Book contents
- Frontmatter
- Contents
- 0 Introduction
- 1 What is Fisher information?
- 2 Fisher information in a vector world
- 3 Extreme physical information
- 4 Derivation of relativistic quantum mechanics
- 5 Classical electrodynamics
- 6 The Einstein field equation of general relativity
- 7 Classical statistical physics
- 8 Power spectral 1 / ƒ noise
- 9 Physical constants and the 1/x probability law
- 10 Constrained-likelihood quantum measurement theory
- 11 Research topics
- 12 EPI and entangled realities: the EPR–Bohm experiment
- 13 Econophysics, with Raymond J. Hawkins
- 14 Growth and transport processes
- 15 Cancer growth, with Robert A. Gatenby
- 16 Summing up
- Appendix A Solutions common to entropy and Fisher I-extremization
- Appendix B Cramer–Rao inequalities for vector data
- Appendix C Cramer–Rao inequality for an imaginary parameter
- Appendix D EPI derivations of Schrödinger wave equation, Newtonian mechanics, and classical virial theorem
- Appendix E Factorization of the Klein–Gordon information
- Appendix F Evaluation of certain integrals
- Appendix G Schrödinger wave equation as a non-relativistic limit
- Appendix H Non-uniqueness of potential A for finite boundaries
- Appendix I Four-dimensional normalization
- Appendix J Transfer matrix method
- Appendix K Numerov method
- References
- Index
14 - Growth and transport processes
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- 0 Introduction
- 1 What is Fisher information?
- 2 Fisher information in a vector world
- 3 Extreme physical information
- 4 Derivation of relativistic quantum mechanics
- 5 Classical electrodynamics
- 6 The Einstein field equation of general relativity
- 7 Classical statistical physics
- 8 Power spectral 1 / ƒ noise
- 9 Physical constants and the 1/x probability law
- 10 Constrained-likelihood quantum measurement theory
- 11 Research topics
- 12 EPI and entangled realities: the EPR–Bohm experiment
- 13 Econophysics, with Raymond J. Hawkins
- 14 Growth and transport processes
- 15 Cancer growth, with Robert A. Gatenby
- 16 Summing up
- Appendix A Solutions common to entropy and Fisher I-extremization
- Appendix B Cramer–Rao inequalities for vector data
- Appendix C Cramer–Rao inequality for an imaginary parameter
- Appendix D EPI derivations of Schrödinger wave equation, Newtonian mechanics, and classical virial theorem
- Appendix E Factorization of the Klein–Gordon information
- Appendix F Evaluation of certain integrals
- Appendix G Schrödinger wave equation as a non-relativistic limit
- Appendix H Non-uniqueness of potential A for finite boundaries
- Appendix I Four-dimensional normalization
- Appendix J Transfer matrix method
- Appendix K Numerov method
- References
- Index
Summary
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.
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- Science from Fisher InformationA Unification, pp. 356 - 391Publisher: Cambridge University PressPrint publication year: 2004