## General Interests

My research is centered on the statistical physics of sediment particle motions and transport in Earth surface systems, including hillslopes and rivers. A growing aspect of my work involves examining the epistemology of Earth surface science within the context of critical rationalism and our 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.

## Current Research

Our current research on the probabilistic physics of sediment particle motions has, in addition to interesting technical aspects, an important philosophical basis. Namely, 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. 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 harmonic oscillators constructed from Hookean springs and dashpots, 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 — relaxing our expectations that a deterministic-like relationship exists between, say, the flux of bed-load sediment and the fluid stress imposed on the streambed, or the flux of sediment on a hillslope and the local land-surface slope. 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 world views are entirely compatible.

## Stuff to Ponder

**Note:** All material appearing in the essays and notes below is fully copyrighted. All rights of ownership are reserved.

Reactions and queries by email are welcomed.

**On the mixed alluvial–bedrock channel problem**

Sediment 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.

Alluvial_Bedrock_Channel

**An explanation of the Shannon entropy, with relevance to sediment transport**

Entropy 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.

Entropy

**The profound implications of the central limit theorem applied to rarefied sediment transport**

The 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.”

Central Limit Theorem

**In celebration of Peter Kirkland Haff: scholar, mentor, friend 1944 — 2024**

A 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.

Peter Kirkland Haff

The obituary on Jesse Haff’s blog provides a wonderful description of Peter — his life, his impact on others, his generosity and kindness:

Peter Haff

**A probabilistic view of the distribution of bubble volumes involving coalescence**

These notes were inspired by conversations with Prof. Kristen Fauria and her PhD student Sarah Ward. The problem pertains to bubbles in magmas.

Bubble Coalescence

**Statistical equilibrium transport of bed load sediment: The role of particle velocity, acceleration and jerk**

From 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.

Statistical Equilibrium Transport of Bed Load

**Estimating π from raindrops**

A delightful way to estimate the number π = 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 π. Here I fill in some of the physical details accompanied by a Monte Carlo code that illustrates the uncertainty in the procedure.

Pi from Raindrops

**The “uncertainty principle” of a Poisson process: An example involving bed load transport**

In 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.

Uncertainty Principle Example

**Interpreting the nominal soil production function as inferred from measurements of cosmogenic radionuclides**

In 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.

Soil Production

**Physical interpretation of the first and second moments — the mean and variance — of a probability distribution**

The 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.

Statistical Moments

**The wells of epistemology**

“The deep wells of science have smaller wells that feed on their knowledge; and these wells have lesser wells, thence goes epistemology.”

Wells of Epistemology

**Cool probabilistic things we typically don’t teach our students about radioactive decay, but should**

Radioactive 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.

Radioactive Decay

**Reviewing manuscripts on my terms**

I suppose this essay is merely encouragement to critically ask ourselves what we are doing and why.

Reviewing Manuscripts

**The joy of watching students demonstrate their intellectual ownership of course material**

My 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.

Ownership

**Four short football stories**

Momentarily 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.”

Football Stories

**Calculus shadow course in EES**

For various reasons our department seriously considered offering our own courses in calculus for our undergraduate majors, but then decided this was not sustainable. Our compromise is something I did years ago — a calculus shadow course. We’ll see how it goes. Course description:

Applications of Calculus in EES

**An example of what a PhD defense should look like and why**

One key takeaway message of this essay is this: In the 21st century let us treat our students with the dignity and respect they deserve — they are our talented colleagues — and not imagine that the PhD defense is intended to be an adversarial rite of passage.

Model for PhD Defenses

**Let us invest in teaching our students about dimensions**

Dimensions and dimensional analysis are cornerstone topics of *all* science. In this essay I offer examples of my favorite, recurring teaching experiences centered on dimensions.

Dimensions_Essay

**The crest of the hill**

I am writing a book based on the material I have been teaching for nearly twenty years in a one-semester course at Vanderbilt. Here the title and contents of the book do not matter so much as the thoughts that emerge during the effort. This essay has two parts. The first part consists of two paragraphs that will land in the Acknowledgments of the book. These paragraphs provide an ideal set up for what I really want to say in the second part of this essay. The link:

The Crest of the Hill

**Six of fifteen outstanding problems in sediment transport science**

Here are the slides of an invited talk on this topic. Following the introductory material the first problem involves formulating a dynamically correct Knudsen number (or suitable analogue) for rarefied transport conditions. The link:

Sediment Transport Problems

**Rain splash transport and the law of large numbers**

So what probabilistic elements underlie a nominally deterministic formula of noise driven sediment transport? Here is an essay on perhaps the “tamest” of surface transport processes: rain splash. The link is here:

Rain Splash and Large Numbers

**Reimagining the meaning and potentialities of ‘geophysics’
**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:

Geophysics

**Catching up to a 21st century view of statistics in the doing and reporting of research in the Earth sciences**

There 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.

21st Century Statistics

**The “uncertainty principle” of a Poisson point process**

The 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.

Uncertainty Principle

**Particle diffusion on a Galton board**

Particle 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.

Gaussian Diffusion on a Galton Board

**The severe limitations of stress-based formulae for describing bed load transport**

Predictions of the flux of bed load particles in a turbulent shear flow typically involve algebraic formulae that relate the flux to the space-time averaged stress on the bed or an averaged near-bed flow velocity. Yet we do not have a clear theoretical basis for such formulae. Click on the link below for a short essay on why the macroscopic fluid-imposed bed stress in turbulent flows is an empirical heuristic and incorrect for what we expect of it.

Limitations of Stress Based Formulae

**The benevolent companionship of failure**

Here are some brief thoughts on the topic of failure, inspired by discussions with students, my reading on the matter, and my own experience.

Companionship of Failure

**The Brickyard in 2020**

In 1963 Bernard K. Forscher published a popular allegorical Letter in *Science* entitled “Chaos in the Brickyard.” Click the link below to see our update on this letter regarding the doing of science.

The Brickyard in 2020

**Probability is a physical thing, not a mathematical thing**

Consider the idea that probability, the measure *p* such that 0 ≤ *p* ≤ 1, is a physical thing rather than a mathematical thing. Click the link below for a short essay exploring the barest essence of this idea.

Probability is Physical

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