Re: [Fis] probability versus power laws

Replying to Koichiro's interesting posting -- yes. I would formulate it
thus: distinguishing local from global, all locales have histories, and,
even very near global equilibrium, would still have some energy gradient
(otherwise we would not distinguish it as a locale, and, internally, it
would not BE one). This gradient will be dissipated as rapidly as
possible, but, because of historicity (a)its steepness would in principle
not be that found in an immature situation, and so, (b) it will be
dissipated in fluctuations at very different rates, with bigger and bigger
jumps being rarer and rarer, AND with biggest jumps becoming rarer with
time. The probabilities would be Cauchy, (or, if we have an observer
looking at a small definite region, lognormal). Globally, equilibrium will
be being attained asymptotically as these locales disappear, with the
probability of fluctuations anywhere approaching a serene Gaussian,
acquiring a definite variance.
     It seems to me further, on the distinction between quantum
probabilities and the historical distribution of contingent events in the
material world, on one interpretation, the more such events there are, as
in an immature situation driven by powerful energy gradients, the
probabilities of decoherences in some small region would be entrained by
the Cauchy, while in a senescent situation, the probabilities of
decoherences as a result of entanglement with material events would move in
the direction of Gaussian.
     On this score it is interesting to note that in biology (and elswhere
in science) one always uses normal (parametric) statistics if possible.
This, in the present context, is clearly based on the ideal of equilibrium,
yet biological systems are (even in senescence)nonequilibrated. Where
have Pearson, Fisher and Neyman taken us?

STAN

>
> Let me jump in the remarks made by Stan from the sideline. Stan wrote:
>
>>In my current(ly evolving) view, power law fluctuations result when
>>different fluctuations are partly reinforced to different degrees by energy
>>gradient dissipations,
>
> Zipf's law on the long-time tail is just another way of saying that the
>time series we consult most often is not ergodic, but historical. If we dare
>to figure out a statistical ensemble of data by cutting a single time series
>of whatever sort in whatever way we want, the power-law behaviors could be
>salvaged if the time series is historical. The power spectrum would diverge
>at the low frequency limit. That's fine. Zipf's law is something reduced
>from something else that is historical, and not the other way around.
>
> Ubiquity of time series that is historical now raises a serious question
>on how we could come up with probability distributions if ever possible.
>According to Max Born's interpretation of quantum mechanics, one may hope to
>obtain the probability density from the square of the absolute value of the
>wavefunction amplitude. This interpretation should certainly be valid in the
>Hilbert space, but not in the ordinary space. On the other hand, if we are
>also interested in thermodynamics, the space we have in mind is the ordinary
>phase space. The distinction between an isolated, closed and open systems
>presumes the underlying ordinary phase space, instead of the Hilbert's.
>Eugene Wigner pointed out about 70 years ago that the probability densities
>that could naturally be connected to Born's in the Hilbert space would not
>necessarily be positive-definite in the ordinary phase space. The malaise
>surrounding us is that if we respect both thermodynamics and quantum
>mechanics on a par, we would lose the basis of what has been called
>probability distributions. Unless the notion of probability distribution is
>available, it would be next to impossible to talk about entropy and
>information in a decent manner.
>
> At this juncture, the second law of thermodynamics as an empirical
>principle would seem to enter without relying upon the notion of
>probability. Fortunately, we can enjoy what is called energy without knowing
>what the probability is. Dissipating energy gradients at the possible
>fastest rate is an expression of the second law without recourse to
>probability distributions. Of course, if we are further fortunate enough to
>have the probability distribution as Boltzmann observed with his H-theorem,
>the second law could certainly be expressed in terms of the probabilistic
>language.
>
> Cheers,
> Koichiro

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Received on Tue Jul 13 22:08:00 2004