Boltzmann distribution in relation to the problem of population migration

Abstract

In this paper, we consider the problem of migration of the population of the Russian Federation among three groups of federal districts. The first group includes the Central and Northwestern Federal Districts, the second group includes the Far Eastern, Ural, and Siberian Federal Districts, and the third group includes the Southern, Volga, and North Caucasian Federal Districts. A model of population dynamics was proposed taking into account the Boltzmann distribution, which describes the distribution of the migrating population among three specified areas. A utility function included in the Boltzmann distribution was proposed, and a measure of the awareness (attractiveness) of the population in relation to the studied areas was also proposed, which is a part of the utility function. In addition, the measure of awareness takes into account the economic factor, that is the proportion of the population living above the subsistence level and the total population in the considered groups of federal districts. It is assumed that the larger the total population in a given territory, the more information about this territory is available. The parameters characterising the coefficients of natural growth and the proportion of the migrating population of the areas under consideration were identified for various values of the parameter characterising the optimal distribution of the population over the three specified areas. Identification of model parameters was implemented in Python 3.5 using the gradient descent method with step splitting. Also, data from official statistical sources was used to implement the identification of parameters. Based on the identification, the values of the Boltzmann distribution and migration flows among the three groups of federal districts were obtained, the model data was compared with real data, and a population forecast for each area was constructed. It was shown that the value of the parameter of optimality influences the population forecast of all the three groups of federal districts.