Theoretical and experimental work (Furbish et al., 2021a, b, c) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling by particle–surface collisions. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic “cost” – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. The analyses leading to these results provide an ideal case study for highlighting three key elements of a statistical mechanics framework for describing sediment particle motions and transport: the merits of probabilistic versus deterministic descriptions of sediment motions, the implications of rarefied versus continuum transport conditions, and the consequences of increasing uncertainty in descriptions of sediment motions and transport that accompany increasing length scales and timescales. We use the analyses of particle energy extraction, the spatial evolution of particle energy states, and the maximum entropy method applied to the generalized Pareto distribution as examples to illustrate the mechanistic yet probabilistic nature of the approach. These examples highlight the idea that the endeavor is not simply about adopting theory or methods of statistical mechanics “off the shelf” but rather involves appealing to the

In three companion papers (Furbish et al., 2021a, b, c) we examine a theoretical formulation of the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. As noted by Furbish et al. (2021a), such motions include the ravel of particles following disturbances (Roering and Gerber, 2005; Doane, 2018; Doane et al., 2019; Roth et al., 2020) or 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 surfaces of talus and scree slopes (Gerber and Scheidegger, 1974; Kirkby and Statham, 1975; Statham, 1976). 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, such that 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 (Risso and Cordero, 2002; Kumaran, 2005, 2006).

The formulation (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 formulation predicts a generalized Pareto distribution of particle travel distances whose form varies with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling by particle–surface collisions. Specifically, the generalized Pareto distribution varies from a bounded form associated with thermal collapse and rapid deposition to an exponential form representing isothermal conditions to a heavy-tailed form associated with net heating of particles and decreased deposition. As described in Furbish et al. (2021b), these varying forms of the generalized Pareto distribution are consistent with laboratory measurements of particle travel distances reported by Gabet and Mendoza (2012) and Furbish et al. (2021b), as well as with field-based measurements of travel distances reported by DiBiase et al. (2017) and Roth et al. (2020). Moreover, as described in Furbish et al. (2021c), 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.

The analyses of rarefied particle motions in these companion papers collectively provide an ideal case study for highlighting key elements of a statistical mechanics framework for describing sediment particle motions and transport. Indeed, as noted in the second companion paper (Furbish et al., 2021b):

We enjoy eating our favorite tortilla chips, and mostly we enjoy them with a well-prepared dip, for example, spicy guacamole. But … [t]he experience then is no longer about the chips – it's about the dip. The chips are just the guacamole delivery system … Similarly, these companion papers nominally concern particle motions on inclined rough surfaces. But these particles are just the delivery system. The dip consists of

We suggest that the timing is ideal for offering perspective on these elements of a statistical mechanics framework. Amidst echoes from the pioneering work of Einstein (1937) on bed load particle motions and that of Culling (1963) on hillslope soil creep, there is a reemerging interest in probabilistic descriptions of sediment motions and transport. For example, recent efforts involve descriptions of (1) bed load particle motions and transport; (2) bed load tracer particle motions, including effects of particle–bed exchanges; (3) nonlocal sediment transport on hillslopes; (4) particle motions in soils, including tracer particles; and (5) rain splash transport (Appendix A). (We also note that there is a parallel interest in describing the statistical physics of relatively dense granular materials, e.g., Bi et al., 2015.) However, this effort is a patchwork of approaches and methods, and to date it mostly has involved kinematic descriptions of sediment motions and transport with limited elucidation of the associated mechanics. We believe that it is important for the philosophical underpinning of this growing effort to be part of the conversation, adding to recent perspectives on bed load transport offered by Ancey (2020a, b). This includes attention to commonalities in the formalism used to describe transport in different settings, for example, in relation to transport on hillslopes and within rivers. The conversation also must include an honest assessment of the expectations and limitations of probabilistic descriptions of transport.

In Sect. 2 we summarize key material from the three companion papers for reference in later sections. In Sect. 3 we step back and consider in general terms the philosophical basis of a statistical mechanics framework for describing sediment motions and transport. In Sect. 4 we return specifically to the problem of rarefied particle motions on hillslopes and use the analysis to illustrate elements of the framework. In Sect. 5 we consider implications of the statistical mechanics description of rarefied particle motions on hillslopes. In several sections we provide historical background on the technical material covered.

With reference to material in Furbish et al. (2021a, b, c), the problem of describing rarefied particle motions on hillslopes is motivated by the entrainment forms of the flux and the Exner equation. Namely, let

The appearance of time

Plot of probability density

The analysis presented in Furbish et al. (2021a) describes the mechanical basis of the disentrainment rate

In mechanical terms the shape and scale parameters

Following Furbish et al. (2021b) we calculate the quantities

Plot of modified exceedance probability

We refer to elements of the analysis summarized here in several sections below. Meanwhile we turn to the philosophy of the statistical mechanics framework.

Although our companion papers are focused on rarefied particle motions on hillslopes, here we purposefully step back and initially consider the broader topic of probabilistic descriptions of sediment motions and transport. Borrowing ideas and wording from a book in preparation, we briefly consider elements of three foundational concepts in the natural sciences that repeatedly appear in the study of complex systems, focusing here on sediment systems: (1) the relation between mechanistic descriptions of sediment behavior and probabilistic versus deterministic formulations of this behavior, (2) differences in rarefied versus continuum conditions;, and (3) the treatment of uncertainties in descriptions of system behavior that grow with increasing length scales and timescales considered. The point of this brief overview is to ask ourselves, at least momentarily, to step out of our comfort zones as informed and conditioned by our different backgrounds in, say, particle mechanics, continuum mechanics, probability and statistics, and, by the length scales and timescales with which we are most familiar. Following this overview, we return to the problem of rarefied particle motions and use it to illustrate the philosophical points involved.

In learning how to describe the behavior of mechanical systems, mostly we are initially exposed to deterministic examples. We study Newton's laws as these pertain to simple particle systems and then move on to the behavior of solids and fluids treated as continuous materials, wrapping our heads around Lagrangian versus Eulerian perspectives. The formalism is unambiguous, and describing the behavior of a well-constrained system is in principle straightforward. Indeed, much of the legacy of geophysics resides in the determinism of continuum mechanics. Perhaps it is therefore natural that we might envision that a mechanistic description of the behavior of a system implies that such a description ought to be, or perhaps only can be, a deterministic one. Such a perception represents a lost opportunity. The most elegant counterpoint example is the field of classical statistical mechanics – devoted specifically to the probabilistic (i.e., non-deterministic) treatment of the behavior of gas particle systems in order to justify the principles of thermodynamics – yet which is no less mechanical in its conceptualization of this behavior than, say, the application of Newton's laws to the behavior of a deterministic system consisting of the interactions of a few billiard balls or involving the motion of a Newtonian fluid subject to specific initial and boundary conditions.

Once steeped in the language of mechanics, we understandably take comfort in mechanistic descriptions of system behavior. Specifically, we invest trust in the underlying foundation, and implied rigor, provided by classical mechanics. This is a good thing. But given the complexity and the uncertainty in describing the behavior of sediment systems, here it is essential to consider the idea that the concepts and language of probability are well suited to the problem of describing this behavior – precisely because of the complexity and uncertainty involved. This involves relaxing our expectations, for example, that a deterministic-like relationship exists between the flux of bed load sediment and the fluid stress imposed on the streambed or between the flux of sediment on a hillslope and the local land-surface slope – particularly when these involve noise-driven processes, as described below. This idea of leaning on probability to describe the behavior of sediment systems is not as straightforward as describing the behavior of idealized gas particle systems. Nonetheless, the objective is the same: to be mechanistic, yet probabilistic. These worldviews are entirely compatible.

To be sure, the extent to which the tools of probability can be fruitfully brought to bear to characterize particle motions and transport varies with the specific process considered and the information we have available to constrain any particular probabilistic description of motions. For example, we know far more about the probabilistic qualities of bed load sediment transport in shear flows based on flume experiments than, say, soil particle transport and mixing associated with bioturbation and granular creep (Appendix A). The objective therefore is to aim at probabilistic descriptions of sediment particle motions and transport that lean on the

The continuum hypothesis – the essential basis of continuum mechanics – stands as a triumph of the physical sciences. (Let us be clear that we are referring to the version of this hypothesis as applied to descriptions of real material systems rather than to the related mathematical idea posed by Georg Cantor, that there is no set of numbers whose size falls between the two infinities associated with the natural numbers and the real numbers.) This hypothesis allows us to envision many solid and fluid materials at our ordinary macroscopic scale of observation as being continuous things whose properties and behavior can be described using that part of the calculus given to continuously differentiable functions – even though when we focus our attention on the scale of the elements of a “continuous” material, that is, at the particle scale, we discover that it is decidedly discontinuous. Indeed, many of the definitions of basic, familiar quantities describing the properties, rheology and motion of real materials – their intensive properties, thermodynamic state variables, rheological coefficients, discharges and fluxes, the divergence of these fluxes, etc. – at the outset assume continuous substances and continuum behavior that involve smooth changes with respect to space and time. That said, this lovely continuum siren is to be avoided as a de facto starting point in descriptions of sediment motions and transport. Many sediment particle motions on Earth's surface are patchy, intermittent and demonstrably rarefied (Furbish et al., 2016b, 2018c; Ancey, 2020a, b) – conditions that are at odds with continuum formulations of these motions.

For these reasons an appropriate strategy involves constructing descriptions of the collective behavior of sediment particles without assuming a continuum behavior at the outset. Indeed, precise definitions of the sediment particle flux and its divergence do not assume continuum conditions (Ancey, 2010; Furbish et al., 2012a, 2016b, 2017; Ancey and Pascal, 2020). Instead, the idea is to develop more general (probabilistic) formulations of this behavior and then ask the following question: under what conditions does a continuum formulation of behavior make sense? As a point of reference, when continuum behavior is assumed at the outset, the Navier–Stokes momentum equation is derived from the Cauchy momentum equation. But when viewed with respect to particle (molecular) behavior, the Navier–Stokes equation is derived from the Boltzmann equation – which is decidedly probabilistic and entirely agnostic to continuum versus rarefied conditions. That is, the Boltzmann equation is equally applicable to both conditions. If the continuum hypothesis is satisfied, then it is natural to adopt the Navier–Stokes formalism. On the other hand, rarefied conditions must be treated probabilistically using methods of statistical mechanics. As described below with respect to sediment particles, this can include the use of continuum-like equations – noting that “continuum-like” means continuously differentiable, not that the particles behave as a continuum, and also noting that such equations apply to ensemble expected conditions, not to individual realizations. This means that we must be careful in interpreting the use of continuous probability distributions and related functions to describe attributes of particle motions (e.g., entrainment rates, travel distances) as in Eqs. (1) and (2).

One of the most important consequences of rarefied transport conditions is this: one cannot expect to predict a well-defined single value of the particle flux from specified, fixed controlling factors. Even under the ideal circumstances of a “perfect” model of the particle flux, such a prediction must be probabilistic. That is, a given set of controlling factors yields a probability distribution of fluxes rather than a single value. Any individual realization therefore can involve a value that may or may not coincide with a statistically expected value, whether this expected value is an empirical outcome or is predicted by a mechanical argument.

Our interest in sediment particle systems spans timescales of less than milliseconds to hundreds of thousands of years. The shortest timescales are represented by, say, observations of the details of particle motions in controlled experiments measured by high-speed imaging. Intermediate timescales are represented by, say, measurements of transport on hillslopes and in rivers on human timescales pertaining to the erosion and deposition of sediment in relation to land-use and river management. Long timescales are represented by our interest in understanding the evolution of hillslope and river systems at geomorphic timescales. Similarly, our interest in sediment systems spans length scales of less than a millimeter to at least tens or hundreds of kilometers. The smallest length scales are represented by differential particle motions during granular creep that are a fraction of a particle diameter or in relation to the initial jiggling of bed load particles prior to entrainment from their microtopographic “pockets”. Intermediate length scales are represented by particle motions involved in the dynamics of river and eolian bedforms – ripples to dunes to megadunes – thence to scales involving, say, intermittent sediment motions from the crest of a hillslope to its base or within the extent of one or two river bends. The largest length scales are represented by the erosion and deposition of sediment over the scale of a hillslope–channel network or a depositional basin.

With increasing scale (length and time) goes increasing uncertainty in our descriptions of sediment motions and the behavior of sediment systems. The essential reasons for this increasing uncertainty reside in the increasing complexity, including heterogeneity, of sediment systems as their size increases, and in the increasing stochasticity, including the patchiness and intermittency, of factors that influence sediment motions and transport as both the system size and the timescale of interest increase. Equally important, with increasing scale our uncertainty grows in relation to the increasing difficulty, and the loss of resolution, associated with observing and measuring quantities that enter into our descriptions of sediment motions and transport – whether these quantities involve features of the sediment itself (e.g., particle sizes and arrangements, details of particle motions), or attributes of the factors influencing sediment transport (e.g., changing fluid motions, surface roughness). Moreover, in approaching climate-change timescales and longer, we can only imagine in probabilistic terms how many of the ingredients of sediment transport might vary (Benda and Dunne, 1997).

In relation to the uncertainty that grows with scale, we also must consider the consequences of differences in our ability to observe and measure quantities representing the dynamics of “fast” versus “slow” systems as viewed relative to the human experience. Focusing specifically on the configuration of a sediment system – a bedform, a river reach, a soil mantled hillslope – a fast system is one for which we can observe and measure attributes of the particle fluxes and associated changes in the system configuration over human timescales. A slow system is one for which the fluxes and changes in configuration are largely imperceptible over these timescales. In simple terms, for a system consisting of

These ideas support a strong case for incorporating concepts and methods of probability – the natural language of uncertainty – into our descriptions of sediment particle motions and transport, tuning the specifics to the demands of different scales. This is as much a philosophical choice as a technical one; it is a choice to make the treatment of uncertainty a key part of the problem at the outset (Ancey and Pascal, 2020; Korup, 2020). Of course the strategies and methods vary with scale, as do the sources of uncertainty, in relation to the transport processes involved and the techniques of observation and measurement used. The purpose of explicitly incorporating probabilistic concepts in describing transport is to use this as a framework to explore, for example, the consequences of patchiness and intermittency of sediment motions in formulations of transport rates or how a predicted transport rate at a specified position and time within a real system actually represents a statistically expected behavior associated with a distribution of possible transport rates. The objective is to illustrate that this approach to uncertainty, combined with aiming at mechanistic, albeit probabilistic, descriptions of sediment particle behavior – avoiding a continuum description at the outset – will move us toward a deeper understanding of sediment particle motions and transport in both experimental and natural systems. We now step through the elements of the probabilistic framework outlined in the preceding sections with specific reference to the problem of rarefied particle motions on hillslopes.

Here we consider three elements of the formulation presented in Furbish et al. (2021a, b, c) to highlight the purpose and merits of a probabilistic description of particle motions and disentrainment. The first concerns our treatment of the extraction of kinetic energy of a particle during particle–surface collisions as a random variable versus appealing to the deterministic idea of fixed coefficients of restitution. The second concerns our use of the Fokker–Planck equation to describe the changing energy states of the particles during their downslope motions, leading to deposition, versus considering only average particle energy conditions. The third concerns our efforts to demonstrate that the generalized Pareto distribution in this problem is a maximum entropy distribution. The objective is to highlight the mechanistic yet probabilistic nature of the analyses.

We start with some background. In classical statistical mechanics the starting point for describing the motions and collective behavior of particles undergoing conservative collisions is the Boltzmann equation. This equation describes the evolution of the joint probability density function of particle positions and velocities in relation to the forces acting on the particles. Depending on the formulation of particle collisions (e.g., Chapman–Enskog theory), the Boltzmann equation leads to the Navier–Stokes equation in the continuum limit of vanishing Knudsen number. For dissipative collisions in granular materials, however, one must incorporate effects of energy losses during particle collisions. In pioneering work, Haff (1983) formulated the hydrodynamic analogue of the Navier–Stokes equations for granular flows. In this formulation he envisioned simple inelastic collisions and appealed to the normal coefficient of restitution to characterize energy losses during collisions, neglecting the details of particle collisions in the scalar treatment. The thermodynamic temperature is replaced with the granular temperature, and the hydrodynamic equations are supplemented with the mechanical energy equation in order to characterize granular flow behavior. One of the key outcomes of this work is Haff's cooling law (Brito and Ernst, 1998; Nie et al., 2002; Brilliantov and Pöschel, 2004; Dominguez and Zenit, 2007; Brilliantov et al., 2018; Yu et al., 2020), which predicts that when the external source of energy is removed the granular temperature decays with time as

Now consider particle–surface collisions in the rarefied particle motion problem. Of interest are downslope motions and travel distances. The energy balance described in Furbish et al. (2021a) thus focuses on the particle kinetic energy state

The quantity

With this description of particle energy extraction in place, it then becomes straightforward to characterize the number of particle–surface collisions per unit distance and in turn the collisional energy loss per unit distance in relation to the surface-parallel velocity

What might an alternative approach look like? One possibility involves using discrete element methods (DEMs) to directly mimic particle motions on rough surfaces, extracting information from the simulations to characterize particle–surface collisions and energy extraction. Such simulations essentially represent numerical analogues of physical experiments. An obvious advantage is speed in examining different conditions, for example, surface slopes, roughness configurations and particle sizes, using the motions of a great number of particles versus the relatively small numbers of particles used in experiments. Add to this the capability to readily extract information on details of motions and collisions that are not accessible in physical experiments, except possibly via high-speed imaging. Disadvantages include the difficulty of mimicking irregular particle shapes and creating realistic surface-roughness textures. (And let us note that informed use of DEMs, for example LAMMPS and LIGGGHTS, is not a plug-and-play endeavor.) Nonetheless, one could potentially learn much in a generic sense about particle–surface interactions from DEMs, particularly if conducted in concert with carefully designed physical experiments. This includes assessing how sensitive macroscopic measures of particle motions (e.g., travel distances) are to variations in controlling factors – for example, particle shape and roughness texture – as these factors are varied. But rather than imagining the need to mimic all details of realistic conditions associated with a hillslope surface prototype, simulations should be designed to examine particle–surface interactions in a manner that allows for generalization of elements of collisional friction.

Herein arises a need for pause in using DEMs. Namely, these methods can quickly generate enormous amounts of numerical information on the details of particle motions and collisions. At risk is using a big numerical hammer to pound on the problem, mimicking particle motions without regard to elucidating general underlying principles of particle behavior, defaulting to descriptions of outcomes without gaining a deeper understanding of the systematics producing them – for example, without learning the mechanical basis of the deposition length scale that sets the pattern of deposition (Sect. 4.1.2) or without learning the form of the distribution of travel distances, the mechanical basis of its parametric values, or its maximum entropy properties (Sect. 4.1.3). This speaks to the relevance of the cliché that much of the power of numerical simulations resides in their use in concert with theory and experiments (Emanuel, 2020). Indeed, the success of numerical simulations that have revolutionized the study of granular gas dynamics grows directly from the grounding of this topic in kinetic and statistical mechanics theory (e.g., Brilliantov and Pöschel, 2004), which both motivates and guides the design of numerical treatments of granular gases. Similar remarks pertain to parallel efforts focused on the behavior of relatively dense granular materials.

Consider a second element of the formulation, which concerns our use of the Fokker–Planck equation to describe the changing energy states of the particles during their downslope motions, leading to deposition, versus considering only average particle energy conditions. We start with some background.

The Fokker–Planck equation represents a triumph of classical statistical mechanics. Although originally developed to describe the time evolution of the distribution of velocities of particles subjected to viscous forces and random forces associated with collisions, the name of this equation now is more generally associated with other observable quantities whose distributions evolve according to an equation having the same form. For example, with reference to the distribution of particle positions rather than velocities, the Fokker–Planck equation historically is referred to as the Smoluchowski equation. It also is referred to as the Kolmogorov forward equation in the context of Markov processes. Although the Fokker–Planck equation can be derived in several ways, perhaps the most general treatment starts with a master equation, a general probabilistic expression of conservation of probability associated with observable states. Like the Fokker–Planck equation, there are several versions of the master equation, sometimes referred to as the Chapman–Kolmogorov equation, depending on the field and application. Here we focus on a continuous form of the master equation as described by Chandrasekhar (1943) and Risken (1984). We start with two familiar examples to develop the essential concepts before turning to the unfamiliar problem of rarefied particle motions addressed in Furbish et al. (2021a). Our objective is to illustrate the statistical mechanics framework of the analysis.

Let

Now, assuming the density

As a point of reference, the Fokker–Planck equation also can be obtained from the Langevin equation, a stochastic differential equation originally used to describe the behavior of Brownian particles. Moreover, in the specific context of bed load sediment transport, Ancey and Heyman (2014) and Heyman et al. (2014) show that for a birth–death Markov process describing the number of active particles within a streambed control volume, the evolution of the probability distribution of this number can be described as a Fokker–Planck equation obtained from the master equation representing transitions among number states. This makes use of the idea (Gardiner, 1983) that the probability distribution of number states can be represented as a mixture of Poisson distributions with varying rates and represents an alternative to the Kramers–Moyal expansion for conditions involving small particle numbers with large relative fluctuations.

Here is a particularly important sidebar. As probabilistic expressions the master equation and the Fokker–Planck equation are entirely agnostic to continuum versus rarefied conditions; they are equally applicable to both. If in an individual system (realization) the continuum hypothesis is satisfied – a condition that is independent of the probabilistic basis of the master equation or the Fokker–Planck equation – then the probabilistic formulation based on ensemble-expected conditions and its continuum counterpart are essentially one and the same. If, however, the continuum hypothesis is not satisfied, then one must appeal to a probabilistic formulation of ensemble-expected conditions (in order to justify the use of continuously differentiable equations), with the proviso that any prediction of the behavior of an individual (rarefied) system is probabilistic in nature. In other words, using the Fokker–Planck equation, Eq. (16), to describe the behavior of a system under rarefied conditions is in effect the same as using a continuum-like equation, where “continuum-like” means continuously differentiable, not that the particles behave as a continuum. Because of the significance of these points, we have reproduced in Appendix C key material from Appendix A presented in Furbish et al. (2018c). We return to the idea of rarefied versus continuum conditions in Sect. 4.2.

Now let

As a point of reference, Furbish et al. (2012b) start with the Fokker–Planck equation given by Eq. (17) to examine the statistical equilibrium distribution of the streamwise velocities of bed load particles. Wu et al. (2020) elaborate this idea by demonstrating that a large proportion of long particle hops experiencing relatively large velocities “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.”

With these ideas in place, we now turn to the problem of rarefied particle motions on rough hillslopes. Here is where we highlight the idea that the endeavor is not simply about adopting theory or methods “off the shelf” but rather involves appealing to the style of thinking of statistical mechanics while tailoring the description to the process.

Because particle motions in this problem are highly rarefied and intermittent, we appeal to the idea of a cohort of particles – a Gibbs-like ensemble of systems, each containing one particle that is subjected to the same physics during downslope motion (Appendix B in Furbish et al., 2021a). Moreover, because the formulation is centered on particle travel distances and therefore on deposition over space rather than time, it aims at describing the evolution of the ensemble distribution of particle energy states with respect to downslope position

With this description of the spatial evolution of the density

Note that the formulation does not involve specifying a threshold energy for deposition … Whereas low-energy particles are on average more likely to become disentrained than are high-energy particles, a set of particles with precisely the same low energy will for probabilistic reasons not be disentrained simultaneously. Each particle experiences a unique set of conditions that disentrain it, and because of this uniqueness of conditions a particle with energy below an arbitrarily assigned threshold can with finite probability be gravitationally reheated to a higher-energy state. For given particle and surface roughness conditions, the formulation treats this aspect of disentrainment as a probabilistic process … [in relation] to the distribution of particle energy states and the probabilistically expected extraction of energy during collisions.

What might an alternative formulation of deposition look like? (Having approached the problem as above, we admit that a description of this sort represents a straw person to criticize. Nonetheless, we also can admit that our early efforts involved thinking of alternatives, so this criticism is not blind.) Suppose we start with the assumption that the deposition rate is inversely proportional to the average particle kinetic energy

Our third example highlighting the purpose and merits of a probabilistic description of particle motions and transport is centered on demonstrating that the generalized Pareto distribution is a maximum entropy distribution. We again start with some background, briefly outlining the origin of the idea of a maximum entropy distribution.

The canonical example of a maximum entropy distribution is the Boltzmann distribution of the energy states

all [micro]states of the assembly are embraced – without overlapping – by the classes [macrostates] described by all different admissible sets of numbers

The present method [of the most probable distribution] admits that, on account of the enormous largeness of the number

The procedure thus amounts to choosing the macrostate containing the greatest number of microstates, each consistent with fixed

From Furbish et al. (2021c),

Jaynes (1957a, 1957b) elaborated the significance of the fact that the Gibbs entropy in statistical mechanics and the Shannon entropy in information theory are essentially one and the same, differing only by a constant. This similarity inspired Jaynes to champion the use of a maximum entropy criterion in choosing a probability distribution, leading to what is now known as the maximum entropy method … [W]hether viewed as a method of statistical mechanics or as one of inferential statistics … it provides an unbiased choice of a distribution by honoring only what is known mechanically about a system. That is, this unbiased choice is a maximally noncommittal choice that is faithful to what we do not know; it is therefore the most reasonable choice in the absence of additional information …

Within this context, there are three notable elements in our effort (Furbish et al., 2021c) to demonstrate that the generalized Pareto distribution as applied to the rarefied motion problem is a maximum entropy distribution. First, in this work we noted that “constraints imposed on the system normally translate to constraints imposed on the moments of the distribution. … [in which] case the method leads to a distribution that is among the exponential family (e.g., exponential, Gaussian). However, applications of the maximum entropy method to non-exponential distributions, including heavy-tailed distributions, are of particular interest in many problems (Peterson et al., 2013).” Moreover, recall that the generalized Pareto distribution has three forms: it is bounded for shape parameter

Second, the versatile form of the generalized Pareto distribution is rather unusual. Numerous well-known distributions of course take the form of a related distribution for certain parametric values. Nonetheless, the generalized Pareto distribution is distinctive in that it has a bounded form (

This result also adds clarity to the idea of nonlocal versus local transport (Metzler and Klafter, 2000; Schumer et al., 2009; Foufoula-Georgiou et al., 2010; Furbish and Roering, 2013). In studies of tracer particle transport, and setting aside the effects of particle rest times, local behavior is associated with a light-tailed distribution of particle displacements during a small interval

Third, focusing on the second part of the quotation above, the maximum entropy method reminds us of the value of the principle of parsimony – appealing to the simplest explanation consistent with available evidence – in the presence of uncertainty. Boltzmann did not know a priori the distribution of gas particle energy states, Eq. (22); he imposed only the constraints of a fixed number of particles and a fixed total energy. The maximum entropy derivation thus honored his understanding of the system, but no more. In effect the derived distribution of energy states – including the foundational assumption that each accessible microstate is equally probable – became a hypothesis to be tested against experimental observations (Tolman, 1938). With respect to applications of the maximum entropy method to sediment particle motions, we “highlight the fact that a distribution thus chosen is not necessarily the `correct' distribution (Furbish et al., 2016a) … [I]t is the most reasonable choice in the absence of additional information … [and in] this sense the maximum entropy method is a formal application of Occam's razor” (Furbish et al., 2021c). We therefore suggest that this represents one viable element of a strategy to deepen our mechanical understanding of attributes of particle motions that we observe, measure and describe statistically. As noted by Ancey (2020b) in relation to bed load transport, “One strength of entropy-based methods is their use of the physical information conveyed by data, thereby enforcing physical consistency … [opening] new avenues of research combining statistical information and physics-based models.” On this point we note that a distribution selected according to a maximum entropy criterion may serve as an ideal prior hypothesis in subsequent analysis, including Bayesian analysis (Jaynes, 1988).

Perhaps it is obvious that in this problem a description of the physics of particle motions cannot meaningfully start from the idea of continuum behavior. Particle motions are patchy and highly intermittent, and in most settings these motions are far from conditions that could be considered continuum-like granular flows. Particle behavior is dominated by particle–surface interactions rather than particle–particle interactions, and the conventional idea of appealing to a Knudsen number to ascertain continuum behavior is irrelevant. Moreover, aside from descriptions of the physics of particle motions, in the absence of continuum conditions we cannot justifiably appeal to familiar continuum-like definitions of the particle flux and its divergence that are based on the assumption that particle number densities and locally averaged velocities are well-defined quantities that vary smoothly over space and time. Yet the particle flux and its divergence are of particular interest in many problems, and we therefore turn our focus to a thorough description of these quantities.

As noted above, precise definitions of the sediment particle flux and its divergence do not assume continuum conditions at the outset (Ancey, 2010; Furbish et al., 2012a, 2016b, 2017; Ancey and Pascal, 2020). For rarefied conditions these definitions are translated into probabilistic expressions, of which the entrainment forms of the flux and the Exner equation, Eqs. (1) and (2), are examples. Of concern, then, is the meaning and use of continuously differentiable functions in these nonlocal expressions, namely, the entrainment rate

Consider the nonlocal expressions, Eqs. (1) and (2). For simplicity of illustration we focus on a single particle size and rewrite these expressions as follows. Let

To illustrate these points, here we consider an idealized situation in which the entrainment rate

The idea of a line source of sediment particles delivered to a hillslope is embodied in the experimental and field-based work of Kirkby and Statham (1975) and Statham (1976) concerning the motions and downslope sorting of particles on scree slopes. Here we consider a simple version of this problem.

These results illustrate that the number

Plot of 10 realizations (colored lines) of normalized number

These plots may be interpreted several ways. First, for a specified delivery rate

The plots in Fig. 3 also may be reinterpreted in terms of variations in the form of the generalized Pareto distribution with respect to hillslope positions

In turn, we compute the normalized time-averaged particle number flux as

Plot of 10 realizations (colored lines) of normalized time-averaged particle number flux

Ancey and Pascal (2020) examine the more general question of estimating the time-averaged flux associated with a Poisson process (compare their Fig. 2 with our Fig. 3). Within the context of measurements of bed load sediment transport, they show how the variability in estimates of the time-averaged flux varies with the measurement interval, and they present a re-sampling (bootstrap) protocol for assessing how the variance of the flux varies with the sampling interval based on an individual realization. As noted below, however, we rarely if ever have time series needed to support this type of analysis when describing slow systems.

With this example in place we offer an explicit definition of the idea of ensemble-expected conditions for a Poisson process. The word “ensemble” refers to a great number

As a point of reference, Benjamin et al. (2020) provide an assessment of efforts to observe and measure rockfall events contributing to cliff erosion and thus to downslope delivery of particles. The frequency and magnitude of these events may vary widely, from the chronic activity of small rockfall events to large infrequent events, depending on the geological and environmental factors that influence the mechanisms of weathering and failure (Luckman, 2013; Strunden et al., 2015; Mair et al., 2020). The frequency of occurrence of rockfall volume typically varies as an approximate inverse power function of volume, where the specific relation depends on the spatial coverage and temporal duration of the data set (Benjamin et al., 2020). Rockfall volumes do not translate directly to particle numbers, both of which are influenced by the geometry of cliff rock fracturing and fragmentation (Domokos et al., 2020; Verdian et al., 2020), and impact shattering (Luckman, 2013). Nonetheless, these observations point to the inherent stochasticity of rockfall over many scales, including variations in intermittency with time (Sect. 4.3.2).

Like a Poisson particle delivery rate (Fig. 3), the number of particles

Plot of 10 realizations of normalized number

For a Poisson process, the deposition events

The idea of distributed entrainment is embodied in the work of Doane (2018) and Doane et al. (2018) concerning nonlocal sediment transport. This work involves numerical simulations of the time evolution of the profiles of steep lateral moraines in the Sierra Nevada, California, for comparison with field-based measurements. It examines entrainment that occurs over the entire moraine profile due to disturbances and the role of vegetation in sediment capacitance – the capture, storage and release of sediment (Furbish et al., 2009a; Lamb et al., 2011, 2013; DiBiase and Lamb, 2013; Doane, 2018; Doane et al., 2018). Here we consider a simple version of this problem involving uniformly random entrainment.

For a uniformly random entrainment rate

If the integral in Eq. (28) does not converge – that is, the mean travel distance is undefined – then the expected particle flux is undefined. This coincides with a shape parameter

As in the preceding example, if the integral in Eq. (28) does not converge – that is, the mean travel distance is undefined – then the expected particle flux is undefined. The implications of this are similar to those described above regarding Poisson entrainment.

Alternatively, for positions

One might anticipate that the balance between expected immigration and emigration rates involving a uniform expected flux, or between the expected deposition and entrainment rates, yields a particle number

With

There is little evidence that numbers

Returning to transport on hillslopes, stabilizing effects may involve local changes in the entrainment rate due to effects of unstable particle configurations, collective entrainment of surface particles by moving particles, and preferential “capture” of moving particles within local low spots or by roughness elements (Furbish et al., 2021b; Roth et al., 2020). Note that the rules of deposition in the particle-based transport model of Tucker and Bradley (2010) inherently provide this sort of stabilizing effect. Moreover, although attention has been given to the role of vegetation in sediment capacitance (Furbish et al., 2010; Lamb et al., 2011, 2013; DiBiase and Lamb, 2013; Doane, 2018; Doane et al., 2018), this topic otherwise is largely unexplored in relation to modulating rates of entrainment and deposition over long timescales. In addition we may imagine a configuration involving (in a Fourier sense) a small-amplitude sinusoidal variation in surface elevation or roughness. The effect of this – including preferential deposition at certain locations – almost certainly would influence the behavior of particles whose motions start upslope, thereby leading to a distribution of travel distances that deviates from an idealized form associated with uniform conditions. We may imagine similar effects on planar surfaces with nominally homogeneous roughness, but with local variations in roughness at the particle and slightly larger scale. However, whether these local effects could be distinguished from the inherent randomness of deposition is an open question. Note also that other processes may operate on hillslope surfaces such that stabilizing – or “smoothing” – influences do not need to be related just to rarefied particle motions as envisioned above. As an unusual example, “the impacts by small distal ejecta fragments. … is the largest contributor to the diffusive [topographic] degradation which controls the equilibrium [size–frequency distribution] of small craters” of the lunar maria (Minton et al., 2019). More generally, the formalism of generalized elastic models used to describe the macroscopic dynamics of fluctuating surfaces due to the competition between processes of surface roughening and relaxation (Pelletier and Turcotte, 1997; Turcotte, 2007) is now being extended to erosional landscapes (Schumer et al., 2017). Whether involving stabilizing effects or not, fluctuations in entrainment and deposition and accompanying variations in the land-surface state about expected conditions are

Here is the key lesson. In the presence of noise-driven processes with rarefied conditions, one must be cautious about predicting behavior in response to fixed continuum-like rates that do not acknowledge noise effects. Individual realizations associated with these effects can involve rich behavior that is not anticipated from a simple deterministic perspective.

Here we consider uncertainty associated with rarefied particle motions on hillslopes viewed as a “slow” system, where changes in hillslope configuration are largely imperceptible over the human timescale (Sect. 3.3). We highlight results from above, that with rarefied transport conditions our descriptions of the particle flux and its divergence pertain to ensemble conditions involving a distribution of possible outcomes, each realization being compatible with the controlling factors. When these factors change over time, individual outcomes reflect a legacy of earlier conditions that is influenced by the rate of change in the controlling factors relative to the intermittency of particle motions. The implication of this result together with preceding material is that landform configurations reflect an inherent variability that is just as important as the expected (average) conditions in characterizing system behavior.

We start by returning to a key starting point described in Sect. 4.2. Namely, despite the continuous forms of the entrainment rate

The simple Poisson processes described in the examples above suffice to illustrate the consequences of rarefied conditions, where intermittency and patchiness add variability about expected conditions. Relative to the expected conditions, this variability may be large when viewed over small timescales. Only in the limit of a large number of particles with averaging over long timescales do predictions of the flux and its divergence approach expected (deterministic-like) values.

For any realization, the flux

More generally, Eqs. (1) and (2) are probabilistic algorithms in which

As outlined in Sect. 4.2.3, however, additional factors may provide stabilizing effects. Focusing on the rate

The analyses above focus on one-dimensional downslope transport. For completeness we note that the particle flux and its divergence more generally involve two-dimensional transport. For example, Williams and Furbish (2021) consider elements of the two-dimensional forms of Eqs. (1) and (2). They show how transverse diffusion of particles arises from particle–surface collisions during downslope travel and how transverse motions influence the downslope particle flux. Clarifying the consequences of two-dimensional rarefied particle transport remains an interesting, open topic.

The factors that control particle delivery rates and entrainment, as well as the conditions that influence particle motions and deposition, change with time at different scales. For example, particle entrainment and surface-roughness texture associated with vegetal sediment capacitance may vary at fire recurrence timescales. Over longer timescales, continuing entrainment with downslope particle motions and deposition may contribute to changes in surface roughness and local land-surface slopes, thus changing the distribution of particle travel distances. At climate-change timescales, particle delivery rates to scree slopes may vary in relation to changing weathering rates and particle release from bedrock. Thus, we must acknowledge an adaptable view of particle delivery and entrainment. Namely, an intermittent “event” during an episode of fire might be represented by the release of sediment from a vegetation capacitor. At the climate-change timescale, in contrast, an “event” may be viewed as consisting of the entirety of the release (or entrainment) of sediment associated with a fire and the period of post-fire recovery to vegetated conditions. Here we consider one element of the consequences of changes in the factors controlling particle motions, in particular the possible mismatch between the timescale over which expected rates of delivery or entrainment change relative to the scale of intermittency in these rates.

Recall that the time-averaged flux eventually converges to the ensemble-expected value with increasing elapsed time and that the rate of convergence decreases with increasing intermittency in the delivery or entrainment of particles (Sect. 4.2.2). However, over intervals that are much shorter than the time required for convergence, the time-averaged flux in an individual realization may differ significantly from the ensemble-averaged value. This is the same as saying that the number of particles moving past a position

Consider for illustration the situation where particles are delivered intermittently to the top of a hillslope as a line source (Sect. 4.2.2). For illustration we specify the ensemble-expected rate

Plot of 10 realizations (colored lines) of normalized number

Specifically, and with reference to Fig. 6d, if by chance during the early part of the series a relatively large number of events occur, then this preconditions the total number

Consider the slopes of the individual realizations in Fig. 6 estimated by projecting lines of varying duration through different parts of the stepped curves. These slopes represent estimates of the particle flux. With increasingly rarefied conditions, notably when the expected rate

In keeping with our philosophical objectives, we begin this section at a high level. This is to reinforce our view that it is important for growing efforts centered on probabilistic descriptions of sediment transport to include the philosophical underpinnings of this work within the conversation.

In 1943 Kurt Lewin first offered his oft quoted maxim that “there is nothing so practical as a good theory” (Lewin, 1943) for providing a framework to guide analyses of complex systems. This basic, lasting principle appears to resonate in many fields, particularly the social sciences (McCain, 2016). More recently, Deutsch (2009, 2011) built from ideas of Karl Popper to strongly argue for the essential guiding role of theory in the development of scientific explanation – that compelling explanations of natural phenomena are “theory laden”. He forcefully rejects the idea of empiricism, that observation of the world alone can suggest which ideas to adopt. In addition, Eugene Wigner provides an important elaboration of Lewin's maxim for the natural sciences. In his classic essay entitled “The unreasonable effectiveness of mathematics in the natural sciences”, Wigner (1960) notes that the triumph of physics resides in principles of invariance (Wigner, 1985) – that the laws of nature are invariant with any suitable transformation of space or time, thereby rendering them independent of initial conditions, position and history yet holding true for all time. He suggests that it is precisely the existence of this invariance that gives us the confidence and inspiration – what he calls the “empirical law of epistemology” – for continuing the endeavor of discovery with growing complexity and uncertainty. Without this invariance, we would lose trust in our use of the laws of physics in different problems and settings – just as we would lose confidence and interest in playing the game of chess if the rules continually changed from one match to the next, precluding any gain in expertise from experience with fixed rules.

Turning these ideas toward sediment systems, we suggest that the statistical mechanics framework outlined herein offers a compelling strategy for examining particle motions and transport, particularly with rarefied conditions and in view of the uncertainty that goes with describing slow systems. This framework has two key elements that embody the points above. First, this framework is grounded in principles and methods for dealing with particle systems, continuum and rarefied, that have been rigorously scrutinized for more than a century. Its principles rest on Wigner's views of invariance, and its familiarity lends confidence for investing trust in its mechanical basis when examining unfamiliar problems outside of classical statistical mechanics. Second, this framework embraces uncertainty at the outset in its use of probability. It offers established ways to formulate expressions of conservation, clear rules for counting and averaging particle states, and the foundational concept of a Gibbs ensemble (Furbish et al., 2012a; Furbish and Schmeeckle, 2013; Bi et al., 2015; Furbish et al., 2018c). Again, we are inspired to invest trust in this formalism applied to unfamiliar systems. In particular, this framework points us in the right direction for examining the physics of rarefied particle motions on hillslopes, wherein we see the behavior of the particle system precisely for what it is – an unusual granular gas. The effort then consists of elucidating a micro-view of the mechanical behavior of the particles during their downslope motions, which, when described probabilistically, leads to a macroscopic view of their collective (emergent) behavior.

The theoretical analysis of particle motions involves threading together elements of statistical mechanics, concepts from granular gas theory, particle collision mechanics, and probability distribution theory (Furbish et al., 2021a, c). Importantly, the analysis leans on the style of thinking of statistical mechanics while recognizing – as a delightfully challenging twist – that it is not about simply adopting, off the shelf, theory and methods from this field. Instead, the work must be tailored to the transport process and scale of interest.

Each of the examples used in Sect. 4.1 to highlight the merits of a probabilistic description of particle motions and disentrainment – particle energy extraction, energy states and the Fokker–Planck equation, and the generalized Pareto distribution as a maximum entropy distribution – represents a direct extension of established concepts in statistical mechanics as applied to both ordinary gases and granular gases. The analyses are not as straightforward as describing the behavior of ideal gas particle systems. Nonetheless, they nicely illustrate the transferability of basic principles, for example, the treatment of dissipative collisions as a random process, the value of appealing to a Gibbs ensemble as applied to a cohort of particles, and the use of an energetic cost to constrain the entropy maximization method. Thus, these examples illustrate elements of a coherent statistical mechanics framework for describing sediment particle motions – that a mechanistic yet probabilistic analysis is possible. Moreover, the maximum entropy analysis specifically offers clarity on particle behavior that is not otherwise accessible. Namely, that all three forms of the generalized Pareto distribution are constrained in the same manner demonstrates that nothing special or unusual changes in the physics of disentrainment in the transition from the bounded form to the heavy-tailed form of the distribution in crossing isothermal conditions. The analyses thus rest on a solid foundation of statistical mechanics. Nonetheless, it is essential that these results be challenged, and, if necessary, culled and replaced with fresh ideas.

With respect to consequences of rarefied versus continuum conditions, herein we focused on descriptions of the particle flux and its divergence (Sect. 4.2). Inasmuch as the particle delivery rate (as a line source) or the entrainment rate can be approximated as a Poisson or intermittent Poisson-like process, then the analysis clearly points to the idea that the flux or its divergence involves a distribution of possible outcomes, not just a single expected value – an idea that is decidedly different from conventional continuum descriptions of these quantities. Note that the descriptions of the flux and its divergence do not depend on the results described above concerning the physics of particle motions. Indeed, the probabilistic nonlocal expressions of the flux and its divergence, Eqs. (1) and (2), are independent of the form of the probability density

Here we reinforce the idea that Eqs. (1) and (2) are probabilistic algorithms. For rarefied conditions the entrainment rate

As written, then, the flux

The examples involving Poisson or intermittent Poisson-like processes described in Sect. 4.2 and 4.3 highlight the inherent variability that goes with noise-driven processes and point to an important consideration in interpreting landform configurations. Namely, for rarefied transport conditions a landform at any instant represents one of many possible realizations

This perspective also induces us, while acknowledging consequences of the noisiness of rarefied systems, to examine the dynamics of competition between roughening and smoothing processes (Schumer et al., 2017). As mentioned in Sect. 4.2.3, this may involve collective entrainment, the sediment capacitance of vegetation and other roughness elements, preferential entrainment and deposition in relation to surface geometry and roughness, effects of particle size sorting, or “smoothing” processes that are not related to rarefied particle motions per se. We suggest that there is value in taking cues from current work on noise-driven bed load transport, including the coupling between moving particles and the streambed state (Ancey et al., 2008; Ancey and Heyman, 2014; Pierce and Hassan, 2020a). Whereas we have focused on local consequences of noisy delivery rates and entrainment, there is a need to systematically examine land-surface behavior in relation to rarefied particle transport.

In their examination of experimental time series of bed load flux, Ancey and Pascal (2020) provide an interesting lesson for considering slow systems. They show that for a noise-driven process the time-averaged flux calculated from an individual realization (Figs. 3 and 4) may differ significantly from the (known) sediment feed rate. We can imagine having information, for example from a sediment deposit, that allows us to estimate the time-averaged sediment delivery rate associated with a slow, noisy system. However, this estimated rate, representing an individual realization, may not coincide with the ensemble-expected rate associated with the extant controlling conditions. In the absence of a high-fidelity time series of the delivery rate analyzed by re-sampling methods (Ancey and Pascal, 2020), the result is unavoidable uncertainty in this averaged rate.

We end with an anecdote to reinforce the starting point of this section. Last fall we wandered into Guilherme Gualda's graduate class on phase transformations in magmatic systems and, to our delight, discovered on the chalkboard a derivation of the Boltzmann distribution of the energy states of atoms in a crystal lattice – complete with a pictorial rendering of the energy macrostates and microstates of a simple example system. The derivation continued with a description of the particle diffusion coefficient containing the Gibbs activation energy, as a direct consequence of the Boltzmann distribution, thence to the Arrhenius equation. We then enjoyed the discussion surrounding the idea that, in practice, one would experimentally determine the diffusion coefficient for a real (i.e., not ideal) system rather than predict it from the statistical mechanics theory for specified atomic constituents and thermodynamic conditions. The students were quick! “So what is the value of the theory?” “Aha!”, Gualda responded with delight, “the theory provides an unambiguous framework for interpreting our experimental data in view of the uncertainties of real systems! For example, in addition to providing a coherent, testable explanation of the phenomenon, the theory points to the appropriate functional form – the logical basis – of the expected relationship in curve fitting. This in turn provides the basis of error assessment – either by classic propagation using the calculus or by Monte Carlo methods – which is particularly valuable given that experimental data often are sparse and of variable quality. And it assigns clear meaning to estimated parameters for comparison with other work.”

It is such a lovely, simple lesson: “There is nothing so practical as a good theory …” and we would add in the case of sediment systems, “… that pays as much attention to fluctuations as it does to expected (mean) values.”

Here we offer a partial list of papers (81) representing recent work on probabilistic elements of sediment motions and transport in five topical areas. These papers contain numerous references to related work, including early probabilistic descriptions of transport and related material in the mathematics and physics literature. Under each heading we list the papers in the order of their appearance.

Although these papers are just a sample, the relative numbers in the five areas accurately reflect the unevenness of efforts among these areas. The notable difference in efforts pertaining to rivers versus hillslopes is in part a direct reflection of the differences in our ability to observe and measure the transport processes. As noted in the main text, we know far more about bed load sediment transport in shear flows based on flume experiments than, say, soil particle transport and mixing associated with bioturbation and granular creep.

Ancey et al. (2006),

Ancey et al. (2008),

Ancey (2010),

Lajeunesse et al. (2010),

Furbish et al. (2012a),

Furbish et al. (2012b),

Furbish et al. (2012c),

Roseberry et al. (2012),

Campagnol et al. (2013),

Furbish and Schmeeckle (2013),

Ancey and Heyman (2014),

Heyman (2014),

Heyman et al. (2014),

Seizilles et al. (2014),

Ancey et al. (2015),

Fathel et al. (2015),

Bohorquez and Ancey (2016),

Fan et al. (2016),

Fathel (2016),

Fathel et al. (2016),

Furbish et al. (2016a),

Furbish et al. (2016b),

Heyman et al. (2016),

Furbish et al. (2017),

Salevan et al. (2017),

Ballio et al. (2018),

Dhont and Ancey (2018),

Lee and Jerolmack (2018),

Ballio et al. (2019),

Ancey (2020a),

Ancey (2020b),

Ancey and Pascal (2020),

Ashley et al. (2020),

Chartrand and Furbish (2021),

Pierce and Hassan (2020a),

Wu et al. (2020).

Hassan and Church (1991),

Ferguson and Wathen (1998),

Parker et al. (2000),

Ferguson and Hoey (2002),

Ferguson et al. (2002),

Nikora et al. (2002),

Wong et al. (2007),

Schumer et al. (2009),

Bradley et al. (2010),

Ganti et al. (2010),

Hill et al. (2010),

Martin et al. (2012),

Hassan et al. (2013),

Phillips et al. (2013),

Voepel et al. (2013),

Martin et al. (2014),

Pelosi et al. (2014),

Phillips and Jerolmack (2014),

Fathel et al. (2016),

Bradley (2017),

Hassan and Bradley (2017),

Liu et al. (2019),

Pierce and Hassan (2020b).

Foufoula-Georgiou et al. (2010),

Furbish and Haff (2010),

Gabet and Mendoza (2012),

Furbish and Roering (2013),

Doane (2018),

Doane et al. (2018),

Doane et al. (2019),

Furbish et al. (2021a),

Furbish et al. (2021b),

Furbish et al. (2021c),

Roth et al. (2020),

Williams and Furbish (2021).

Furbish et al. (2009b),

Furbish et al. (2018a),

Furbish et al. (2018b),

Furbish et al. (2018c),

Gray et al. (2020).

Furbish et al. (2007),

Furbish et al. (2009a),

Dunne et al. (2010),

Furbish et al. (2016b),

Sochan et al. (2019).

Consider the entrainment forms of the flux and the Exner equation, Eqs. (1) and (2). Here we show that Eq. (2) is consistent with the divergence expressed as

We start by writing

Consider an alternative formulation. With

The material in this appendix is mostly extracted directly from Appendix A in Furbish et al. (2018c). Our aim is to further illustrate the significance of rarefied versus continuum conditions, and the interpretation of the Fokker–Planck equation applied to these conditions. We focus on the familiar example of Brownian motion, the initial formal description of which is separately attributable to Einstein (1905) and von Smoluchowski (1906). For additional background, Schumer et al. (2009) provide a particularly clear description of the Lagrangian perspective of particle motions and its relation to the Eulerian perspective of particle behavior as embodied in the Fokker–Planck equation.

Plot of coordinate position

Histograms of particle positions

With reference to Fig. C1,
let

Let us now imagine an arbitrarily great number

Now select a system with a modest number

Let us now consider a great number

To complete the picture, suppose that the

Histogram of particle positions

With respect to developments in the text, the Fokker–Planck equation describes the time evolution of the probability density

Here we provide an explicit definition of an ensemble-expected value or state and its relation to a time-averaged value. Recall that the Poisson distribution, Eq. (25), describes the probability that

Let us imagine, as did Gibbs (1902), a great number

Now, at any time

The essential idea is this. For any individual system there is one possible outcome

Consider the situation where particles are delivered as a line source (

Whereas the ensemble distribution represented by Eq. (D10) evolves with time

Plot of time evolution of

The data plotted in Fig. 2 are available from sources described in Furbish et al. (2021b).

The reported work represents an intellectual co-conspiracy between the authors growing from the PhD work of THD. DJF wrote the paper with critical review and input from THD.

The authors declare that they have no conflict of interest.

We appreciate continuing discussions with Peter Haff and Sarah Williams concerning descriptions of Earth-surface systems. Shawn Chartrand and Tom Dunne offered useful reactions to an earlier draft. We appreciate reviews of our work provided by Peter Haff, Joris Heyman and an anonymous referee.

This research has been supported by the National Science Foundation (grant nos. EAR-1420831 and EAR-1735992).

This paper was edited by Eric Lajeunesse and reviewed by Joris Heyman and one anonymous referee.