doesn't even have mean/expectation or variance or
any moments, is even remotely similar to my recollection with
probability density function fX(x) = 1/[pi(1 + x^2)]. The beta
distribution, which includes the standard uniform distribution, is a
power law distribution in x and in 1 - x as factors except that for the
standard uniform distribution the 1 - x factor disappears (is "zeroed
out"). The logistic distribution involves exp(x) in numerator and 1 +
exp(x) in denominator of its pdf. The Student's t distribution has a
power law in (1 + y^2/k) for k > 0 integer (k is the degrees of
freedom) with negative exponent greater than 1 in magnitude.
Osher Doctorow
.
|
|
|
|
|
|
Related Articles |
|
|
