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Physical model of (infinite) tetration=(NON-isotropic) turbulence
#7
If there is a statistic process whose distribution w function logarithm is Poisson distribution:

with mean value .

Here

Is the Euler tree function. k denotes branches. is a probability of state k.

Then the process w istself probability distribution (unnormed) is :

with, mean value

If there is a physical (or mathematical process ) whose probality distribution (distribution function) is:

then its entropy S is defined as mean value of logarithm of logarithm of its distribution function:



For w_k as log Poisson process entropy is then:



If we assume that negative sign can (?) be moved inside entropy mean ,than



But and where is a branch of Lambert function.





But on the right we now have Poisson process 1 event with mean . Since entropy is also mean value <>, than



For example, if the

if then

if , then

if , then

Such interpretation of entropy allows negative, positive and complex values of entropy .

When is entropy of such process 0 ?

if then

Based on the above, every complex number w can be viewed as entropy of some log Poisson process and . Obviously, the fluctuations around mean value in such log Poisson process depend on fluctuations of z. Are there any inherent fluctuations of complex numbers?.

A process whose mean is I described Here

The result was:








if we took instead of x, mean was .

From above , then is entropy S of some log-Poisson process, while is entropy of negative log Poisson process such that

If some process has entropy then from above average value of such process is



So in this specific case (iteration of logarithm of absolute value of 1/x) , mean value of process and entropy are the same.

Would that mean that this iteration of logarithm of absolute value of (x) has log Poisson distribution? and in general iterations could be looked upon as statistical mathematical processes? Each try n would give a random value of iteration and the mean is then the same as limit of sum of iteration values as n->oo divided by n as it goes to infinity.

It would be interesting to find higher moments and also the structure functions which involve statistical deviations between 2 values of iteration as a function of "distance" n2-n1 between them.

Since log Poisson processes are involved in intermittency of turbulence, might be that Tree and Lambert functions are as well.

Ivars
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