Entropy and its relationship to allometry

Researchers have found that the metabolisms of organisms appear to scale proportionally to a 3/4 power of their mass. Mathematics in this article suggests that the capacity of an isotropically radiating energy supply scales up by a 4/3 power as its size (and therefore the degrees of freedom of its circulatory system) increases. Accordingly, cellular metabolism scales inversely by a 3/4 power, likely to prevent the 4/3 scaling up of the energy supply overheating the cell. The same 4/3 power scaling may explain cosmological dark energy.

Isotropy, entropy, and energy scaling

Two principles explain emergence. First, in the Receipts reference frame, Deg(S) = 4/3 Deg(R), where Supply S is an isotropic radiative energy source, Receipt R receives Ss energy, and Deg is a systems degrees of freedom based on its mean path length. Ss 1/3 more degrees of freedom relative to R enables Rs growth and increasing complexity. Second, rho(R) = Deg(R) times rho(r), where rho(R) represents the collective rate of R and rho(r) represents the rate of an individual in R: as Deg(R) increases due to the first principle, the multiplier effect of networking in R increases. A universe like ours with isotropic energy distribution, in which both principles are operative, is therefore predisposed to exhibit emergence, and, for reasons shown, a ubiquitous role for the natural logarithm.

Lexical growth, entropy and the benefits of networking

If each node of an idealized network has an equal capacity to efficiently exchange benefits, then the networks capacity to use energy is scaled by the average amount of energy required to connect any two of its nodes. The scaling factor equals textit{e}, and the networks entropy is $ln(n)$. Networking emerges in consequence of nodes minimizing the ratio of their energy use to the benefits obtained for such use, and their connectability. Networking leads to nested hierarchical clustering, which multiplies a networks capacity to use its energy to benefit its nodes. Network entropy multiplies a nodes capacity. For a real network in which the nodes have the capacity to exchange benefits, network entropy may be estimated as $C log_L(n)$, where the base of the log is the path length $L$, and $C$ is the clustering coefficient. Since $n$, $L$ and $C$ can be calculated for real networks, network entropy for real networks can be calculated and can reveal aspects of emergence and also of economic, biological, conceptual and other networks, such as the relationship between rates of lexical growth and divergence, and the economic benefit of adding customers to a commercial communications network. textit{Entropy dating} can help estimate the age of network processes, such as the growth of hierarchical society and of language.

A theory of intelligence: networked problem solving in animal societies

A societys single emergent, increasing intelligence arises partly from the thermodynamic advantages of networking the innate intelligence of different individuals, and partly from the accumulation of solved problems. Economic growth is proportional to the square of the network entropy of a societys population times the network entropy of the number of the societys solved problems.