Phase change

Following Feynman's (Reference Feynman2000) identification of information as a form of free energy, an ‘entropy’ can be defined across a vector of driving system parameters ${\bf J}$ as the Legendre transform of H(X,  Y,  Z,  L _D):

(1)$$S \equiv H({\bf J}) - {\bf J} \cdot \nabla _{\bf J} H.$$

Then, in first order, it is also possible to apply an analogue to the Onsager approximation of non-equilibrium thermodynamics in which system dynamics are determined by gradients of S in the components of ${\bf J}$ (de Groot & Mazur Reference de Groot and Mazur1984):

(2)$${\rm d}J_t^i \approx \left( {\sum\limits_k {\rm \mu} _{i,k} \partial S/\partial J_t^k} \right){\rm d}t + {\rm \sigma} _i J_t^i {\rm d}B_t. $$

This is a standard construction for which μ_i,k is a kind of diffusion matrix, in this approximation. The last term characterizes stochastic ‘volatility’: the σ_i are magnitude parameters, and dB _t represents a noise that may not be the usual Brownian white noise which is supposed to have uniform strength at all frequencies.

Setting the expectations of these equations to zero, it is assumed here, produces a relatively large set of nss, each indexed by some set j = 1,  2, …, j _max and each characterized by a joint source uncertainty having value H _j. Noise effects ensure that the nss are not inherently unstable.

Assuming metabolic free energy is available from the embedding environment at some index of intensity ρ, it is possible to define a pseudoprobability for state q as

(3)$$P_q = \displaystyle{{\exp ( - H_q /{\rm \kappa \rho} )} \over {\sum\nolimits_j {\exp ( - H_j /{\rm \kappa \rho} )}}}, $$

where κ is an index of loss due to thermodynamic Second Law effects. We can iteratively use that pseudoprobability to define a ‘higher free energy’ Morse Function F (Pettini Reference Pettini2007) in terms of the denominator sum,

(4)$$\exp ( - F/{\rm \kappa \rho} ) \equiv \sum\limits_j \exp ( - H_j /{\rm \kappa \rho} ).$$

Arguing by abduction from physical theory, changes in ρ will, as in the case of ordinary phase transitions at different temperatures, be associated with profound – and highly punctuated – evolutionary transitions (Eldredge & Gould Reference Eldredge, Gould and Schopf1972; Gould Reference Gould2002). As in the case of physical phase changes, higher values of the ‘temperature’ ρ will be associated with richer cognitive structures. These transitions, it is important to realize, are indexed by the symmetries of the underlying cognitive gene expression groupoids (Wallace Reference Wallace2012) and define entirely new pathways along which organisms develop. The approach represents extension of group symmetry breaking arguments from physical systems to groupoid symmetry breaking/making in biological systems (Landau & Lifshitz Reference Landau and Lifshitz2007; Pettini Reference Pettini2007). This is no small thing.

Ratchet dynamics

It is possible to formally describe an economic-like ratchet that, although constrained by evolutionary limitations, is usually characterized as a ‘self-referential’ dynamic. Goldenfeld & Wose (Reference Goldenfeld and Wose2011) describe the underlying mechanism as involving self-referential dynamics in which the update rules changes during the evolution of the system, a function of its state and history.

The evolutionary ‘economic-like’ ratchet we propose involves the interaction of metabolic energy cycles and the organisms that utilize them. The simplest way to explore such mechanisms is via an extension of equation (4).

First, assume that the right-hand side of equation (4) can be decently approximated as an integral. Second, suppose that the maximum possible value of H, say H _max, is much greater than κρ as a consequence of entropic loss. Third, suppose that ρ → ρ + Δ, Δ  ≪  ρ.

This leads to an expression for the free energy index F of the form

(5)$$\eqalign{& \exp \left[ { - \displaystyle{F \over {{\rm \kappa} ({\rm \rho} + \Delta )}}} \right] \approx \int_0^{H_{{\rm max}}} \exp [ - H/{\rm \kappa} ({\rm \rho} + \Delta )]{\rm d}H \cr & \quad \approx \int_0^\infty \exp [ - H/{\rm \kappa} ({\rm \rho} + \Delta )]{\rm d}H = {\rm \kappa} ({\rm \rho} + \Delta ).} $$

Defining another entropy in the free energy measure F as ${\rm {\cal S}} \equiv F(\Delta ) - \Delta {\rm d}F/{\rm d}\Delta $ allows use of an iterated stochastic Onsager approximation for the dynamics of Δ in the gradient ${\rm d}{\cal S}/{\rm d}\Delta $ (de Groot & Mazur Reference de Groot and Mazur1984). The resulting stochastic differential equation is

(6)$$\eqalign{& {\rm d}\Delta _t = \displaystyle{{{\rm \mu} \Delta _t {\rm \kappa}} \over {{\rm \rho} + \Delta _t}} {\rm d}t + {\rm \sigma} \Delta _t {\rm d}W_t \cr & \quad \approx \displaystyle{{{\rm \mu \kappa}} \over {\rm \rho}} \Delta _t {\rm d}t + {\rm \sigma} \Delta _t {\rm d}W_t,} $$

where μ is another ‘diffusion coefficient’, dW _t represents Brownian white noise, σ determines the magnitude of the volatility, and we use the condition that Δ  ≪  ρ.

Applying the Ito Chain Rule (Protter Reference Protter1990) to log [Δ] produces the SDE

(7)$${\rm d}\log [\Delta _t ] = \left( {\displaystyle{{{\rm \mu \kappa}} \over {\rm \rho}} - \displaystyle{1 \over 2}{\rm \sigma} ^2} \right){\rm d}t + {\rm \sigma d}W_t. $$

Invoking the Stochastic Stabilization Theorem (Mao Reference Mao2007; Appleby et al. Reference Appleby, Mao and Rodkina2008),

$$\mathop {\lim} \limits_{t \to \infty} \displaystyle{{\log [ \vert \Delta _t \vert ]} \over t} \to\, \lt 0$$

almost surely unless

(8)$$\eqalign{& \displaystyle{{{\rm \mu \kappa}} \over {\rm \rho}}\, \gt\, \displaystyle{1 \over 2}{\rm \sigma} ^2, \cr & {\rm \rho}\, \lt\, \displaystyle{{2{\rm \mu \kappa}} \over {{\rm \sigma} ^2}}.} $$

The essential point is that there will be an upper limit to ρ in this version of the ratchet. Above that ceiling, other things being equal, Δ_t  →  0.

This mechanism might constrain the maximum possible per cent of oxygen in Earth's atmosphere as a result of the aerobic transition.

Conversely, environmentally driven onset of a negative ratchet may explain patterns of local oxygen depletion in aquatic environments: increase in σ triggers a decline in ρ that in turn increases σ.

Another possible inference, however, might well be that increasing ‘volatility’ as a consequence of global warming can lower global oxygen concentration, perhaps triggering a downward ratchet to Mars-like conditions. The scenario would involve something like economically driven desertification of the tropics followed, through various mechanisms, by large-scale ocean eutrophication and massive plankton die-off.

Mathematical appendix: groupoids

Given a pairing, connection by a meaningful path to the same basepoint, it is possible to define ‘natural’ end-point maps α(g) = a _j,  β(g) = a _k from the set of morphisms G into A, and a formally associative product in the groupoid g ₁ g ₂ provided α(g ₁ g ₂) = α(g ₁),  β(g ₁ g ₂) = β(g ₂), and β(g ₁) = α(g ₂). Then the product is defined, and associative, i.e. (g ₁ g ₂)g ₃ = g ₁(g ₂ g ₃), with inverse defined by g = (a _j,  a _k), g ⁻¹ ≡ (a _k,  a _j).

In addition, there are natural left and right identity elements λ_g,  ρ_g such that λ_g g = g = gρ_g.

An orbit of the groupoid G over A is an equivalence class for the relation a _j ~ Ga _k if and only if there is a groupoid element g with α(g) = a _j and β(g) = a _k. A groupoid is called transitive if it has just one orbit. The transitive groupoids are the building blocks of groupoids in that there is a natural decomposition of the base space of a general groupoid into orbits. Over each orbit, there is a transitive groupoid, and the disjoint union of these transitive groupoids is the original groupoid. Conversely, the disjoint union of groupoids is itself a groupoid.

The isotropy group of a ∈ X consists of those g in G with α(g) = a = β(g). These groups prove fundamental to classifying groupoids.

If G is any groupoid over A, the map (α,  β) : G → A × A is a morphism from G to the pair groupoid of A. The image of (α,  β) is the orbit equivalence relation ~G, and the functional kernel is the union of the isotropy groups. if f : X → Y is a function, then the kernel of f, ker(f) = [(x ₁,  x ₂) ∈ X × X : f(x ₁) = f(x ₂)] defines an equivalence relation.

Groupoids may have additional structure. As Weinstein (Reference Weinstein1996) explains, a groupoid G is a topological groupoid over a base space X if G and X are topological spaces and α,  β and multiplication are continuous maps. A criticism sometimes applied to groupoid theory is that their classification up to isomorphism is nothing other than the classification of equivalence relations via the orbit equivalence relation and groups via the isotropy groups. The imposition of a compatible topological structure produces a non-trivial interaction between the two structures. Below we will introduce a metric structure on manifolds of related information sources, producing such interaction.

In essence, a groupoid is a category in which all morphisms have an inverse, here defined in terms of connection by a meaningful path of an information source dual to a cognitive process.

As Weinstein (Reference Weinstein1996) points out, the morphism (α,  β) suggests another way of looking at groupoids. A groupoid over A identifies not only which elements of A are equivalent to one another (isomorphic), but it also parameterizes the different ways (isomorphisms) in which two elements can be equivalent, i.e. all possible information sources dual to some cognitive process. Given the information theoretic characterization of cognition presented above, this produces a full modular cognitive network in a highly natural manner.

Brown (Reference Brown1987) describes the basic structure as follows:

EXAMPLE 1. A disjoint union [of groups] $G = \cup _{\rm \lambda}\,\, G_{\rm \lambda}, \,{\rm \lambda} \in \Lambda $, is a groupoid: the product ab is defined if and only if a, b belong to the same G _λ, and ab is then just the product in the group G _λ. There is an identity 1_λ for each λ ∈ Λ. The maps α,  β coincide and map G _λ to λ, λ ∈ Λ.

EXAMPLE 2. An equivalence relation R on [a set] X becomes a groupoid with α,  β : R → X the two projections, and product (x, y)(y, z) = (x, z) whenever (x, y), (y, z) ∈ R. There is an identity, namely (x, x), for each x ∈ X…

Weinstein (Reference Weinstein1996) makes the following fundamental point:

Almost every interesting equivalence relation on a space B arises in a natural way as the orbit equivalence relation of some groupoid G over B. Instead of dealing directly with the orbit space B/G as an object in the category S _map of sets and mappings, one should consider instead the groupoid G itself as an object in the category G _htp of groupoids and homotopy classes of morphisms.

It is, in fact, possible to explore homotopy in paths generated by information sources.