Acta Scientiarum Naturalium Universitatis Pekinensis ›› 2023, Vol. 59 ›› Issue (2): 326-330.DOI: 10.13209/j.0479-8023.2022.118

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Boltzmann Model of Growth Curve of the β Index for Traffic Networks

CHEN Yanguang   

  1. Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871
  • Received:2022-02-25 Revised:2022-10-20 Online:2023-03-20 Published:2023-03-20
  • Contact: CHEN Yanguang, E-mail: chenyg(at)



  1. 北京大学城市与环境学院, 北京 100871
  • 通讯作者: 陈彦光, E-mail: chenyg(at)
  • 基金资助:


Based on the verified empirical models of urbanization level and traffic network, a set of mathematical models of traffic network development are proposed. With the help of mathematical deduction, a Boltzmann equation is derived for the β index of traffic networks from the common logistic function of urbanization level growth, and a quadratic Boltzmann equation is derived for the β index from the quadratic logistic function of urbanization curve. The former is applicable to European and American countries and China’s southeast coastal areas, while the latter is applicable to mainland China, especially northern China. The main functions of the models are as follows: the first is to explain and predict the development of traffic networks, and the second is to research the stage division of a growth process or even the spatial dynamics of a traffic network.

Key words: traffic network, urbanization, Boltzmann equation, logistic function, allometric scaling, per capita income


从已被验证的城市化水平和交通网络的经验模型出发, 提出交通网络β指数的数学模型。基于城市化水平的一次logistic函数, 借助数学演绎法, 推导出交通网络β指数增长曲线的Boltzmann方程; 基于城市化水平的二次logistic函数, 推导出交通网络β指数增长曲线的二次 Boltzmann方程。前者适用于欧美国家和中国东南沿海地区, 后者则适用于中国大陆地区, 特别是中国北方地区。该模型具有两个主要功能: 1) 解释和预测交通网络的发展; 2) 有助于交通网络增长过程的分段研究。

关键词: 交通网络, 城市化, Boltzmann方程, logistic函数, 异速标度, 人均收入