We examine a theoretical formulation of the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces using measurements of particle travel distances obtained from laboratory and field-based experiments, supplemented with high-speed imaging and audio recordings that highlight effects of particle–surface collisions. The formulation, presented in a companion paper (Furbish et al., 2021a), is based on a description of the kinetic energy balance of a cohort of particles treated as a rarefied granular gas, as well as 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 due to conversion of potential to kinetic energy and frictional cooling due to particle–surface collisions. For a given particle size and shape these behaviors vary from a bounded distribution representing rapid thermal collapse with small slopes or large surface roughness, to an exponential distribution representing approximately isothermal conditions, to a heavy-tailed distribution representing net heating of particles with large slopes. 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. This energy conversion is strongly influenced by particle shape, although 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.

As described in our first companion paper (Furbish et al., 2021a), we are focused on rarefied motions of particles which, once entrained, travel downslope over the land surface. This notably includes the dry ravel of particles down rough hillslopes following disturbances (Roering and Gerber, 2005; Doane, 2018; Doane et al., 2019; Roth et al., 2020) or upon their release from obstacles (e.g., vegetation) following failure of the obstacles (Lamb et al., 2011, 2013; DiBiase and Lamb, 2013; DiBiase et al., 2017; Doane et al., 2018, 2019), and the motions of rockfall material over the rough surfaces of talus and scree slopes (Gerber and Scheidegger, 1974; Kirkby and Statham, 1975; Statham, 1976; Tesson et al., 2020). By “rarefied motions” we are referring to the situation in which moving particles may frequently interact with the surface but rarely interact with each other. Thus, rarefied particle motions are distinct from granular flows. Although this idea is most applicable to processes such as rockfall and the subsequent motions of the rock material over talus or scree slopes, our description of the motions of individual particles nonetheless may be entirely relevant to conditions that are not strictly rarefied (e.g., ravel involving many particles) but where during the collective motions of many particles the effects of particle–surface interactions dominate over effects of particle–particle interactions in determining the behavior of the particles – akin to granular shear flows at high Knudsen number (Kumaran, 2005, 2006). We note that laboratory experiments (Kirkby and Statham, 1975; Gabet and Mendoza, 2012) and field-based experiments (DiBiase et al., 2017; Roth et al., 2020) designed to mimic particle motions and travel distances on hillslopes effectively focus on rarefied conditions.

The formulation of rarefied particle motions presented in the first companion paper (Furbish et al., 2021a) 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. The particle energy balance involves gravitational heating with conversion of potential to kinetic energy, frictional cooling associated with particle–surface collisions, and an apparent heating associated with preferential deposition of low-energy particles. Deposition probabilistically occurs with frictional cooling in relation to the distribution of particle energy states as this distribution varies downslope. The Kirkby number

The purpose of this second companion paper is to present an analysis of several data sets concerning particle motions on rough surfaces, as viewed through the lens of the theory presented in the first companion paper (Furbish et al., 2021a). In Sect. 2 we summarize the context for our work provided by recent probabilistic descriptions of the flux and the Exner equation (Furbish and Haff, 2010; Furbish and Roering, 2013), and then step through essential elements of the mechanical basis of the theory leading to the generalized Pareto distribution of particle travel distances. In Sect. 3 we compare the formulation with the laboratory measurements of particle travel distances on rough surfaces reported by Gabet and Mendoza (2012) and Kirkby and Statham (1975). We also report new laboratory experiments designed to clarify how the size and shape of particles influence their motions and disentrainment based on high-speed imaging. In Sect. 4 we compare the formulation with the field-based measurements of travel distances reported by DiBiase et al. (2017) and Roth et al. (2020).

Particle travel distances from both the laboratory and field-based experiments are consistent with the generalized Pareto distribution and provide compelling evidence for the full range of predicted behaviors, from rapid thermal collapse to approximately isothermal conditions to net heating of particles. Nonetheless, 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. In the third companion paper (Furbish et al., 2021b) we show that the generalized Pareto distribution in this problem is a maximum entropy distribution (Jaynes, 1957a, b) constrained by a fixed energetic “cost” – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstates – the many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost – the generalized Pareto distribution represents the most probable arrangement. In the fourth companion paper (Furbish and Doane, 2021) we step back and examine the philosophical underpinning of the statistical mechanics framework for describing sediment particle motions and transport.

The problem of describing rarefied particle motions on hillslopes is motivated by the entrainment forms of the flux and the Exner equation. Namely, let

For completeness we note that the formulation above involving continuous functions can be recast into a discrete form that is useful for considering situations in which conditions influencing particle motions, for example the surface slope and roughness texture, change in the downslope direction. Let

As summarized next, the analysis presented in Furbish et al. (2021a) describes the mechanical basis for the disentrainment rates

Definition diagram of surface inclined at angle

Consider a rough, inclined surface with uniform slope angle

In particular, let

Namely, conservation of the total energy of the particle cohort is given by

Conservation of particle mass is given by

The ensemble-averaged energy satisfies the expression

Simultaneous solution of Eqs. (5), (9) and (10) using Eq. (11) leads to the disentrainment rate function

With reference to the presentation in the first companion paper (Furbish et al., 2021a) and for the purpose of presenting results below, let

Plot of dimensionless probability density

Behavior of the generalized Pareto distribution associated with the shape parameter

The dimensionless disentrainment rate is

The distribution

With

For reference to data fitting presented below, a binomial expansion of Eq. (13) gives

Also note that in fitting the data to the generalized Pareto distribution, Eq. (13), we use the dimensional form of the exceedance probability, Eq. (14). Specifically, we estimate values of the exceedance probability as

The quantities

This hypothesis formally enters the formulation via Eq. (6). Namely, from this relation we may write

In turn, Furbish et al. (2021a) suggest that the quantity

The experiments reported by Gabet and Mendoza (2012) involved launching spherical, sub-angular 1 cm particles with initial velocity

Because the same particle is launched each time with the same initial velocity

We choose

Plot of exceedance probability

Plot of logarithm of exceedance probability

The data are reasonably well fit by the theoretical curves, where we plot the data twice (Figs. 3 and 4)
in order to highlight several points. Estimated parametric values are provided in Table 2,
where estimates of the variability in

Fitted and estimated values of the parameters for the data shown in Figs. 3 and 4 with coefficients of variation in parentheses.

For data involving an estimated shape parameter

For data involving an estimated shape factor

Further note that for all cases less than 24

Plot of reciprocal of average travel distance

Here is an interesting sidebar. Following Samson et al. (1999) we plot the reciprocal,

The experiments reported by Kirkby and Statham (1975) involved dropping particles onto an inclined surface with fixed roughness composed of particles of different sizes, thus giving different ratios of particle to roughness size. For different slopes, the particles were dropped from different initial heights

For

The data are well fit by Eq. (38) (Fig. 6). Estimated parametric values are provided in Table 3.

Plot of average travel distance

Fitted values of the parameters for the data shown in Fig. 6.

We emphasize that other choices of the quantities

We conducted two sets of experiments. The first set was aimed at demonstrating the basis for treating the proportion of energy extraction,

Image of rounded (left), small (center) and angular (right) test particles on the concrete surface with granular roughness texture.

In the second set of experiments we launched particles of varying size and angularity onto the rough surface, then measured their travel distances for several slope angles. The slopes were

Image of pendulum catapult. A particle is placed on the low-friction (glossy cardboard) cradle at the base of the pendulum arms (

As a point of reference, the vertical rebounds of ordinary spherical glass marbles following their impacts on the hard slate reveal no surprises. These collinear collisions give a normal coefficient of restitution of

In contrast, the rebounds of natural particles from the hard slate reveal how noncollinear collisions strongly influence the rebound angle and normal rebound height. If

To illustrate this we write a normalized form of

Plot of

Plot of

For the hard slate surface, spheres show a small variance about the mean value. Probability is then redistributed toward

For the rough experimental surface,

Average energy partitioning as a proportion of initial energy

For each particle–surface combination, the largest rebound heights provide an estimate of the (ordinary) normal coefficient of restitution

Slope-parallel launch velocities

High-speed imaging of particle motions launched by the catapult onto the rough surface provides estimates of initial surface-parallel particle velocities

Plots of exceedance probability versus travel distance for the Vanderbilt experiments over six values of slope

For all surface slopes, the large rounded particles systematically travel farther than the large angular particles, and the small particles typically travel distances similar to those of the large angular particles (Fig. 11).
The data are reasonably fit by the theoretical curves, notably at small and large

Fitted and estimated values of the parameters for the data shown in Fig. 11 with coefficients of variation in parentheses.

Note that, for reference below, two values of

For the lower slopes

For a given slope, values of the friction factor

Video and audio recordings (Supplement, Vanderbilt University Institutional Repository,

One of the more compelling results appearing in several of the videos is when the translational kinetic energy of a particle at first impact is converted to translational kinetic energy involving transverse motion and rotational kinetic energy. Then during a second or third collision, rotational energy is converted back to vertical motion, thence to gravitational potential energy. The likelihood of this occurring increases with particle angularity, where noncollinear collisions are the rule rather than exception, and pointy particle corners lead to unusual collision configurations. The file “Angular_all_rotational.avi” shows a particularly strong conversion of translational to rotational motion with the initial collision on hard slate. The file “Angular_rotational_to_vertical.avi” shows conversion of rotational to vertical motion with the second collision. The file “Semiangular_rotational_die.avi” shows rapid cessation of motion following conversion of rotational to vertical motion.

The geometry of a noncollinear collision following the vertical drop of an angular particle is different from that of a particle at relatively small incident angle. Nonetheless, the strong conversion of translational to rotational motion associated with the former is analogous to the behavior of a particle that experiences stick during a small incident angle collision with conversion of translational to rotational energy (Appendix E in Furbish et al., 2021a).

The files “Angular_18%slope.avi” and “Angular_28%slope.avi” show examples of angular particles launched from the catapult onto the rough surface. Although the surfaces in these videos appear flat because of camera alignment, the slopes are

Audio recordings of particle–surface interactions during their downslope motions reveal the distinctive clickety-click sounds of collisions (“Bouncing.m4a”), which are markedly different from the sounds emitted by particles that are either gently or forcefully made to slide on the rough surface (“Sliding.m4a”). These clickety-click sounds occur with high frequency, particularly when particles are in a tumbling (nominally “rolling”) mode, giving way to decreasing frequency when particles undergo runaway bouncing motions. In contrast, sliding motions emit continuous scraping sounds. The key result of these recordings is to audibly reinforce the idea that motions involve collisional friction rather than a sliding Coulomb-like behavior, except in association with the brief collision impulses as described in collision mechanics theory (Brach, 1991; Stronge, 2000).

Further analyses of these detailed particle motions in relation to downslope and cross-slope motions are to be reported elsewhere.

The field-based experiments reported by DiBiase et al. (2017) involved launching three different sizes of particles down a natural hillslope surface. The particle size classes included diameters

As with the data of Gabet and Mendoza (2012), we again fit the parameters

Plot of exceedance probability versus travel distance for experiments described by DiBiase et al. (2017) showing small (black circles), medium (gray circles) and large (open circles) particles together with fitted distributions (lines).

The data are reasonably well fit by the theoretical curves (Fig. 12).
Estimated parametric values are provided in Table 7,
where estimates of the variability in

Fitted and estimated values of the parameters for the data shown in Fig. 12 with coefficients of variation in parentheses.

For all particle diameters, the Kirkby number

In contrast to the ambiguity of an exponential versus a Pareto fit to the data of Gabet and Mendoza (2012) for

Based on reported particle velocities, values of the initial average kinetic energies

The field-based experiments reported by Roth et al. (2020) involved dropping three different sizes of particles on eight natural hillslope surfaces in the Oregon Coast Range. Five of the hillslopes were covered with natural vegetation (designated by V) and included slope angles of 0, 14, 20, 24 and 39

The surfaces of the vegetated hillslopes had a layer of duff, woody debris and banana slugs beneath small plants (e.g., ferns) and trees. The surfaces of the burned hillslopes had little vegetal cover and were markedly smoother than the vegetated sites. Further details of the experiments, including measurements of surface roughness, are provided by Roth et al. (2020). Banana slugs, whose locomotive energetic costs are constrained by the shear-thinning rheology of their mucus (Lauga and Hosoi, 2006), appear as slow-moving Dirac functions in the power spectra of surface elevation. None were injured during the experiments.

As above, we fit the parameters

Plot of logarithm of exceedance probability

Fitted and estimated values of the parameters for the data reported by Roth et al. (2020) as shown in Figs. 13 and 14 with coefficients of variation in parentheses.

Plot of logarithm of exceedance probability

The data for the V sites are reasonably well fit by the theoretical curves (Fig. 13).
In these two examples as well as in cases not plotted, the estimated parameter

Plot of exceedance probability versus travel distance for experiment B28M described by Roth et al. (2020) showing data (circles) fit to mixed distribution (black line) composed of sum of two distributions (gray lines) weighted by proportions

The steepest burned site, B28, offers a further interesting perspective on particle motions. On this steep and relatively smooth hillslope, exceedance probabilities associated with all three particle sizes cannot be fit with reasonable fidelity by individual curves. Rather, the data suggest a mingling of two particle behaviors – rapid cooling for many particles and runaway heating for a second group leading to a pronounced heavy tail (Fig. 15) – in effect giving a mixed distribution. Namely, let

Fitted and estimated values of the parameters for the data shown in Fig. 15.

For the three particle sizes, exceedance probabilities are well matched by the weighted sum of a nearly exponential distribution (

Using the same data set, Roth et al. (2020) directly calculate the disentrainment rate function

We emphasize at the outset a key point in comparing travel distances measured in experiments (laboratory or field) with theoretical distributions. By definition a sample of measured values drawn from a distribution possesses a finite sample mean and variance, regardless of the form of the underlying distribution. If the underlying distribution possesses a finite mean and variance (e.g., an exponential distribution), then the calculated sample average and variance are unbiased estimates of the underlying parametric values. If the mean or variance of the underlying distribution is undefined (e.g., the generalized Pareto distribution for

In contrast, estimates of the parameters

Plot of modified exceedance probability

With these points in mind, we suggest that the fits presented above are consistent with the idea that each of the data sets represents a specific case of the generalized Pareto distribution. To further illustrate this idea we calculate the following quantities:

The bounded form of the generalized Pareto distribution

From an empirical point of view the data are consistent with the generalized Pareto distribution and reflect the predicted variation in behavior from rapid thermal collapse to approximately isothermal conditions to net heating of particles. Nonetheless we proceed by asking whether the estimated values of the quantities

The laboratory measurements with zero slope merit particular attention. The Kirkby number

The field-based measurements of DiBiase et al. (2017, Table 7) and Roth et al. (2020, Table 8) suggest that the friction factor

Plot of friction factor

Values of

As summarized in Sect. 2.4, scaling suggests that the factor

Turning to the quantity

Plot of factor

The data from Gabet and Mendoza (2012) support the idea that

Recall that

To reinforce the idea of a mixed distribution, consider the example of angular and rounded particles from the Vanderbilt experiments for

Plot of exceedance probability versus travel distance for Vanderbilt University experiment with

The laboratory and field-based measurements of particle travel distances presented above provide clear evidence that these distances are well described by a generalized Pareto distribution, where the form of the distribution reflects variations in particle behavior associated with the balance between gravitational heating and frictional cooling by particle–surface collisions. These behaviors vary from a bounded 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. Here we reiterate a point made in the first companion paper (Furbish et al., 2021a). Namely, we do not choose the generalized Pareto distribution in the empirical manner of selecting a distribution based on goodness-of-fit criteria applied to data sets. Rather, this distribution is dictated by the probabilistic physics of the problem; it 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.

The experiments involving high-speed imaging of particle motions reinforce what we intuitively already understand. Relative to a spherical particle, a rounded non-spherical particle is more likely to experience a noncollinear collision that converts the translational energy of free fall into transverse motion and rotational energy; an angular particle is more likely than is a rounded particle to experience such conversions. The effect of this behavior is a systematic increase in the proportion

The essence of rapid thermal collapse (

Although not essential to the fitting of particle travel distances to the generalized Pareto distribution, it nonetheless is desirable to have a clearer mechanical interpretation of the quantities

We suggest that, in designing and conducting particle launching experiments, we have a propensity to select pretty particles, and rounded (if not spherical) particles are pretty. This is not a bad thing. But it skews our view of particle motions toward the behavior of rounded particles. The experiments clearly demonstrate that particle angularity matters in the disentrainment process, specifically the likelihood of converting translational to rotational energy and the decreasing extraction of energy by collisional friction (Williams and Furbish, 2021).

Here in essence are the shortcomings of the formulation and its application to the experimental data sets of particle travel distances. We do not understand the transient (probabilistic) physics soon after launch from the catapult as particle motions become randomized with the onset of particle–surface collisions, so there is uncertainty in choosing the truncation distance and the associated particle energy state. Similarly, little is known about the distribution

We reemphasize that the work reported here is aimed at a probabilistic description of expected particle travel distances. This is a part of a larger effort to understand and inform the essence of the ingredients of nonlocal formulations of transport. We are not suggesting that the results presented here can be immediately cast as a nonlocal formulation of transport. But in order to progress beyond current formulations, the probabilistic physics of particle motions merits closer examination. For example, this level of understanding provides the basis for justifying a Taylor expansion of the convolution (Furbish and Haff, 2010) to form a local Fokker–Planck-like description of transport assuming an exponential-like distribution of travel distances – with clarity regarding the limitations of this description. Furthermore, we have focused on the energetics of particles in motion. But this is one of two ingredients of nonlocal formulations. The other involves the probabilistic physics and energetics of particle entrainment – a particularly difficult ingredient to constrain because of the difficulty of observing the entrainment process and because we do not yet know how to properly simulate this process. For this we must rely on theory and measurements of tracer particles in ways that have yet to be designed.

We end with a philosophical point. We enjoy eating our favorite tortilla chips, and mostly we enjoy them with a well-prepared dip, for example, spicy guacamole. But let us be honest. The experience then is no longer about the chips – it's about the dip. The chips are just the guacamole delivery system. (Yumm.) Similarly, these companion papers nominally concern particle motions on inclined rough surfaces. But these particles are just the delivery system. The dip consists of the coherent statistical mechanics framework for describing the particle motions and a demonstration that such a framework, albeit with rough edges, is possible. This represents a solid basis for subsequent efforts aimed at replication, falsification and refinement or replacement and possibly for fresh ideas concerning particle motions more generally.

Here we demonstrate the basis for using visual fits of the exceedance probability plots to illustrate the behavior of the generalized Pareto distribution, and we provide context for interpreting these fits. We work with the dimensionless form of the distribution for comparison with Fig. 2 and pursue a straightforward Monte Carlo analysis.

First, let

Second, among the methods for estimating the values of

Now consider a sample size of

Plots of estimated exceedance probabilities

Here is an important sidebar. In the presence of an exact theory that predicts the values of

We therefore reemphasize our objective. At this stage of our work we are aimed at reasonable estimates of the shape and scale parameters in order to demonstrate the existence of the behaviors – rapid thermal collapse, isothermal conditions, and net heating of particles – represented by the generalized Pareto distribution. Refined values of these parameters are not needed until we possess a clearer understanding of the mechanics of deposition. Semi-log plots highlight deviations in the tails and provide a clear sense of the concavity that discriminates between cooling and heating. Log–log plots highlight deviations near the origin and provide a sense of the log-linear decay of the tails for heavy-tailed conditions. The variability in the tails of the distribution as outlined above emphasizes the importance of avoiding over-fitting of the tails in visual fitting (or in any other method of fitting).

For comparison with our fits, we return to dimensional quantities and compute the MLE values of

One alternatively can choose, say, a nonlinear least-squares fitting algorithm that weights various parts of data differently, emphasizing or deemphasizing values near the origin or in the tails. We suggest, however, that this is just a rule-based version of visual fitting. We also note that visual fitting is not directly influenced by censorship, although the form of the censored tail can never be known (Ballio et al., 2019). Bringing more sophisticated techniques to bear (e.g., Hosking and Wallis, 1987; Castillo and Hadi, 1997; Cramer and Schmiedt, 2011; Giles et al., 2013a, b; Pak and Mahmoudi, 2018) to refine estimates of

We end with a cautionary note: parametric values of a heavy-tailed distribution estimated from a data set with

Plots of exceedance probability

Histograms of maximum likelihood estimates of shape parameter

Plots of exceedance probability

Histograms of maximum likelihood estimates of shape parameter

Fitted and estimated values of the parameters for the data reported by Gabet and Mendoza (2012), the Vanderbilt experiments, DiBiase et al. (2017), and Roth et al. (2020).

Example of fit (gray line) based on MLE values of

Consider as an example the initial exceedance probability plot for the angular particles on a flat surface (Fig. B1a),
which shows a clear inflection at about 5 cm. In this example, high-speed imaging of particles launched from the catapult reveals that the particles consistently travel

Plot of exceedance probability

For these reasons we truncate the plots at the inflection position and then recalculate exceedance probabilities with reduced

Of interest is how uncertainty in the estimates of the shape and scale parameters,

We perform a straightforward Mont Carlo analysis. Assuming values of

We emphasize that these calculations provide a sense of the variability

In general, calculated coefficients of variation decrease with increasing surface slope. Coefficients of variation are on the order of 10 % or more for smaller slopes in the experiments reported by Gabet and Mendoza (2012) and in the Vanderbilt experiments. Coefficients of variation generally are on the order of 1 % or smaller in the field-based experiments of DeBiase et al. (2017) and Roth et al. (2020) involving steep slopes.

Let

For completeness, here we show plots of the friction factor

Values of

Plot of friction factor

Values of

Plot of factor

It is well known that a Pareto distribution with positive shape parameter can be obtained as a mixture of exponential distributions whose rate parameters are distributed as a gamma distribution. This result suggests an interesting physical interpretation of the Pareto distribution of particle travel distances, and it also may indicate a strategy for clarifying how particle size and shape in concert with surface roughness influence the extraction of particle energy and the likelihood of deposition. For completeness we therefore offer the following.

Recall that for an exponential distribution of travel distances

The expected value

Because

Plot of

As an example, for a value of

Inasmuch as the quantities

Emmanuel Gabet provided the data described in Sect. 3.1. The data described in Sect. 4.1 and 4.2 are available from DiBiase et al. (2017) and Roth et al. (2020). The data described in Sect. 3.3, including video and audio files, and the MATLAB/GNU Octave code described in Appendix A are archived and readily accessible via the Vanderbilt University Institutional Repository (

All authors contributed to the conceptualization of the problem and its technical elements. SGW led the Vanderbilt University experiments. DJF wrote the paper with critical review and input from the other authors.

The authors declare that they have no conflict of interest.

We appreciate critical discussions with Angel Abbott (1990–2018) concerning rarefied particle motions on the spectacular scree slopes within Martian craters, with Jonathan Gilligan concerning parameter estimation, and with Mark Schmeeckle concerning collision mechanics. Emmanuel Gabet generously provided the experimental data in Figs. 3 and 4. We appreciate the help of Brandt Gibson in setting up the experimental slope and the help of Rachel Bain in coding the Q–Q algorithm. We appreciate reviews of our work provided by Rachel Glade and Joris Heyman.

This research has been supported by the National Science Foundation (grant nos. EAR-1420831 and EAR-1735992 to David Jon Furbish, CNS-1831770 to Joshua J. Roering, and EAR-1625311 to Danica L. Roth).

This paper was edited by Eric Lajeunesse and reviewed by Rachel C. Glade and Joris Heyman.