My research and writing efforts are centered on the statistical physics of sediment particle motions and transport in Earth surface systems, including hillslopes and rivers. I am increasingly interested in the epistemology of Earth surface science in view of its loose connection to critical rationalism and its tepid relationship with probability. I formerly taught courses on transport processes in Earth and environmental systems, fluid dynamics, probability and statistics, and Earth surface processes.<\/p>\n
Bluesky: @davidjonfurbish.bsky.social
\nMastodon: @PhilSciProf@mastodon.online<\/p>\n
Sediment transport research is now steadily moving beyond its mostly semi-empirical engineering legacy. Here is key perspective on framing the topic in a manner that refocuses on clarifying the essential physics of transport at the outset.<\/p>\n
The literature now clearly points to the idea that, possibly except for dense granular creep\/flow, sediment transport on hillslopes and in rivers occurs under rarefied conditions. Transport does not satisfy the continuum hypothesis as is conventionally assumed or asserted. From a theoretical point of view the clearest defensible starting point is to view rarefied transport in terms of the phase transitions involved. To wit:<\/p>\n
In its barest essence, rarefied sediment transport consists of phase transitions of a granular material: the sublimation (entrainment) and deposition (disentrainment) of the material, where intervening particle motions are directionally biased owing to gravity or fluid forces. This phenomenon occurs at the interface between a granular solid that contains interstitial fluid, and a fluid that intermittently contains the sublimated solid, where individual particles reside only briefly in a state of motion close to the interface. In this view of things the physics of the phase transitions becomes central in describing transport, where the stochastic process of entrainment regulates the availability of particles and the stochastic process of disentrainment modulates the bias of the particle motions via particle–surface interactions. Importantly, these phase transitions do not constitute a continuum behavior.<\/p>\n
This point of view acknowledges all attributes of particle motions, start to stop. It highlights that any rigorous kinematic definitions of the particle flux and its divergence (e.g. the particle activity forms and the entrainment forms of the flux and the Exner equation) necessarily start with a (probabilistic) nonlocal formulation of particle motions. Moreover, this point of view highlights the statistical mechanics of particle–surface interactions in explanations of motions at small scales, and it points to the dominant role of entrainment (i.e. particle availability) in setting transport rates when viewed at large space and time scales, where uncertainties in the factors controlling transport unavoidably increase.<\/p>\n
Statistical Physics of Rarefied Sediment Particle Motions and Transport:<\/strong> Applications to Hillslopes and Rivers<\/strong><\/em><\/p>\n Posting this preview of material from my next (unfinished) book might be a mistake. But eh… what the hell. It represents real progress in understanding. And given the current state of domestic and world affairs amidst an irreversible climate crisis, who knows what doing science will even look like by the time I finish the book?<\/p>\n Contents<\/strong> Title Page<\/strong> These sections (pdf link below) provide a clear idea of the material covered in the book, including its philosophical basis. Although I originally committed an early version of the book to a well-known university press, I eventually decided instead to prepare it as an ebook, consistent with my views on open access, to be freely accessible to all. Although it currently exceeds 450 pages without figures, I still have much to do and won’t offer a target date. (Pieces of the book appear as essays on this webpage below.)<\/p>\n The objective of the book is to provide a framework for describing sediment particle motions and transport in a manner that integrates principles and methods of probability with mechanical considerations of particle motions. The book therefore covers the topic of sediment transport in a manner that is mostly unfamiliar to folks in the Earth-surface processes community. Nonetheless, I suspect it will resonate with students and early career people eager for a fresh view of this complex and often confusing topic. To be sure, the book unapologetically breaks from current dogmatic styles of thinking and analysis in the field. Probabilistic Concepts I (Chapter 2)<\/strong> Oh gosh! Nuts and bolts!<\/p>\n These chapters illustrate how the principles and methods of probability are essential, foundational elements of rational explanations of sediment transport. Starting with an outline of classical versus frequentist views of probability, the chapters cover interpretations and applications of discrete and continuous probability distributions, and the algebra and calculus of random variables. Highlights include applications of the law of the unconscious statistician (LOTUS), the profound physical implications of the central limit theorem, and Gibbs-like ensemble averaging applied to sediment systems.<\/p>\n Sidebar:<\/em> In presenting the material in these chapters, I don’t hide my contempt for the dogma of 20th century frequentist statistics centered on the bullshit idea of “statistical significance” in hypothesis testing and related inferential methods. Basic Stochastic Processes (Chapter 4)<\/strong><\/p>\n Highlights of this chapter include: counting processes (Poisson, simple renewal, and additive Levy processes); basic Markov processes (birth-death processes, and Wiener, Langevin and Ornstein-Uhlenbeck processes); and survival analysis (focused on particle disentrainment).<\/p>\n Together with Chapters 2 and 3, this fourth chapter completes the basic probabilistic (stochastic) material needed to launch into more advanced foundational topics — kinematics of the particle flux, master equations, advection and diffusion, considerations of entropy, etc. — thence to the second part of the book focused on applications. Kinematics of the Particle Flux (Chapter 6)<\/strong><\/p>\n Everybody in the business thinks they understand the sediment particle flux, but nobody actually does lol \ud83d\ude42<\/p>\n This chapter is particularly fun because, judging from colleague reactions, several of the foundational concepts are decidedly counterintuitive to those steeped in the continuum framework that is conventionally adopted for describing sediment transport. Master Equations (Chapter 7)<\/strong><\/p>\n Of all the topics covered in the book, the formalism of a master equation quintessentially embodies the power and beauty of the principles and methods of statistical physics as applied to sediment particle systems. Herein we see that a master equation really is what its name says it is — a universal “truth” describing conservation of probability — where e.g. the behavior of Brownian particles, sediment particles and the stars of the Milky Way Galaxy are all described by the same equation, the Fokker-Planck equation. Yes… Brownian particles are like rarefied bed load particles are like stars — all experiencing velocity proportional (Stokes) friction! lol \ud83d\ude42<\/p>\n Bonus:<\/em> This chapter provides the first completely rigorous derivation and explanation of the activity form of the Exner equation — highlighting its stochastic qualities under rarefied transport conditions — thus avoiding unphysical consequences of conventional misapplications of the continuum hypothesis. Particle Advection and Diffusion (Chapter 8)<\/strong><\/p>\n (Note:<\/em> I plan to post this chapter next. Thereafter I probably will mostly stop posting completed material.)<\/p>\n Rain Splash Transport (Chapter 11)<\/strong><\/p>\n Rain splash transport has been described in the literature as the “simplest of all geomorphic processes.”<\/p>\n sure ok lol \ud83d\ude42<\/p>\n This chapter of the book is soooo fun. One of Earth’s clearest examples of the elements and consequences of unsteady, rarefied, nonlocal transport conditions.<\/p>\n “Indeed, an explanation of how rain splash transport works requires<\/em> a probabilistic description; no other possibility exists.” Note:<\/strong> All material appearing in the essays and notes below is fully copyrighted. All rights of ownership are reserved.<\/p>\n To be sure<\/strong> “Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.” (Bertrand Russell, 1929)<\/p>\n “When unsure what something is [the meaning of probability], it often pays to ask what it does.” (David Wallace, The Emergent Multiverse: Quantum Theory according to the Everett Interpretation<\/em>, 2012)<\/p>\n “In our view the things we call “classical probabilities” can be seen as originating in the quantum probabilities that govern the microscopic world, suitably propagated by physical processes so as to be relevant on classical scales.” (A. Albrecht and D. Phillips, Phys. Rev. D<\/em>, 90, 123514, 2014)<\/p>\n Serious snark footnotes<\/strong> The sediment particle flux: An illustration of the pronounced consequences of the small numbers of particles involved in rarefied (non-continuum) transport conditions<\/strong> Bed load particle velocities: The essential role of particle–bed collisions in the Langevin-like equation<\/strong> On the mixed alluvial–bedrock channel problem<\/strong> An explanation of the Shannon entropy, with relevance to sediment transport<\/strong> The profound implications of the central limit theorem applied to rarefied sediment transport<\/strong> In celebration of Peter Kirkland Haff: scholar, mentor, friend 1944 — 2024<\/strong> Statistical equilibrium transport of bed load sediment: The role of particle velocity, acceleration and jerk<\/strong> Estimating \u03c0 from raindrops<\/strong> The “uncertainty principle” of a Poisson process: An example involving bed load transport<\/strong> Interpreting the nominal soil production function as inferred from measurements of cosmogenic radionuclides<\/strong> Physical interpretation of the first and second moments — the mean and variance — of a probability distribution<\/strong> Cool probabilistic things we typically don’t teach our students about radioactive decay, but should<\/strong> Reviewing manuscripts on my terms<\/strong> The joy of watching students demonstrate their intellectual ownership of course material<\/strong> Four short football stories<\/strong> An example of what a PhD defense should look like and why<\/strong> Let us invest in teaching our students about dimensions<\/strong> Reimagining the meaning and potentialities of ‘geophysics’ Catching up to a 21st century view of statistics in the doing and reporting of research in the Earth sciences<\/strong> The “uncertainty principle” of a Poisson point process<\/strong> Particle diffusion on a Galton board<\/strong> The benevolent companionship of failure<\/strong> The Brickyard in 2020<\/strong> General Interests My research and writing efforts are centered on the statistical physics of sediment particle motions and transport in Earth surface systems, including hillslopes and rivers. I am increasingly interested in the epistemology of Earth surface science in view of its loose connection to critical rationalism and its tepid relationship with probability. I formerly…<\/p>\n","protected":false},"author":1930,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-3","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/pages\/3","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/users\/1930"}],"replies":[{"embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/comments?post=3"}],"version-history":[{"count":367,"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/pages\/3\/revisions"}],"predecessor-version":[{"id":444,"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/pages\/3\/revisions\/444"}],"wp:attachment":[{"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/media?parent=3"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/my.vanderbilt.edu\/davidjonfurbish\/wp-json\/wp\/v2\/tags?post=3"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nThe current table of Contents lists the topics to be covered, although the headings for the chapters in the second part of the book (Applications) are not yet included.
\neBook Contents July 2026<\/a><\/p>\n
\nPreface<\/strong>
\nIntroduction (Chapter 1)<\/strong><\/p>\n
\nPreview 2.0<\/a><\/p>\n
\nProbabilistic Concepts II (Chapter 3)<\/strong><\/p>\n
\nPreview Chapters 2 and 3<\/a><\/p>\n
\nPreview Chapter 4<\/a><\/p>\n
\nPreview Chapter 6<\/a><\/p>\n
\nPreview Chapter 7<\/a><\/p>\n
\nPreview Chapter 11<\/a><\/p>\nStuff to Ponder<\/h3>\n
\nI cannot think of an applied mathematics that is more beautiful and far-reaching, or philosophically wilder, than probability. No, nonlinear dynamics and chaos people, it’s not even close \ud83d\ude42<\/p>\n
\nI usually compose footnotes in a manner befitting their normally intended purpose: to support or elaborate material in the main text. But I also view certain footnotes as opportunities to drop serious snark. Here is one of my favorite recently composed footnotes, with context. It appears in Chapter 8 (Particle Advection and Diffusion) of the draft ebook, and it concerns the nonsense of the popular “stream power model” of land-surface evolution.
\nFootnotes<\/a><\/p>\n
\nAdopting a continuum framework to describe rarefied transport imposes a deterministic style of thinking consisting of the concepts and language of continuum mechanics. Such descriptions, however, are entirely misleading. In contrast, the probabilistic view outlined herein fully embraces the mechanics of sediment particle motions and transport. It acknowledges, relative to continuum conditions, the small numbers of particles involved during transport, and the pronounced consequences of these small numbers. Then the variability in realizations of the particle transport rate is viewed as an inherent feature of the transport process where, for a specific set of controlling factors, any one of an ensemble of possible realizations could occur, each entirely consistent with the physics involved. Expectations of behavior are thus probabilistic, where variability about an average state is just as important as the average in terms of characterizing how the process works.
\nFlux with Small Numbers<\/a><\/p>\n
\nThe formalism of the Langevin equation and the associated Fokker–Planck equation provides a way to highlight the essential role of particle–bed collisions in modulating bed load particle velocities. In this essay we start with a brief primer on Brownian particle motion and the Langevin equation, emphasizing the ergodic behavior of the particle in relation to thermal equilibrium. We then illustrate an application of a Langevin-like equation to describe the velocity distribution of bed load particles involving continuous particle motions. In the case of discontinuous, non-ergodic particle motions, effects of turbulence and nonlinear particle–fluid coupling point to state-dependent diffusion of particle velocity states in the Fokker–Planck equation.
\nLangevin Equation<\/a><\/p>\n
\nSediment transport and bedrock abrasion are stochastic processes, whether viewed at the short time scales of particle motions or at longer time scales during which significant downcutting by abrasion occurs. In this situation it is unclear how to reconcile descriptions of transport and abrasion viewed at experimental time scales with the inherent variability that exists in the wild, and which cannot be empirically constrained with confidence owing to our limited ability to measure things over long time scales. The problem therefore is inherently probabilistic. We can only aim at the statistical likelihood of outcomes based on defensible probabilistic descriptions of transport and abrasion whose physics is suitably coarsened to the length and time scales of interest. Herein I offer a straightforward starting point for conceptualizing the mixed alluvial–bedrock part of the problem. Mechanistic descriptions of the abrasion part of the problem likewise require a statistical rethinking, although I only briefly comment on this point without elaboration.
\nAlluvial_Bedrock_Channel<\/a><\/p>\n
\nEntropy seems to be a popular but enigmatic concept from science that is frequently used to qualitatively explain, at a high level, the behavior and configurations of systems, yet which is frequently misunderstood in practice. To complicate things, there are two principal definitions of entropy: the Gibbs entropy from statistical mechanics and thermodynamics, and the Shannon entropy from information theory. This essay unfolds the concept of entropy, with particular relevance to problems in sediment transport.
\nEntropy<\/a><\/p>\n
\nThe central limit theorem is among the crown jewels of mathematics and science. Here is the conclusion of this essay: “Let us end with a simple but profound truth attributable to the central limit theorem. Sediment particles experience varying velocities and displacements during transport — a hallmark of their behavior. Regardless of the detailed physics involved, formulations of rarefied sediment transport that do not explicitly acknowledge the existence and effects of particle diffusion are wrong. Particle diffusion is an inherent feature of transport — a probabilistic consequence of varying particle velocities and displacements — and its effects must figure into formulations of transport.”
\nCentral Limit Theorem<\/a><\/p>\n
\nA legion of beautiful stars watching over Peter Haff’s beloved Mojave Desert twinkled extra last night — a brief spacetime ripple — in celebration of an extraordinary person.
\nPeter Kirkland Haff<\/a>
\nThe obituary on Jesse Haff’s blog provides a wonderful description of Peter — his life, his impact on others, his generosity and kindness:
\nPeter Haff<\/a><\/p>\n
\nFrom a statistical mechanics point of view, thermodynamic equilibrium of an ordinary gas coincides with a condition in which the Maxwell-Boltzmann distributions of particle energies and speeds are stationary. This is manifest macroscopically as fixed thermodynamic state variables — pressure and temperature. In pursuing a statistical mechanics description of sediment particles transported as bed load, an intriguing possibility is that an analogue of thermodynamic equilibrium exists. Currently our simplest description of equilibrium bed load transport is that the particles collectively experience zero acceleration. Here we present a brief qualitative description of this problem.
\nStatistical Equilibrium Transport of Bed Load<\/a><\/p>\n
\nA delightful way to estimate the number \u03c0 = 3.14159… involves simultaneously counting raindrop impacts on a circular sensor and a square sensor during a rainstorm. Experimental demonstrations of this idea occasionally are posted on various websites. Perhaps understandably, the explanations provided with these demonstrations focus on the experimental measurements and calculations, and offer little regarding the physical basis of why the procedure leads to estimates of \u03c0. Here I fill in some of the physical details accompanied by a Monte Carlo code that illustrates the uncertainty in the procedure.
\nPi from Raindrops<\/a><\/p>\n
\nIn a separate essay posted on this webpage I describe the well known “uncertainty principle” of a Poisson point process. Here I offer a delightful example of this principle involving bed load transport, with important practical as well as theoretical implications. This example involves data that Madeline Allen and Shawn Chartrand analyzed.
\nUncertainty Principle Example<\/a><\/p>\n
\nIn notes prepared for colleagues I examine the foundational elements of this procedure for inferring the nominal soil production function relating the rate of production to the soil thickness. I explain why one must be skeptical of the procedure — how it likely leads to spurious results under the transient conditions of varying soil thickness that mostly exist in the wild, yielding empirical curves whose forms are largely determined by the attenuation length of cosmogenic radionuclide production in the soil — regardless of the form of any underlying “true” function relating the soil production rate to soil thickness. In effect the procedure uses values of the independent variable, the soil thickness, to create the values of the dependent variable, the production rate — a statistics no-no. I then show why the soil production rate might be empirically determined only when variations in soil thickness are sufficiently slow that quasi-steady conditions are maintained, and I explain why the procedure is unlikely to reveal a non-monotonic relationship between the production rate and soil thickness, if it exists. (Note:<\/em> Much of the material presented on the Wikipedia page, “Soil production function,” is patently wrong.)
\nSoil Production<\/a><\/p>\n
\nThe idea of a statistical moment comes from physics. Indeed, the history of the development of mathematical probability tells us that physics has often conditioned our views and interpretations of probability. And probability is of course a foundational element of certain fields of physics, notably statistical mechanics and quantum mechanics. In this short essay I offer a well-known physical interpretation of the mean and variance of a probability distribution centered on computing the torque of a system.
\nStatistical Moments<\/a><\/p>\n
\nRadioactive decay is a rich topic whose implications and applications appear in many fields of science. Moreover, because of its familiarity, radioactive decay is a nice entry into the broader topic of stochastic processes. The idea of a Poisson process in particular is a lovely starting point for considering a variety of stochastic processes that occur in natural and engineered systems across many scales.
\nRadioactive Decay<\/a><\/p>\n
\nI suppose this essay is merely encouragement to critically ask ourselves what we are doing and why.
\nReviewing Manuscripts<\/a><\/p>\n
\nMy courses in Transport Processes, Fluid Dynamics, and Probability and Statistics involve take-home exercises and individual student projects. We discuss the idea that I am less interested in “correct” answers and far more interested in seeing how their thinking unfolds in demonstrating intellectual ownership of the material. This essay offers a fun example of what I mean by this. It’s a bit technical, maybe… but I think it makes the point.
\nOwnership<\/a><\/p>\n
\nMomentarily stepping outside of academics, I was inspired by Luke Zaleski’s 2017 article in GQ Magazine, “What kind of father lets his son play football?” to offer a brief essay entitled “Four short football stories.” It’s personal. The first one starts with: “As a sophomore in high school I watched my close friend, Bob Warren, break his neck playing football.”
\nFootball Stories<\/a><\/p>\n
\nOne key takeaway message of this essay is this: In the 21st century let us treat our students with the dignity and respect they deserve \u2014 they are our talented colleagues \u2014 and not imagine that the PhD defense is intended to be an adversarial rite of passage.
\nModel for PhD Defenses<\/a><\/p>\n
\nDimensions and dimensional analysis are cornerstone topics of all<\/em> science. In this essay I offer examples of my favorite, recurring teaching experiences centered on dimensions.
\nDimensions_Essay<\/a><\/p>\n
\n<\/strong>This short essay was inspired by comments of Douglas Jerolmack (UPenn) on Twitter concerning the traditional narrow meaning of ‘geophysics’ versus the intellectual opportunities that a broader perspective might inspire. The link is here:
\nGeophysics<\/a><\/p>\n
\nThere is a compelling need to reexamine our views and use of statistics in the Earth sciences, and press toward a more informed, measured use of statistical methods in data analysis. This involves moving beyond the false premise that hypothesis testing can be reduced to the dichotomous choice of “significant” or “not significant” decided by arbitrary statistical thresholds, paying increasing attention to the “don’ts” of statistics, and crafting well-reasoned descriptive statistics and analyses with full explanation. Our statistics courses must cover the probabilistic foundation of statistics, not just its applications, giving students the needed insight and thus confidence to critically evaluate their own work as well as what is presented in the literature. Click the link below for an essay on this topic.
\n21st Century Statistics<\/a><\/p>\n
\nThe concept of a Poisson point process is a beautiful thing, with important applications throughout the sciences. This concept involves an “uncertainty principle,” which, although not rivaling Heisenberg’s in its importance, nonetheless is delightful in its implications. Click the link below for a one-page essay that illustrates this uncertainty principle.
\nUncertainty Principle<\/a><\/p>\n
\nParticle motions on a Galton board, also known as a quincunx or bean machine, have inspired the design of toys, descriptions of sediment particle motions, and theories of the statistical physics of a Lorentz gas. Click the link below for a short essay that illustrates particle diffusion on a Galton board, with implications for sediment particle transport.
\nGaussian Diffusion on a Galton Board<\/a><\/p>\n
\nHere are some brief thoughts on the topic of failure, inspired by discussions with students, my reading on the matter, and my own experience.
\nCompanionship of Failure<\/a><\/p>\n
\nIn 1963 Bernard K. Forscher published a popular allegorical Letter in Science<\/em> entitled “Chaos in the Brickyard.” Click the link below to see our update on this letter regarding the doing of science.
\nThe Brickyard in 2020<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"