If there is a statistic process whose distribution w function logarithm \( \ln w \) is Poisson distribution:
\( \ln w_k=T_k(z)*e^{-T_k(z)} \) with mean value \( <\ln w_k>= T_k(z) \).
Here \( T(z)=\sum_{n=1}^{\infty}{n^{n-1}*z^n/n! \)
Is the Euler tree function. k denotes branches. \( w_k \) is a probability of state k.
Then the process w istself probability distribution (unnormed) is :
\( w_k = e^{T_k(z)*e^{-T_k(z)}} \) with, mean value \( <w>=e^{T_k(z)} \)
If there is a physical (or mathematical process ) whose probality distribution (distribution function) is:
\( w_k \) then its entropy S is defined as mean value of logarithm of logarithm of its distribution function:
\( S= -<\ln w_k> = -<\ln (e^{T_k(z)*e^{-T_k(z)}})> \)
For w_k as log Poisson process entropy is then:
\( S= -<T_k(z)*e^{-T_k(z)}> \)
If we assume that negative sign can (?) be moved inside entropy mean ,than
\( S= <-T_k(z)*e^{-T_k(z)}> \)
But \( T_k(z) = -W_k(-z) \) and where \( W_k (-z) \) is a branch of Lambert function.
\( W_0(-z) = \sum_{n=1}^{\infty}{(-n)^{n-1}*(-z)^n/n! \)
\( S = -z_k=<W_k(-z) *e^{W_k(-z)}> \)
But on the right we now have Poisson process 1 event with mean \( W_k (-z) \) . Since entropy is also mean value <>, than
\( S = W_k(-z) \)
For example, if \( z=\pi/2 \) the \( S= W_k(-\pi/2) = I*\pi/2 \)
if \( z=-1 \) then \( S= W_k(1) = \Omega \)
if \( z=\ln2/2 \), then \( S=W_k(-\ln2/2) =\ln 2 \)
if \( z=1/e \) , then \( S=W_k(-1/e) = 1 \)
Such interpretation of entropy allows negative, positive and complex values of entropy \( S=W_k(-z) \).
When is entropy of such process 0 ?
if \( z=0 \) then \( S = W_k(0) = 0 \)
Based on the above, every complex number w can be viewed as entropy of some log Poisson process and \( w=S= W_k(-z) \). 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 \( -\Omega \) I described Here
The result was:
\( -\Omega \)
\( f(x) = \ln(x) \text{ if } x>0 \)
\( f(x) = \ln(-x) \text{ if }x<0 \)
\( \lim_{n\to\infty}\frac{\sum_{n=1}^\infty f^{\circ n}(x)}{n}= -\Omega=-0.567143..=\ln(\Omega) \)
if we took \( 1/x \) instead of x, mean was \( \Omega \).
From above , \( \Omega \) then is entropy S of some log-Poisson process, while \( -\Omega \) is entropy of negative log Poisson process such that \( S= - W(-z) = T(z) \)
If some process has entropy \( S=\Omega=-<\ln\Omega> \) then from above average value of such process is
\( <w> = 0,567143 = \Omega \)
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
\( \ln w_k=T_k(z)*e^{-T_k(z)} \) with mean value \( <\ln w_k>= T_k(z) \).
Here \( T(z)=\sum_{n=1}^{\infty}{n^{n-1}*z^n/n! \)
Is the Euler tree function. k denotes branches. \( w_k \) is a probability of state k.
Then the process w istself probability distribution (unnormed) is :
\( w_k = e^{T_k(z)*e^{-T_k(z)}} \) with, mean value \( <w>=e^{T_k(z)} \)
If there is a physical (or mathematical process ) whose probality distribution (distribution function) is:
\( w_k \) then its entropy S is defined as mean value of logarithm of logarithm of its distribution function:
\( S= -<\ln w_k> = -<\ln (e^{T_k(z)*e^{-T_k(z)}})> \)
For w_k as log Poisson process entropy is then:
\( S= -<T_k(z)*e^{-T_k(z)}> \)
If we assume that negative sign can (?) be moved inside entropy mean ,than
\( S= <-T_k(z)*e^{-T_k(z)}> \)
But \( T_k(z) = -W_k(-z) \) and where \( W_k (-z) \) is a branch of Lambert function.
\( W_0(-z) = \sum_{n=1}^{\infty}{(-n)^{n-1}*(-z)^n/n! \)
\( S = -z_k=<W_k(-z) *e^{W_k(-z)}> \)
But on the right we now have Poisson process 1 event with mean \( W_k (-z) \) . Since entropy is also mean value <>, than
\( S = W_k(-z) \)
For example, if \( z=\pi/2 \) the \( S= W_k(-\pi/2) = I*\pi/2 \)
if \( z=-1 \) then \( S= W_k(1) = \Omega \)
if \( z=\ln2/2 \), then \( S=W_k(-\ln2/2) =\ln 2 \)
if \( z=1/e \) , then \( S=W_k(-1/e) = 1 \)
Such interpretation of entropy allows negative, positive and complex values of entropy \( S=W_k(-z) \).
When is entropy of such process 0 ?
if \( z=0 \) then \( S = W_k(0) = 0 \)
Based on the above, every complex number w can be viewed as entropy of some log Poisson process and \( w=S= W_k(-z) \). 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 \( -\Omega \) I described Here
The result was:
\( -\Omega \)
\( f(x) = \ln(x) \text{ if } x>0 \)
\( f(x) = \ln(-x) \text{ if }x<0 \)
\( \lim_{n\to\infty}\frac{\sum_{n=1}^\infty f^{\circ n}(x)}{n}= -\Omega=-0.567143..=\ln(\Omega) \)
if we took \( 1/x \) instead of x, mean was \( \Omega \).
From above , \( \Omega \) then is entropy S of some log-Poisson process, while \( -\Omega \) is entropy of negative log Poisson process such that \( S= - W(-z) = T(z) \)
If some process has entropy \( S=\Omega=-<\ln\Omega> \) then from above average value of such process is
\( <w> = 0,567143 = \Omega \)
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