Power laws of complex systems from extreme physical information.

Many complex systems obey allometric, or power, laws y=Y x(a) . Here y > or = 0 is the measured value of some system attribute a , Y> or =0 is a constant, and x is a stochastic variable. Remarkably, for many living systems the exponent a is limited to values n/4 , n=0, +/-1, +/-2.... Here x is the mass of a randomly selected creature in the population. These quarter-power laws hold for many attributes, such as pulse rate (n=-1) . Allometry has, in the past, been theoretically justified on a case-by-case basis. An ultimate goal is to find a common cause for allometry of all types and for both living and nonliving systems. The principle I-J=extremum of extreme physical information is found to provide such a cause. It describes the flow of Fisher information J-->I from an attribute value a on the cell level to its exterior observation y . Data y are formed via a system channel function y identical to f (x,a) , with f (x,a) to be found. Extremizing the difference I-J through variation of f (x,a) results in a general allometric law f (x,a) identical to y=Y x(a) . Darwinian evolution is presumed to cause a second extremization of I-J , now with respect to the choice of a . The solution is a=n/4 , n=0,+/-1,+/-2..., defining the particular powers of biological allometry. Under special circumstances, the model predicts that such biological systems are controlled by only two distinct intracellular information sources. These sources are conjectured to be cellular DNA and cellular transmembrane ion gradients.