• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Physical model of (infinite) tetration=(NON-isotropic) turbulence Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 03/08/2009, 10:35 PM (This post was last modified: 03/11/2009, 07:59 PM by Ivars.) 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 $=e^{T_k(z)}$ 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: $S= -$ 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. $W_0(-z) = \sum_{n=1}^{\infty}{(-n)^{n-1}*(-z)^n/n!$ $S = -z_k=$ 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 $ = 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 « Next Oldest | Next Newest »

 Messages In This Thread Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/13/2009, 07:27 AM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/18/2009, 07:29 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/20/2009, 10:31 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/25/2009, 10:27 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by bo198214 - 02/26/2009, 12:01 AM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/26/2009, 06:53 PM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 03/17/2009, 08:51 AM RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 03/17/2009, 08:52 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Improved infinite composition method tommy1729 2 24 06/14/2021, 04:17 AM Last Post: JmsNxn [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 2,564 03/20/2018, 12:16 AM Last Post: tommy1729 [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 18,817 12/28/2014, 10:22 PM Last Post: MphLee Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 8,256 05/06/2014, 09:47 PM Last Post: tommy1729 Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 10,959 07/17/2013, 09:46 AM Last Post: Gottfried Wonderful new form of infinite series; easy solve tetration JmsNxn 1 6,180 09/06/2012, 02:01 AM Last Post: JmsNxn the infinite operator, is there any research into this? JmsNxn 2 7,918 07/15/2011, 02:23 AM Last Post: JmsNxn Infinite tetration of the imaginary unit GFR 40 82,012 06/26/2011, 08:06 AM Last Post: bo198214 Infinite Pentation (and x-srt-x) andydude 20 38,627 05/31/2011, 10:29 PM Last Post: bo198214 Infinite tetration fractal pictures bo198214 15 33,435 07/02/2010, 07:22 AM Last Post: bo198214

Users browsing this thread: 1 Guest(s)