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From microscopic social force models to macroscopic continuum models for pedestrian flow

Liangze Yang, Hui Yu, Jie Du

TL;DR

The paper creates a rigorous micro-to-macro bridge for pedestrian flow by starting from a microscopic social force model with reactive route choice, deriving a mesoscopic mean-field kinetic equation, and obtaining macroscopic continuum models via hydrodynamic scaling. It identifies two main regimes for interaction kernels: a nonlocal, weak-coupling case and a local, strong-coupling case, yielding corresponding closures in the macroscopic momentum equation. Numerical experiments show strong agreement between the particle model and the derived macroscopic model, including scenarios with obstacles and fundamental-diagram behavior, and demonstrate the usefulness of calibrating macroscopic parameters from microscopic dynamics. This bottom-up framework enables efficient simulations and theoretical analysis of high-density crowd dynamics with practical implications for safety design and crowd management.

Abstract

The pedestrian flow is one of the most complex systems, involving large populations of interacting agents. Models at microscopic and macroscopic scales offer different advantages for studying related problems. In general, microscopic models can describe interaction forces at the individual level. Macroscopic models, on the other hand, provide analytical insights into global interactions and long-term overall dynamics, along with efficient numerical simulations and predictions. However, the relationship between models at different scales has rarely been explored. In this study, based on the original microscopic social force model with a reactive optimal route choice strategy, we first derive kinetic equations at the mesoscopic level. By varying the interaction force in different scenarios, we then derive several continuum models at the macroscopic level. Finally, numerical examples are given to evaluate the behaviors of the social force model and our continuum models.

From microscopic social force models to macroscopic continuum models for pedestrian flow

TL;DR

The paper creates a rigorous micro-to-macro bridge for pedestrian flow by starting from a microscopic social force model with reactive route choice, deriving a mesoscopic mean-field kinetic equation, and obtaining macroscopic continuum models via hydrodynamic scaling. It identifies two main regimes for interaction kernels: a nonlocal, weak-coupling case and a local, strong-coupling case, yielding corresponding closures in the macroscopic momentum equation. Numerical experiments show strong agreement between the particle model and the derived macroscopic model, including scenarios with obstacles and fundamental-diagram behavior, and demonstrate the usefulness of calibrating macroscopic parameters from microscopic dynamics. This bottom-up framework enables efficient simulations and theoretical analysis of high-density crowd dynamics with practical implications for safety design and crowd management.

Abstract

The pedestrian flow is one of the most complex systems, involving large populations of interacting agents. Models at microscopic and macroscopic scales offer different advantages for studying related problems. In general, microscopic models can describe interaction forces at the individual level. Macroscopic models, on the other hand, provide analytical insights into global interactions and long-term overall dynamics, along with efficient numerical simulations and predictions. However, the relationship between models at different scales has rarely been explored. In this study, based on the original microscopic social force model with a reactive optimal route choice strategy, we first derive kinetic equations at the mesoscopic level. By varying the interaction force in different scenarios, we then derive several continuum models at the macroscopic level. Finally, numerical examples are given to evaluate the behaviors of the social force model and our continuum models.
Paper Structure (13 sections, 3 theorems, 55 equations, 9 figures, 1 table)

This paper contains 13 sections, 3 theorems, 55 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Microscopic social force model
  3. Social force model coupled with an optimal route choice
  4. Asymptotic behaviors with a constant desired velocity
  5. Mean Field models and hydrodynamic Limits
  6. Hydrodynamic scaling
  7. The interaction kernels
  8. Macroscopic model
  9. Numerical examples
  10. Comparison between microscopic and macroscopic models
  11. Simulations without/with obstacles
  12. Characteristics of pedestrian flow
  13. Conclusion

Key Result

Proposition 2.1

Let $(\bm{x}_{i}(t), \bm{v}_{i}(t))$ be the solution to the social force model sf1 with $\bm{u}^{\rm e}$ being a constant vector. The following estimates hold: Hence, the system satisfies the mass conservation, and $M_1(t)$ converges to $\bm{u}^{\rm e}$ exponentially fast as $t\to \infty$.

Figures (9)

  • Figure 1: Diagram of the direction in the social force model.
  • Figure 1: The interaction kernels $\Phi(\bm{x})$ and $\Psi(\bm{x})$ with different range and $\epsilon=0.01$.
  • Figure 1: Three different modeling domains.
  • Figure 2: The number of left pedestrians against the simulation time $t$.
  • Figure 3: The evolution of the pedestrian dynamic by the microscopic social force model at different times in Example 2.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Proposition 2.1
  • Proof 1
  • Theorem 4.1
  • Proof 2
  • Theorem 4.2
  • Proof 3
  • Remark 4.1