Again f(1) = 1.
Using Mathematica 4.2 to solve equation (8) for RC gives
RC = [-----------Q----/ʒ---∏∙ ]y/z,
LambertW [Q exp(z/my)]
(14)
clumped into the same ‘universality class’, having fixed expo-
nents at transition (e.g. Binney, 1986). Biological and social
phenomena may be far more complicated:
If we suppose the system of interest to be a mix of sub-
groups with different values of some significant renormaliza-
tion parameter m in the expression for f (R, m), according to
a distribution ρ(m), then we expect the first expression in
equation (1) to generalize as
where
Q ≡ (z/my)2-1/y[KCKC - K)]1/y.
The transcendental function LambertW(x) is defined by the
relation
H[KR,JR] =< f(R,m) > H[K,J]
≡ H[K, J] f(R, m)ρ(m)dm.
LambertW (x) exp(LambertW (x)) = x.
It arises in the theory of random networks and in renormal-
ization strategies for quantum field theories.
An asymptotic relation for f (R) would be of particular bi-
ological interest, implying that ‘language richness’ increases
to a limiting value with population growth. Such a pattern
is broadly consistent with calculations of the degree of allelic
heterozygosity as a function of population size under a bal-
ance between genetic drift and neutral mutation (Hartl and
Clark, 1997; Ridley, 1996). Taking
(17)
If f (R) = 1 + m log(R) then, given any distribution for m,
we simply obtain
< f(R) >= 1+ < m > log(R)
(18)
f (R) = exp[m(R - 1)/R]
(15)
gives a system which begins at 1 when R=1, and approaches
the asymptotic limit exp(m) as R → ∞. Mathematica 4.2
finds
R = my/z
C LambertW [A]
(16)
where
A≡ (my/z) exp(my/z)[21/y[KC/(KC - K)]-1/y]y/z.
These developments indicate the possibility of taking the
theory significantly beyond arguments by abduction from sim-
ple physical models, although the notorious difficulty of im-
plementing information theory existence arguments will un-
doubtedly persist.
3. Universality class distribution. Physical sys-
tems undergoing phase transition usually have relatively
pure renormalization properties, with quite different systems
where < m > is simply the mean of m over that distribu-
tion.
Other forms of f(R) having more complicated dependencies
on the distributed parameter or parameters, like the power
law Rδ , do not produce such a simple result. Taking ρ(δ) as
a normal distribution, for example, gives
< Rδ >= R<δ> exp[(1∕2)(log(Rσ))2],
(19)
where σ2 is the distribution variance. The renormalization
properties of this function can be determined from equation
(8), and is left to the reader as an exercise, best done in
Mathematica 4.2 or above.
Thus the information dynamic phase transition properties
of mixed systems will not in general be simply related to those
of a single subcomponent, a matter of possible empirical im-
portance: If sets of relevant parameters defining renormaliza-
tion universality classes are indeed distributed, experiments
observing pure phase changes may be very difficult. Tun-
ing among different possible renormalization strategies in re-
sponse to external pressures would result in even greater am-
biguity in recognizing and classifying information dynamic
phase transitions.
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