Non-Maxwellian Kinetic Modelling of City Size Distribution

To extend the kinetic formulation of city size distribution introduced in [1], the non-Maxwellian kinetic modeling is introduced in the present study, in which a variable collision kernel is used in the underlying kinetic equation of Boltzmann type. By resorting to the well-known grazing asymptotic, a kinetic Fokker-Planck counterpart is obtained. The equilibrium of the Fokker-Planck equation belongs to the class of generalized Gamma distributions. Numerical test shows good fit of the generalized Gamma distribution with the city size distribution of China.


Introduction
The city size distribution has been long recognised to satisfy a very simple distribution law since Zipf, which is attributed to the generic least effort principle of human behavior [2]. Denote the number of cities having a population size between v and , with 1 ≈ γ and Fokker-Planck type equations for the size distribution of cities are obtained, by introducing interactions based on some migration rule among cities. This model demonstrated that the city size distribution is kinetically related to some factors such as the rate or the tendency of migration of the inhabitants. As noticed in [1] [14], the reasons behind migration are very complex, and it is quite difficult to select one or another reason as dominant. The different choices in the parameters of the kinetic interactions may explain the origin of different effects or even a mixture of effects, which give in the limit a distribution that can be closer to a Pareto or Zipf law, or a lognormal density or others. In any case, the kinetic modeling considered in this framework is useful to clarify the formation of various distributions in terms of various different microscopic interactions.
We further mention that kinetic models were originally used to describe the dynamics of rarefied gas by constructing a Boltzmann-type equation to analyse the effects of the discrete structure of gas molecules [15]. In recent years, various kinetic models have been developed to study the social and economic interactions in multi-agent systems, for example, in social sciences, the statistical description of wealth distribution [16] [17] [18], opinion formation [19] [20] [21], knowledge formation [22], belief formation [23], criminality [24] and so on. Note that a multi-agent system is often composed of "agents" rather than particles; a kinetic model is used to describe the collective behaviour of individuals in a multi-agent system.
At the kinetic level in [1] [14], the Boltzmann collision operator has been selected to be of Maxwellian type as in the classical kinetic theory, that is, the collision kernel is chosen as a constant that does not depend on the "relative velocity of the molecules". In the context of city size distribution, the Maxwellian hypothesis corresponds to the strong assumption that the migration rate between agents (cities) does not depend on the amount of inhabitants, thus a constant collision kernel is used. This is a simplification of the sophisticated problem, such that, it could be more easily handled from the mathematical point of view. To extend the kinetic formulation in [1] [14], in the present study, we introduce in the underlying kinetic equation of Boltzmann type a variable collision kernel as in [25].
The arrangement of the rest of the paper is as follows. We will show the detailed kinetic modeling of the problem in Section 2, and derive the quasi-invariant limit in Section 3. Finally, we will carry out some numerical tests to validate the model in Section 4. Note that for the quasi-invariant limit, we will show the Journal of Applied Mathematics and Physics

Kinetic Modeling of City Size Distribution
To study the evolution of the city size distribution by kinetic models [1], one first needs to specify the microscopic "collision" rules to describe the change of the population of a city. Consider a multi-agent system in which all agents (cities) are assumed to be indistinguishable [26]. A city's state at any instant of time 0 t ≥ is completely characterized by its number of inhabitants v. To avoid inessential difficulties, we can simply assume that v + ∈  although it is clear that v is a natural number. Consequently, the distribution of the multi-agent system, the city size distribution, can be fully characterized by an unknown probability density function Follow [1], we assume that the number of residents of a city will essentially increase with the inflow of immigrants and decrease with the outflow of emigrants.
At the same time, due to some uncontrollable factors, the population will change for some other uncertain reasons and show random fluctuations. Hence, the microscopic variation of the city size v is the result of three different contributes where  * , v v : the number of inhabitants of a city before and after a microscopic interaction process, respectively;  z + ∈  : the amount of population which can migrate towards a city from the environment (the multi-agent system). This value is usually sampled by a certain given distribution function ( ) z  , which characterizes the environment itself;  η : a random variable with zero mean and bounded variance, that is, the rate of variation of the city size v consequent to internal and external mechanism, respectively. More precise description of them will be prescribed in below.
Internal mechanism. For ( ) E v related to the internal mechanism, we use the concept of "value function" originally used in the study of the distribution of wealth by Kahneman and Twersky [27]: losses weigh heavier than gains in the change of the value function, that is, the value function is concave in the domain of gains and convex in the domain of losses, thus considerably steeper for losses than for gains. In the collision (2.1), the function ( ) in which the value v defines an ideal city size, ε is a small positive parameter introduced to represent the strength of the interaction, 0 1 < < λ and 0 1 < ≤ δ are used to quantify the intensity of migration rates near the ideal city size v .
For more explanation on the choice of the value function, we refer [28]. Here, in order to simplify the model, we first consider the ideal size of all cities in the whole system as a given value. However, for different countries, this value may depend on the history, political system or cultural background of different countries, or other factors. It is obvious that E is bounded, for some positive parameters µ and 0 1 < ≤ α characterizing the intensity of immigration rate.
With the interaction rule (2.1), the variation in time of ( ) , f v t satisfies a linear Boltzmann-like equation [26], which can be written in weak form, for all In (2.3), the notation ⋅ denotes mathematical expectation taking into account the presence of the random variable η in (2.1). And the function ( ) v χ denotes the collision kernel, which assigns to the interaction a certain probability to occur. Note that in [1], the simplification of the Maxwell molecules, leading to a constant interaction kernel χ , has been assumed. To extend, we notice that the distribution of city size in a country has a close relationship with the national conditions of the country. It is also closely related to many factors such as the

Quasi-Invariant Limit: The Fokker-Planck Equation
In order to describe the development of city size distribution more accurately and intuitively, we carry out the quasi-invariant limit. In this Section, we illustrate the main steps leading from Equation (2.5) to its Fokker-Planck limit. To avoid inessential difficulties, we will assume that the environmental distribution  has a certain number of bounded moments, more precisely.
Now, we can resort to a scaling of time to observe an evolution of the average value independent of ε . Setting , then the evolution of the average value for Since (3.5) means that the second term vanishes as 0 → ε , one obtains in the limit a closed form for the evolution of the mean value.
It can be observed that the evolution of the mean ( ) The above analysis can be used to justify the passage from the kinetic model Therefore, in terms of powers of ε , we easily obtain the expression where the remainder term , d . 2 2 Providing the boundary terms produced by the integration by parts vanish, Equation (3.12) coincides with the weak form of the Fokker-Planck equation Without loss of generality, we will simplify Equation (3.13) by assuming 1 : , : , : , : Thus, the resulting Fokker-Planck equation takes the form As exhaustively discussed in Ref. [29] [30], the right boundary conditions that guarantee mass conservation are the so-called no-flux boundary conditions given by where the positive constant C has been chosen to normalize the equilibrium distribution. It is not difficult to discover that

Numerical Tests
In this section, we will use statistical data to verify the validity of the model. Journal of Applied Mathematics and Physics From this probability distribution, we noticed that cities with a population of 1 million to 2 million are the majority. The number of cities with a population of more than 3 million decreases with the increase of the number of people contained, and the rate of decrease also changes from a sharp decline to a slow convergence to zero with the increase of the number of people. To fit the data with our model, we take a set of parameters

Conclusion and Perspectives
In this paper, we introduced non-Maxwellian kinetic modeling, in which a variable collision kernel is used in the underlying kinetic equation of Boltzmann type, to explain the evolution of city size in China. By resorting to the well-known quasi-invariant asymptotic, we obtain a kinetic Fokker-Planck counterpart and the steady-state of city size which is defined as the generalized Gamma distribution. Numerical test shows good fit of the generalized Gamma distribution with the city size distribution of China. However, further understanding of the role of each parameter, for example, the ideal city size v , is still open. It would also be interesting to investigate the trend of the city size distribution under the effect of fast urbanisation of China in recent and next several years.