Path-integral representation of diluted pedestrian dynamics

TL;DR

This paper introduces a path-integral formalism for modeling pedestrian dynamics, providing a trajectory-based probabilistic framework that captures variability and rare events, and connects to existing Langevin models.

Contribution

It develops a novel path-integral approach for pedestrian motion, linking it to Langevin models and enabling analysis of rare events in crowd dynamics.

Findings

Path-integral formalism accurately models pedestrian trajectories.

Quantitative connection to previous Langevin models for corridor flow.

Method enables evaluation of rare-event probabilities like U-turns.

Abstract

We frame the issue of pedestrian dynamics modeling in terms of path-integrals, a formalism originally introduced in quantum mechanics to account for the behavior of quantum particles, later extended to quantum field theories and to statistical physics. Path-integration enables a trajectory-centric representation of the pedestrian motion, directly providing the probability of observing a given trajectory. This appears as the most natural language to describe the statistical properties of pedestrian dynamics in generic settings. In a given venue, individual trajectories can belong to many possible usage patterns and, within each of them, they can display wide variability. We provide first a primer on path-integration, and we introduce and discuss the path-integral functional probability measure for pedestrian dynamics in the diluted limit. As an illustrative example, we connect the path-integral description to a Langevin model that we developed previously for a particular crowd flow condition (the flow in a narrow corridor). Building on our previous real-life measurements, we provide a quantitatively correct path-integral representation for this condition. Finally, we show how the path-integral formalism can be used to evaluate the probability of rare-events (in the case of the corridor, U-turns).