Order in a multidimensional system

Abstract

We show that any convex K-dimensional system has a level of order R that is proportional to its level of Fisher information I. The proportionality constant is 1/8 the square of the longest chord connecting two surface points of the system. This result follows solely from the requirement that R decrease under small perturbations caused by a coarse graining of the system. The form for R is generally unitless, allowing the order for different phenomena, or different representations (e.g., using time vs frequency) of a given phenomenom, to be compared objectively. Order R is also invariant to uniform magnification of the system. The monotonic contraction properties of R and I define an arrow of time and imply that they are entropies, in addition to their usual status as informations. This also removes the need for data, and therefore an observer, in derivations of nonparticipatory phenomena that utilize I. Simple graphical examples of the new order measure show that it measures as well the level of "complexity" in the system. Finally, an application to cell growth during enforced distortion shows that a single hydrocarbon chain can be distorted into a membrane having equal order or complexity. Such membranes are prime constituents of living cells.