Non-Maxwellian Dynamic Model Of City Size Distribution

Since the last century,the problem of city's size distribution has attracted the attention of many researchers,who studied the law of its distribution through data statistics information and hoped to study the motivation of its formation through modeling method.This paper proposes to define the rules of urban population change from the perspectives of population flow between cities and immigration,and the Non-Maxwellian Boltzmann dynamic model was combined with the size distribution of city to explore the basic mechanism of city agglomeration distribution behaviour.Under reasonable scale transformation,a Fokker-Planck equation with variable diffusion and drift coefficients is obtained from the Boltzmann equation.We can use it to describe the evolution of city size over time.The results show that the stable state of city size belongs to the class of generalized Gamma distributions,which is proved to be reasonable by using the recent city population data of China.In addition,for the large-time behaviour of the solution to the Fokker-Planck equation,by means of the classical entropy method,it is proved that the solution of the Fokker-Planck equation for the initial state satisfies certain conditions shows exponential convergence towards the equilibrium on L1(R+).