Emergent Detailed Balance in Human Mobility under Temporal Coarse-Graining

Abstract

A fundamental question in nonequilibrium statistical physics is whether effective equilibrium behavior can emerge at coarse-grained scales in strongly driven systems. Here, we investigate this question in the context of human mobility by analyzing five years of intercity flow data covering millions of travelers. While short-term flows are highly asymmetric, temporal coarse-graining reveals that over half of all city pairs converge toward effective flow balance, with normalized directional imbalance decaying as a power law. The remaining pairs either exhibit persistent drift-dominated currents or a crossover between these two extremes. A stochastic model decomposing mobility into directional drift and correlated fluctuations quantitatively captures the coexistence of all three regimes. Directly measured variance scaling of the fluctuation process confirms near-diffusive behavior with regime-dependent deviations. These results demonstrate that large-scale mobility networks exhibit a scale-dependent transition from broken to restored flow symmetry, with direct implications for modeling transport and spreading dynamics.

Figures (4)

• Figure 1: Weekly mobility flows between Beijing and Baoding. (a) The bidirectional flows exhibit similar temporal patterns, with clear disruptions during the COVID-19 and holiday peaks after 2023. (b) The net flow reveals strong fluctuations, frequently reaching 30--40% of the total bidirectional volume.
• Figure 2: (a) Ensemble-averaged normalized imbalance $\langle\langle R^{(\tau)} \rangle \rangle$ as a function of aggregation window $\tau$. The dashed line indicates a power-law fit with exponent $-0.30$. (b) Normalized imbalance trajectories for a representative sample of city pairs, illustrating heterogeneous behavior.
• Figure 3: Three empirical regimes of directional imbalance revealed by the scaling behavior. (a) Regime-averaged normalized imbalance $\langle \langle R^{(\tau)} \rangle \rangle$ as a function of the aggregation window $\tau$ for city pairs classified into three regimes: fluctuation-dominated decay (blue), crossover to plateau (orange), and drift-dominated saturation (red). Shaded regions indicate the interquartile range (25--75%). The dashed line shows the theoretical prediction for uncorrelated fluctuations (slope = -0.5). (b) Distribution of decay exponents for fluctuation-dominated links. (c) Distribution of goodness-of-fit, measured by $R^2$, for the power-law fits. (d) Growth of cumulative bidirectional traffic with aggregation window $\tau$, showing the approximately linear scaling $S_{ij}^{(\tau)} \propto \tau$. Flows are normalized by their maximum value for comparison across city pairs. (e) Distribution of plateau values of $R^{(\tau)}$ for drift-dominated and crossover links. (f) Distribution of mean weekly flow for the three regimes.
• Figure 4: Spatial distribution of mobility regimes. (a) Network map of the 2,000 city pairs with the largest weekly mean flows. Colors indicate the three transport regimes identified in Fig. \ref{['fig3']} (fluctuation-dominated, crossover, and drift-dominated). Node size is proportional to city population from the 2020 census. (b) The twenty city pairs with the largest mean flows. (c) The Guangdong Bay Area, corresponding to several high-traffic links highlighted in panel (b).