Most particle motions on Earth's surface are fundamentally stochastic and often occur under rarefied transport conditions. Every particle makes a unique path along the bed, similar but distinct from the paths of all other particles in motion, until it loses enough kinetic energy to become disentrained. The details of a particle's motion are determined by the amount of energy added and extracted during each moment of travel. Thus, particle motions physically reflect the complex energy dynamics at play and are a building block of morphodynamic theory. A full appreciation of this energy balance is needed to properly describe the motion of particles and associated disentrainment under different transport conditions. Often multidimensional behaviors occur during transport as both a result of and influence on these particle scale energy dynamics. One such phenomenon is that of particle-scale random walking during transport which results in diffusion over short timescales in both the downstream and transverse directions. We have adopted the Galton board as the fundamental conceptual model on which to create a mechanistic yet probabilistic formulation of particle diffusion. Here we provide a data set of two-dimensional particle travel distances supplemented with high-speed videos of particle-surface collisions collected during laboratory experiments to characterize the influence of shedding fluid vortices and angularity on collisional distances and two-dimensional travel for particles at low Reynolds numbers. Such a description is consistent with diffusion from the top-down and may be distinct from the bottom-up, or surface roughness, controlled random walking that other studies have explored. Preliminary analysis shows that spherical particles experience jiggling motions resulting in transverse displacement in the absence of surface roughness and this behavior is further exaggerated for particles of natural angularity. We hope to clarify the influence of the particle Reynolds number in top-down and bottom-up spreading.

During the last century, descriptions of sediment transport on the surface of Earth have been mostly deterministic and strongly influenced by concepts from continuum mechanics. The assumption that particle motions on hillslopes and in rivers satisfy the continuum hypothesis has provided an important foundation for this topic. Recent studies, however, have recognized that bed load and hillslope sediment transport conditions often are rarefied and do not satisfy continuum assumptions, therein pointing to the need for new ways of describing particle motions and transport. The problem of rarefied sediment transport is probabilistic in nature, and emerging methods for describing particle motions hark back to the pioneering work of Einstein (1938), who conceptualized bed load transport as a probabilistic problem. Here we provide a data set of particle travel distances and supplemental high-speed videos of particle-surface collisions collected during laboratory experiments to assess a theoretical formulation of the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. The formulation is based on a description of the kinetic energy balance of a cohort of particles treated as a rarefied granular gas, and a description of particle deposition that depends on the energy state of the particles. Both laboratory and field-based measurements are consistent with a generalized Pareto distribution of travel distances and predicted variations in behavior associated with the balance between gravitational heating and frictional cooling by particle-surface collisions. These behaviors vary from a truncated distribution associated with rapid thermal collapse to an exponential distribution representing approximately isothermal conditions to a heavy-tailed distribution associated with net heating of particles. The transition to a heavy-tailed distribution likely involves an increasing conversion of translational to rotational kinetic energy leading to larger travel distances with decreasing effectiveness of collisional friction. The analysis points to the need for further clarity concerning how particle size and shape in concert with surface roughness influence the extraction of particle energy and the likelihood of deposition.