Competition and the canonical ensemble

A macroscopic approach to demography

A canonical/lognormal model for human demography is established, specifying the net maternity function and the age distribution for mothers of new-borns using a single macroscopic parameter vector of dimension five. The age distribution of mothers is canonical, while the net maternity function normalizes to a lognormal density. Comparison of an actual population with the model serves to identify anomalies in the population which may be indicative of phase transitions or influences from levels outside the demographic. Tracking the time development of the parameter vector may be used to predict the future state of a population, or to interpolate for data missing from the record. In accordance with classical theoretical considerations of Backman, Prigogine, et al., it emerges that the logarithm of a mother's age is the most fundamental time variable for demographic purposes.

Demography and the canonical ensemble

The Gibbs canonical ensemble of statistical mechanics is used to describe the probability distribution of the age classes of mothers of new-borns in an age-structured population. The Malthusian parameter emerges as a Lagrange multiplier corresponding to a generation time constraint, while a new perturbation parameter appears as the Lagrange multiplier corresponding to a maternity constraint. Classical Lotka stability reduces to the unperturbed case of the more general canonical ensemble model. The model is used in a case study of the female (peninsular) Malaysian population of 1970. The Malthusian parameter and perturbation are calculated easily by linear regression. Use of the model identifies an anomaly in the population due to the effects of World War II.