Generalized generalized entropy

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Maximum entropy approach to central limit distributions of correlated variables 2008. Stefan Thurner and Rudolf Hanel SFI working papers/08-05-020

These are all strong Main result: positive feedback loop with 25-50 year lags, kappa, Silk road dependent:

Largest City --50--> qMLE --50y--> Internecine War --25y--> Largest City
(logged)     \-> Kappa   \                                          
                    \-----> Silk roads

In a joint test-project using the Chandler and other data for China, 900AD-1970, Thurner and Hanel are collaborating with Douglas R. White and Laurent Tambayong to examine the behavior of these data comparing Tsallis and their more generalized forms of entropy. These are just some rough notes on the thermodynamic variables, some characterizing the distribution, some the thermodynamics.

T temperature = Tsallis parameter kappa - kappaBef-->50 years time lag -->silk roads trade adj rsq .52 linear correlation (p<.001)
  PLUS q,largest city-->50 years time lag -->silk roads trade rsq +.25 linear correlation (p=.01)
  largest city-->25 years time lag-->q rsq .14 quadratic correlation (p=.06)
~ T&H 1/beta ~ Q?
U internal energy = mean city size (as computed from the fit parameters you gave us: kappa, Tsallis q)
T&H SQ ~ Generalized generalized entropy
T&H alpha and beta are Lagrange multipliers
? T&H zeta \zeta ~ alternate to alpha as normalization parameter? 
? T&H W ~ work - includes warfare ("signal")
? T&H theta, alpha  ~ used to normalize?
\epsilon ~ generalized exponential?
zeta = normalization for the distribution   p= 1/zeta (1-(1-q)/kappa*x))^(1/1-q)

(zeta is an alternative normalization, not relevant here. It becomes relevant when you have situations when certain scaling properties are fulfilled)

T&H note that you gave us the fit parameters for cumulative fits: we express kappa and q in terms of pdf!  s. 

A General Theory of Complex Living Systems: Exploring the Demand Side of Dynamics Graeme Donald Snooks] - Published 2008 in Complexity 13(6): 12-20.

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