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.
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 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 — 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.
Sarah Williams (PhD) Earth and Environmental Sciences. Physics and implications of bed load particle diffusion
Stuff to Ponder
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 are analyzing for a forthcoming paper.
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.
Probabilistic versus continuum descriptions of rarefied sediment particle motions and transport: 2. Rain splash
The transport of sediment particles by rain splash is an ideal example to illustrate consequences of rarefied transport conditions using a probabilistic description that is not conditioned by the conventional assumption of continuum behavior. This document contains a preview — the abstract and introduction — of a paper I am working on with colleagues. Rain splash transport is perhaps the “tamest” surface transport process imaginable — the “simplest of all geomorphic processes” (Dunne et al., 2010, JGR-ES, 115). But as a noise driven process, it is not as tame or predictable as we might imagine.
Preview Rain Splash
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.
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
Probabilistic versus continuum descriptions of rarefied sediment particle motions and transport: 1. Problem statement and theory
Descriptions of sediment transport in hillslope and riverine settings conventionally appeal to a continuum formalism, yet the continuum hypothesis is rarely if ever satisfied. Clearer explanations of transport must instead adopt principles and methods of statistical physics. This document contains a preview — the abstract and introduction — of a paper I am working on with colleagues. We are aimed at clarifying the meaning of the sediment particle flux, distinguishing between ensemble expected conditions versus what occurs in any realization, and why the Exner equation represents noise — a Levy process with drift — at small space and time scales.
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.
Reviewing manuscripts on my terms
I suppose this essay is merely encouragement to critically ask ourselves what we are doing and why.
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.
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.”
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.
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:
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.
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