^{1}

^{1}

^{*}

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.

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 [

R ( v ) = ∫ v ∞ h ( w ) d w .

Zipf found that empirically, R ( v ) ∼ 1 / v γ , with γ ≈ 1 , especially when focusing on large cities, that is, when v is big. As well-known in city size distribution literatures, this empirical law is quite close to reality for most societies across time [

Moreover, although very accurate in recovering the city size distribution, these contributions do not, in general, give the mechanism for the formation in time of this behavior, thus a theoretical derivation of Zipf’s law for cities has been the object of many studies, see for example [

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 [

At the kinetic level in [

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 asymptotic procedure leading from the kinetic description of Boltzmann type to the Fokker-Planck equation. The equilibrium of the Fokker-Planck equation belongs to the class of generalized Gamma distributions. The model test is based on a collection of 332 cities (prefecture-level administrative regions) in China in the 2019 statistical Yearbook, which fits well with the generalized Gamma distribution.

To study the evolution of the city size distribution by kinetic models [

Follow [

v * = v − E ( v ) v + I E ( v ) z + η v , (2.1)

where

• v , v * : the number of inhabitants of a city before and after a microscopic interaction process, respectively;

• z ∈ ℝ + : 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 E ( z ) , which characterizes the environment itself;

• η : a random variable with zero mean and bounded variance, that is, 〈 η 〉 = 0 , 〈 η 2 〉 = σ with σ > 0 suitably small (such that v * to be positive);

• E ( v ) , I E ( v ) : 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 [

E ε ( v ) = λ e ε ( s δ − 1 ) / δ − 1 e ε ( s δ − 1 ) / δ + 1 , s = v v ¯ , (2.2)

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 < λ < 1 and 0 < δ ≤ 1 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 [

External mechanism. A non-negative function I E ( v ) can be used to describe a measure of the immigration rate. For simplicity, following [

I E ( v ) = μ v α 1 + v α

for some positive parameters μ and 0 < α ≤ 1 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 [

d d t ∫ ℝ + φ ( v ) f ( v , t ) d v = 〈 ∫ ℝ + 2 χ ( v ) ( φ ( v * ) − φ ( v ) ) f ( v , t ) E ( z ) d v d z 〉 , (2.3)

In (2.3), the notation 〈 ⋅ 〉 denotes mathematical expectation taking into account the presence of the random variable η in (2.1). And the function χ ( v ) denotes the collision kernel, which assigns to the interaction a certain probability to occur. Note that in [

According to the National Bureau of Statistics (NBS) of China, the size of population of China’s cities has been constantly expanding over the past 70 years, with large, small and medium-sized cities distributed across the country. Among them, small cities attract people because of the low threshold of Hukou. Due to people’s pursuit for better quality of life, many people are willing to live in big cities such as Beijing or Shanghai which can provide people with more employment opportunities, better economic income and higher education, etc. However, big cities are already too crowded, in recent years, with China’s urbanisation, medium-sized cities are also attractive for their new opportunities and living conditions. The migration between different cities is much often than ever. There is strong evidence that the population mobility is greater in cities with a large population, such as Beijing, Shanghai, compared with smaller cities such as Yibin, Xining etc. On the other side, due to geographic and historic reasons, the number of cities (prefecture-level administrative regions) is relatively fixed, so the interactions for v small or near zero should be excluded. Thus, the collision frequency may proportional to the city size v. Hence, to elaborate this behavior, it seems natural to consider a variable collision kernel that

χ ( v ) = κ ⋅ v α , (2.4)

where the constants κ > 0 and 0 < α < 1 . This kernel assigns a high probability of interactions for cities with large population, and low probability of interactions for cities with population v close to zero. By taking into account this new assumption, we consider in the following that f ( v , t ) satisfies the linear kinetic model

d d t ∫ ℝ + φ ( v ) f ( v , t ) d v = κ 〈 ∫ ℝ + 2 v α ( φ ( v * ) − φ ( v ) ) f ( v , t ) E ( z ) d v d z 〉 . (2.5)

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 E has a certain number of bounded moments, more precisely.

M β : = ∫ ℝ + z β E ( z ) d z ≤ ∞ , 0 ≤ β ≤ 4, (3.1)

meanwhile, we introduce the notation

m a ( t ) : = ∫ ℝ + v a f ( v , t ) d v , a > 0. (3.2)

It’s obvious that the kinetic equation is mass preserving by taking φ ( v ) = 1 in (2.5).

For the quasi-invariant limit, one assumes that a single interaction determines only an extremely small change of the value v. Therefore, a small parameter ε is introduced and we consider the scaling

I E → ε I E , η → ε η . (3.3)

At this point, under the effect of ε , the interaction will only produce a very small change to the population size of a city. Obviously, the conservation of “mass” of the system still holds under the scaling. To observe the evolution of the mean value, in (2.5), take φ ( v ) = v , there is

d m 1 ( t ) d t = κ [ − ∫ ℝ + v α + 1 E ε ( v ) f ( v , t ) d v + ε I E M 1 m α ( t ) ] . (3.4)

Next, denote A ε ( v ) : = 1 ε E ε ( v ) , then

A ε ( v ) = λ ε e ε ( ( v / v ¯ ) δ − 1 ) / δ − 1 e ε ( ( v / v ¯ ) δ − 1 ) / δ + 1 → λ 2 δ ( ( v v ¯ ) δ − 1 ) as ε → 0. (3.5)

Now, we can resort to a scaling of time to observe an evolution of the average value independent of ε . Setting τ = ε t , f ε ( v , τ ) = f ( v , t ) , then the evolution of the average value for f ε ( v , τ ) satisfies

d d τ ∫ ℝ + v f ε ( v , τ ) d v = κ ∫ ℝ + ( I E M 1 − λ v 2 δ ( ( v v ¯ ) δ − 1 ) ) v α f ε ( v , τ ) d v + κ ∫ ℝ + ( λ 2 δ ( ( v v ¯ ) δ − 1 ) − 1 ε E ε ( v ) ) v α + 1 f ε ( v , τ ) d v . (3.6)

Since (3.5) means that the second term vanishes as ε → 0 , one obtains in the limit a closed form for the evolution of the mean value.

d d τ ∫ ℝ + v f ε ( v , τ ) d v = κ ∫ ℝ + ( I E M 1 − λ v 2 δ ( ( v v ¯ ) δ − 1 ) ) v α f ε ( v , τ ) d v . (3.7)

It can be observed that the evolution of the mean ∫ ℝ + v f ε ( v , τ ) d v does not depend on ε . Since for ε ≪ 1 the microscopic interactions produce a very small change of the value v, a finite variation of the mean density can be observed only if agents in the system undergo a huge number of interactions in a fixed period of time to restore the original evolution. Similarly, with this scaling, one obtains in the limit a closed form for the evolution of the second moment

d d τ ∫ ℝ + v 2 f ε ( v , τ ) d v = κ ∫ ℝ + ( σ v 2 + 2 I E M 1 v − λ v 2 δ ( ( v v ¯ ) δ − 1 ) ) v α f ε ( v , τ ) d v (3.8)

The above analysis can be used to justify the passage from the kinetic model (2.5) to its continuous counterpart given by a Fokker-Planck type equation. Given a smooth function φ ( v ) , let us expand in Taylor series φ ( v * ) around φ ( v ) . First, by the scaling (3.3), it holds

〈 v * − v 〉 = − ε A ε ( v ) v + ε I E z , (3.9)

〈 ( v * − v ) 2 〉 = ε σ v 2 + ε 2 ( A ε 2 ( v ) v 2 + I E 2 z 2 − 2 I E A ε ( v ) v z ) . (3.10)

Therefore, in terms of powers of ε , we easily obtain the expression

〈 φ ( v * ) − φ ( v ) 〉 = ε [ φ ′ ( v ) ( I E z − A ε ( v ) v ) + 1 2 σ v 2 φ ″ ( v ) ] + R ε ( v , z ) , (3.11)

where the remainder term R ε ( v , z ) vanishes at the order ε 3 2 as ε → 0 . Therefore, as ε → 0 , we can obtain that in consequence of the scaling (3.3) the weak form of the kinetic model (2.5) is well approximated by the weak form of a linear Fokker-Planck equation

d d τ ∫ ℝ + φ ( v ) f ε ( v , τ ) d v = κ ∫ ℝ + [ φ ′ ( v ) ( I E M 1 − λ v 2 δ ( ( v v ¯ ) δ − 1 ) ) + 1 2 σ v 2 φ ″ ( v ) ] v α f ε ( v , τ ) d v . (3.12)

Providing the boundary terms produced by the integration by parts vanish, Equation (3.12) coincides with the weak form of the Fokker-Planck equation

∂ g ( v , τ ) ∂ τ = κ σ 2 ∂ 2 ∂ v 2 ( v 2 + α g ( v , τ ) ) + κ ∂ ∂ v [ ( λ v 2 δ ( ( v v ¯ ) δ − 1 ) − I E M 1 ) v α g ( v , τ ) ] . (3.13)

Without loss of generality, we will simplify Equation (3.13) by assuming

θ : = κ σ 2 , μ : = κ λ 2 δ v ¯ δ , ξ : = κ λ 2 δ , ς : = κ I E M 1 . (3.14)

Thus, the resulting Fokker-Planck equation takes the form

∂ g ( v , τ ) ∂ τ = θ ∂ 2 ∂ v 2 ( v 2 + α g ( v , τ ) ) + ∂ ∂ v ( ( μ v 1 + δ − ξ v − ς ) v α g ( v , τ ) ) . (3.15)

As exhaustively discussed in Ref. [

θ d d v ( v 2 + α g ( v , τ ) ) + ( ( μ v 1 + δ − ξ v − ς ) v α g ( v , τ ) ) | v = 0 , + ∞ = 0 , t > 0. (3.16)

With these no-flux boundary conditions, we can obtain the explicit stationary solution of the Fokker-Planck Equation (3.13) by solving the ordinary differential equation of first order

θ d d v ( v 2 + α g ( v ) ) + ( μ v 1 + δ − ξ v − ς ) v α g ( v ) = 0. (3.17)

Using h ( v ) = v 2 + α g ( v ) in (3.17) as unknown function, separation of variables gives as unique solution to (3.17) the function

g ∞ ( v ) = C ⋅ v ξ θ − 2 − α ⋅ exp ( − ς θ v − μ θ δ v δ ) = C ⋅ v λ δ σ − 2 − α ⋅ exp ( − 2 I E M 1 σ v − λ σ δ 2 ( v v ¯ ) δ ) , (3.18)

where the positive constant C has been chosen to normalize the equilibrium distribution. It is not difficult to discover that g ∞ ( v ) tends to 0 as v → 0 and v → ∞ . In other words, the city population size distribution obtained under the Non-Maxwellian collision does not exist with too little or too much population, which is more consistent with the real situation, and g ∞ ( v ) is close to the generalized Gamma density as v → ∞ .

In this section, we will use statistical data to verify the validity of the model. Here, we chose data from the Statistical Yearbook 2019 (27 provinces and 4 municipalities directly under the Central Government) released by the National Bureau of Statistics of China with the population of 332 cities (prefecture-level cities) in 2018. The histogram of the city size distribution is shown in

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

δ = 0.5 , λ = 0.8 , I E = 0.6 , M 1 = 20 , σ = 0.5 , v ¯ = 400 , α = 0.2.

The equilibrium distribution of both Maxwellian model [

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.

The research is partially supported by the National Science Foundation of China (Grant Nos. 11871335 and 11971008).

The authors declare no conflicts of interest regarding the publication of this paper.

Yu, L.J. and Liao, J. (2021) Non-Maxwellian Kinetic Modelling of City Size Distribution. Journal of Applied Mathematics and Physics, 9, 1329-1339. https://doi.org/10.4236/jamp.2021.96090