Introduction

The population components of certain systems evolve according to what are called “growth processes” in biology and “transport processes” in physics. These systems are generally “open” to exterior effects such as, e.g., uniform or random energy inputs. Thus they do not necessarily follow the closed-system model of Sec. 1.2.1. However, as noted in Sec. 1.2.3, the Cramer–Rao inequality and the definition (1.9) of Fisher information hold for open systems as well. It will result that EPI can be applied to these problems as well, provided that the defining form (1.9) of Fisher information is used.

These growth processes are described by probability laws that obey first-order differential equations in the time. Examples are the Boltzmann transport equation in statistical mechanics, the rate equations describing the populations of atomic energy levels in the gas of a laser cavity, the Lotka–Volterra equations of ecology, the equation of genetic change in genetics, the equations of molecular growth in chemistry, the equations of RNA cell replication in biological cell growth, and the master equation of macroeconomics. These equations of growth describe diverse phenomena, and yet share a similar form. This leads one to suspect that they can be derived from a common viewpoint. The connection is provided by information (not energy; Sec. 1.1), and they are all derived by a single use of EPI.

Definitions

Consider a generally open system containing N kinds of “particles,” at population levels mn, n = 1, …, N.