Re: [Fis] probability versus power laws

Folks,

   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 01:57:00 2004