Isotropy, entropy, and energy scaling
  Two principles explain emergence. First, in the Receipt's reference frame, Deg(S) = 4/3 Deg(R), where Supply S is an isotropic radiative energy source, Receipt R receives S's energy, and Deg is a system's degrees of freedom based on its mean path length. S's 1/3 more degrees of freedom relative to R enables R's 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. 
Robert Shour