(6)
where the sum/integral is over all possible elements of the
largest available symmetry groupoid. By the arguments of
Ash above, compound sources, formed by the union of un-
derlying transitive groupoids, being more complex, generally
having richer alphabets, as it were, will all have higher free-
energy-density-equivalents than those of the base (transitive)
groupoids.
Let
ZD ≡ exp[-HDj K].
j
(7)
We now define the Groupoid free energy of the system, FD ,
at inverse normalized metabolic energy density K , as
Fd [K] ≡--1log[ZD [K]],
K
(8)
again following the standard arguments from statistical
physics (again, Landau and Lifshitz, 2007, or Feynman, 2000).
The groupoid free energy construct permits introduction of
important ideas from statistical physics.
5.3 Spontaneous symmetry breaking
We have expressed the probability of an information source
in terms of its relation to a fixed, scaled, available (inverse)
metabolic free energy density, seen as a kind of equivalent (in-
verse) system temperature. This gives a statistical thermo-
dynamic path leading to definition of a ‘higher’ free energy
construct - FD [K] - to which we now apply Landau’s fun-
damental heuristic phase transition argument (Landau and
Lifshitz 2007; Skierski et al. 1989; Pettini 2007). See, in
particular, Pettini (2007) for details.
The essence of Landau’s insight was that second order phase
transitions were usually in the context of a significant sym-
metry change in the physical states of a system, with one
phase being far more symmetric than the other. A sym-
metry is lost in the transition, a phenomenon called spon-
taneous symmetry breaking, and symmetry changes are in-
herently punctuated. The greatest possible set of symmetries
in a physical system is that of the Hamiltonian describing
its energy states. Usually states accessible at lower temper-
atures will lack the symmetries available at higher tempera-
tures, so that the lower temperature phase is less symmetric:
The randomization of higher temperatures - in this case lim-
ited by available metabolic free energy densities - ensures that
higher symmetry/energy states - mixed transitive groupoid
structures - will then be accessible to the system. Absent
high metabolic free energy rates and densities, however, only
the simplest transitive groupoid structures can be manifest.
A full treatment from this perspective requires invocation of
groupoid representations, no small matter (e.g., Buneci, 2003;
Bos 2006).
Somewhat more rigorously, the biological renormalization
schemes of the Appendix to Wallace and Wallace (2008) may
now be imposed on FD [K] itself, leading to a spectrum of
highly punctuated transitions in the overall system of devel-
opmental information sources.
Most deeply, however, an extended version of Pettini’s
(2007) Morse-Theory-based topological hypothesis can now
be invoked, i.e., that changes in underlying groupoid struc-
ture are a necessary (but not sufficient) consequence of phase
changes in FD [K]. Necessity, but not sufficiency, is important,
as it, in theory, allows mixed groupoid symmetries.
The essential insight is that the single simply connected
giant component of Ciliberti et al. is unlikely to be the full
story, and that more complete models will likely be plagued
- or graced - by highly punctuated dynamics.
6 Tunable epigenetic catalysis
Incorporating the influence of embedding contexts - epige-
netic effects - is most elegantly done by invoking the Joint
Asymptotic Equipartition Theorem (JAEPT) and the ex-
tensions of Network Information Theory in equations (6-8)
(Cover and Thomas, 1991). For example, given an embed-
ding contextual information source, say Z , that affects de-
velopment, then the dual cognitive source uncertainty HDi
is replaced by a joint uncertainty H (XDi , Z). The objects
of interest then become the jointly typical dual sequences
yn = (xn, zn), where x is associated with cognitive gene ex-
pression and z with the embedding context. Restricting con-
sideration of x and z to those sequences that are in fact jointly
typical allows use of the information transmitted from Z to
X as the splitting criterion.
One important inference is that, from the information the-
ory ‘chain rule’ (Cover and Thomas, 1991),
H(X,Y) =H(X)+H(Y|X) ≤H(X)+H(Y),
while there are approximately exp[nH (X)] typical X se-
quences, and exp[nH (Z)] typical Z sequences, and hence
exp[n(H (x) + H(Y ))] independent joint sequences, there are
only about exp[nH (X, Z)] ≤ exp[n(H (X) + H(Y ))] jointly
typical sequences, so that the effect of the embedding con-
text, in this model, is to lower the relative free energy of a
particular developmental channel.