A high‐resolution data set obtained from high‐speed imaging of coarse sand particles transported as bed load allows us to confidently describe the forms and qualities of the ensemble distributions of particle velocities, accelerations, hop distances, and traveltimes. Autocorrelation functions of frame‐averaged values (and the decay of these functions) support the idea that the forms of these distributions become time invariant within the 5 s imaging interval. Distributions of streamwise and cross‐stream particle velocities are exponential, consistent with previous experiments and theory. Importantly, streamwise particle velocities possess a “light” tail, where the largest velocities are limited by near‐bed fluid velocities. Distributions of streamwise and cross‐stream particle accelerations are Laplace in form and are centered on zero, consistent with equilibrium transport conditions. The majority of particle hops, measured start to stop, involve short displacements, and streamwise hop distances possess a Weibull distribution. In contrast to previous work, the distribution of traveltimes is exponential, consistent with a fixed temporal disentrainment rate. The Weibull distribution of hop distances is consistent with a decreasing spatial disentrainment rate and is related to the exponential distribution of traveltimes. By taking into account the effects of experimental censorship associated with a finite sampling window, the relationship between streamwise hop distances and traveltimes, Lx∼Tpα, likely involves an exponent of α ∼ 2. These experimental results—an exponential distribution of traveltimes Tp and a Weibull distribution of hop distances Lx with shape parameter k < 1—are consistent with a nonlinear relationship between these quantities with α > 1.

We describe the most likely forms of the probability distributions of bed load particle velocities, accelerations, hop distances, and travel times, in a manner that formally appeals to inferential statistics while honoring mechanical and kinematic constraints imposed by equilibrium transport conditions. The analysis is based on E. Jaynes's elaboration of the implications of the similarity between the Gibbs entropy in statistical mechanics and the Shannon entropy in information theory. By maximizing the information entropy of a distribution subject to known constraints on its moments, our choice of the form of the distribution is unbiased. The analysis suggests that particle velocities and travel times are exponentially distributed and that particle accelerations follow a Laplace distribution with zero mean. Particle hop distances, viewed alone, ought to be distributed exponentially. However, the covariance between hop distances and travel times precludes this result. Instead, the covariance structure suggests that hop distances follow a Weibull distribution. These distributions are consistent with high‐resolution measurements obtained from high‐speed imaging of bed load particle motions. The analysis brings us closer to choosing distributions based on our mechanical insight.

The ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker–Planck equation. The diffusive term approximates nonlocal transport. It ‘looks’ upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High‐resolution measurements of bedload particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land surface slope. Particle diffusion during both bedload and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds‐averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle‐borne substances. Copyright © 2016 John Wiley & Sons, Ltd.

Bedload sediment transport is the basic physical ingredient of river evolution. Formulae exist for estimating transport rates, but the diffusive contribution to the sediment flux, and the associated spreading rate of tracer particles, are not clearly understood. The start-and-stop motions of sediment particles transported as bedload on a streambed mimic aspects of the Einstein-Smoluchowski description of the random-walk motions of Brownian particles. Using this touchstone description, recent work suggests the presence of anomalous diffusion, where the particle spreading rate differs from the linear dependence with time of Brownian behavior. We demonstrate that conventional measures of particle spreading reveal different attributes of bedload particle behavior depending on details of the calculation. When we view particle motions over start-and-stop timescales obtained from high-speed (250 Hz) imaging of coarse-sand particles, high-resolution measurements reveal ballistic-like behavior at the shortest (10 2 s) timescale, followed by apparent anomalous behavior due to correlated random walks in transition to normal diffusion (> 10 1 s) -similar to Brownian particle behavior but involving distinctly different physics. However, when treated as a 'virtual plume' over this timescale range, particles exhibit inhomogeneous diffusive behavior because both the mean and the variance of particle travel distances increase nonlinearly with increasing travel times, a behavior that is unrelated to anomalous diffusion or to Brownian-like behavior. Our results indicate that care is needed in suggesting anomalous behavior when appealing to conventional measures of diffusion formulated for ideal particle systems.

Macroecology strives to identify ecological patterns on broad spatial and temporal scales. One such pattern, Rapoport's rule, describes the tendency of species' latitudinal ranges to increase with increasing latitude. Several mechanisms have been proposed to explain this rule. Some invoke climate, either through glaciation driving differential extinction of northern species or through increased seasonal variability at higher latitudes causing higher thermal tolerances and subsequently larger ranges. Alternatively, continental tapering or higher interspecific competition at lower latitudes may be responsible. Assessing the incidence of Rapoport's rule through deep time can help to distinguish between competing explanations. Using fossil occurrence data from the Palaeobiology Database, we test these hypotheses by evaluating mammalian compliance with the rule throughout the Caenozoic of North America. Adherence to Rapoport's rule primarily coincides with periods of intense cooling and increased seasonality, suggesting that extinctions caused by changing climate may have played an important role in erecting the latitudinal gradients in range sizes seen today.