Generalization of Hop Distance‐Time Scaling and Particle Velocity Distributions via a Two‐Regime Formalism of Bedload Particle Motions

To date, there is no consensus on the probability distribution of particle velocities during bedload transport, with some studies suggesting an exponential‐like distribution while others a Gaussian‐like distribution. Yet, the form of this distribution is key for the determination of sediment flux and the dispersion characteristics of tracers in rivers. Combining theoretical analysis of the Fokker‐Planck equation for particle motions, numerical simulations of the corresponding Langevin equation, and measurements of motion in high‐speed imagery from particle‐tracking experiments, we examine the statistics of bedload particle trajectories, revealing a two‐regime distance‐time (L‐Tp) scaling for the particle hops (measured from start to stop). We show that particles of short hop distances scale as L ~ Tp2 giving rise to the Weibull‐like front of the hop distance distribution, while particles of long hop distances transition to a different scaling regime of L ~ Tp leading to the exponential‐like tail of the hop distance distribution. By demonstrating that the predominance of mostly long hop particles results in a Gaussian‐like velocity distribution, while a mixture of both short and long hop distance particles leads to an exponential‐like velocity distribution, we argue that the form of the probability distribution of particle velocities can depend on the physical environment within which particle transport occurs, explaining and unifying disparate views on particle velocity statistics reported in the literature. Plain Language Summary The complex motion of sediment particles on a riverbed dictates the rate of bedload sediment transport with important implications for water quality, biotic life, and infrastructure safety. Inferring characteristics of individual particle trajectories from collective statistics of particle motion attributes (velocities, accelerations, hop distances, and travel times) is a challenge with significant theoretical and practical implications, including probabilistic formulations of transport, correct interpretation of experimental measurements, and comparison of disparate environmental transport conditions in rivers. Here, we document and physically explain two distinct regimes of particle motion, namely that short and long hop distance particles exhibit different dynamics, particle statistics, and hop distance‐travel time scaling regimes. This finding explains and unifies disparate views on particle velocity statistics reported in th