# Administrative Information
- **Class Meeting**: Friday, October 23, 2026, 1:00–1:50pm — **Coolbaugh 212**
- Previous meeting: [[MATH310F26(Day 23) - Transport moves, diffusion spreads]]
- 📄 [[Teaching/MATH310/MATH310F26/Administration/MATH310_F26_Syllabus_1.0.pdf|MATH 310 F26 Syllabus, v1.0]]
> [!info]- Admin. Notes
> Expect an LR and WB for each lecture. WBs target 15-minute engagements (longer early on); LRs are short, ~5 minutes.
>
> - 🎯 **[LR] [Day 22 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121959)** · **[Statement](https://elearning.mines.edu/courses/81632/quizzes/121960)** — due **Friday 10/23, 11:59pm**
> - 🎯 **[LR] [Day 23 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121971)** — 4 questions drawn from Wednesday's material, unlimited attempts, highest kept — due **Friday 10/30, 11:59pm**
> - 🎯 **[LR] [Day 23 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121972)** — one free response, no wrong answers — due **Friday 10/30, 11:59pm**
> - 🎯 **[LR] [Day 24 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/122140)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Friday 10/30, 11:59pm**
> - 🎯 **[LR] [Day 24 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/122141)** — one free response, no wrong answers — due **Friday 10/30, 11:59pm**
> - 🎯 **[MR] [Media Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121701)** · **[Statement](https://elearning.mines.edu/courses/81632/quizzes/121702)** · [[The Breakdown of Linear Stability Analysis and Critical Slowing Down|the page]] — due **Friday 10/23, 11:59pm**. **IT has cleared `scottastrong.org`** for the campus network (announced today), so the applets should now load on campus; email if they don't.
> - 📝 **[MP3 — Initial Model and Test (first results)](https://elearning.mines.edu/courses/81632/assignments/590872)** and **[MA2 — Source and Background](https://elearning.mines.edu/courses/81632/assignments/590873)** — due **Monday 10/26, 11:59pm** ([[MATH310F26 - Project Work|Project Work]]; today if you want, more time on request). Another round of feedback follows.
> - 🕒 **Office block for 310:** Friday 3pm this week; more blocks next week.
> - 🎯 **[WB] [Workbook 11](https://elearning.mines.edu/courses/81632/assignments/588368)** · [[MATH310F26_WB11.pdf|blank]] · [[MATH310F26(Workbook 11 Companion) - Three outcomes, two moments|companion]] — due **Friday 10/30, 11:59pm**
> - 🎬 [[MATH310F26 - Random numbers, two videos]] — the guided videos for this unit, if you haven't yet
> - 🖼️ [[MATH310F26 - Convolution, in pictures]] — the convolution integral's meaning, told with the walk's own objects: two steps as $p*p$, $n$ steps as $p^{*n}\to$ Gaussian, the stamp-the-normal integral, moving averages, kernels, and why Fourier keeps showing up; four-language code at the end
---
# Deliverables
- 🎯 **[LR] [Day 24 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/122140)** — 4 questions drawn from today's material, unlimited attempts, highest kept — due **Friday 10/30, 11:59pm**
- 🎯 **[LR] [Day 24 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/122141)** — one free response, no wrong answers — due **Friday 10/30, 11:59pm**
- 📝 **[MP3](https://elearning.mines.edu/courses/81632/assignments/590872)** and **[MA2](https://elearning.mines.edu/courses/81632/assignments/590873)** — due **Monday 10/26, 11:59pm**
Questions and statement are due Friday 10/30 at 11:59 pm (moved from Monday 10/26; the pair went live late).
---
# Lecture Boards + Transcript + GenAI
**Twenty-fourth meeting, October 23 — the continuum transport–diffusion process, with applications.** Board header: *24 | 10/23/26, The continuum transport-diffusion process w/ App.* Family weekend in Golden, sun before the rain, and the end of the random-walk sequence: "there's a bunch of gobbledygook in my brain that I wanna try to make sure to get out to you before I should wrap it up," so the hour is deliberately frenetic and tied down with pictures. One more talk on Monday, probably; then **discrete-time processes**, with the goal of putting randomness onto them so that *deterministic* and *discrete* join up; after that, about five lecture slots that will mostly become group work in support of the projects. Prizes are [Starburst](https://en.wikipedia.org/wiki/Starburst_(confectionery)) — a bet lost to the summer numerical-science class. The content, in order: the missing $2\pi$ restored (Tony); micro, macro, and the *[meso](https://en.wikipedia.org/wiki/Mesoscopic_physics)* in between; the normal distribution matched coefficient by coefficient to the walkers' Gaussian — *why* the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem); the [convolution integral](https://en.wikipedia.org/wiki/Convolution) that generalizes the delta start to any start; two applets; a [3Blue1Brown](https://en.wikipedia.org/wiki/3Blue1Brown) clip on convolution; and a slide run from [Gaussian blur](https://en.wikipedia.org/wiki/Gaussian_blur) to edge-preserving *nonlinear* blur, via a [fingerprint](https://en.wikipedia.org/wiki/Fingerprint).
## Recall, with an edit
For the walk process with $\Delta x,\Delta t\searrow 0$, the proportion of walkers at $(x,t)$ obeys
$(1)\qquad \frac{\partial u}{\partial t}=\frac{D}{2}\,\frac{\partial^{2}u}{\partial x^{2}}-c\,\frac{\partial u}{\partial x},$
where $D$ is the **[diffusivity](https://en.wikipedia.org/wiki/Mass_diffusivity)** and — not written last time, but shown by the $f(x-ct)$ solutions and by its units — $c$ is the **speed of drift**. Think of (1) as the **macroscopic representation of the microscopic walkers.** And if $u(x,0)=\delta(x)$, Fourier can show
$u(x,t)=\frac{1}{\sqrt{2\pi D t}}\;e^{-\dfrac{(x-ct)^{2}}{2Dt}}=g(x,t),$
with the $\sqrt{2\pi}$ **missing from last time** — Tony pointed it out. It was a judgment call about where the $2\pi$ lives, made on the fly from how this course placed the $2\pi$ in its [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform); a second look after class settled it: the $2\pi$ belongs. "It's just a [normalization constant](https://en.wikipedia.org/wiki/Normalizing_constant), a scale factor" — the one that makes the area one.
## Micro, macro, meso
A thought from the dog walk that is part of getting ready for class: ten or fifteen years ago the big topic in computational and applied math was **[multi-scale modeling](https://en.wikipedia.org/wiki/Multiscale_modeling)** — honoring the fact that at the lowest scale you can identify particles, or walkers, or cars, or stock prices, and that if you squint well enough there is a level at which you think instead about *blobs* of walkers, cars, prices. The continuum limit is exactly the move from the **microscopic** scale to the **macroscopic**. Why pull it off? Some things are easier to see or to program at one scale than the other. An uncomfortable example, from a documentary half-watched while moving around the house — the [Astroworld crowd disaster](https://en.wikipedia.org/wiki/Astroworld_Festival_crowd_crush): people are not waves, and even globs of people are not waves, but at a certain density a crowd carries a **wave through it**, a macroscopic behavior that formed over the microscopic people — and that was part of what went wrong. The word for what lives between macro and micro is **meso**, and mesoscale behavior is always an interesting place; on the math side the usual pair is the *atomistic* perspective and the *continuum* perspective.
## Why the central limit theorem, with random walks
If $X\sim\mathcal{N}(\mu,\sigma^{2})$ — and it is worth highlighting that a $\mu$ and a $\sigma^{2}$ are floating around in there — the probability density is
$(2)\qquad \rho(x;\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\;e^{-\dfrac{(x-\mu)^{2}}{2\sigma^{2}}}.$
Put your walkers on a checkerboard or a [Plinko](https://en.wikipedia.org/wiki/Plinko) system, let tons of them out, and squint so you stop seeing the histogram boxes: that is the continuum limit, and with the finite-difference approximations it shows that the transport–diffusion equation *emerges*. Pile the walkers at $x=0$ at time zero, let them evolve, and they immediately take on the profile Fourier says. Now look at that profile next to (2) and spot the correspondences. **$\mu$ is easiest:** $(x-\mu)$ against $(x-ct)$ gives
$\mu=ct,$
and $c$ fundamentally contains the *difference* of the left and right probabilities — the **bias** — so a nonzero bias moves the mean in time. Geometrically (the demonstration again: "I'm a normal distribution, here's my handle, here's my maximum") the lump, the central tendency, drifts: **the mean is transported.** The other rogue element, $\sigma^{2}$, matches
$\sigma^{2}=Dt,\qquad\text{so}\qquad \sigma=\sqrt{Dt},$
which is what was said about walkers being poor explorers: the characteristic width, the [standard deviation](https://en.wikipedia.org/wiki/Standard_deviation), grows **sublinearly**. So why does the central limit theorem appear? There are a variety of ways to say it, but with the stats we have in the can: squint at the process in a blobby way and you get *exactly* a Gaussian distribution of walkers, transported in time — the same thing the central limit theorem was telling us about the same process.
## Fourier also generalizes this
Call the Gaussian $g(x,t)$. All the walkers started at one spot; you might not want that. The result, whose derivation takes multiple lectures and whose meaning takes a lecture of its own:
$u(x,t)=\int_{-\infty}^{\infty}u(x,0)\,g(x-p,t)\,dp.$
Read it in pieces. The left side is the oracle: the proportion of walkers at any location, at any time you care to sample. Inside the crazy integral, $u(x,0)$ — forget the rest for a second — is the **initial distribution of walkers**, and now they don't have to be all in one place. Break the room into three groups: group one crams into an infinitesimal space here (continuum thinking), group two there, group three there; that is $u$ at time zero, and then everybody starts walking. Or draw any shape, more here and fewer there. Three deltas, say,
$u(x,0)=\delta(x)+\delta(x-1)+\delta(x+1),$
and $g$ is **the normal from just one $\delta(x)$**. The integral is a **convolution integral**: it *mixes* the initial distribution of walkers with the normal, over all space, to get $u(x,t)$. How, with the instructor as the normal and three objects as walker groups: inside the integral $g$ carries $x-p$, and $p$ is the variable of integration, running over positives and negatives — so the normal gets **shifted everywhere** along the line. Out where there are no walkers it multiplies against zero; sliding on, its tail starts to see a group, then it is mushing over the top of them — "yo, walkers, spread out" — then it is between groups and sees nothing, then it finds the next group and tells them to be more like the normal distribution. (Andrew: "and the integral is making that happen.") It stamps the normal on every point and tells whatever is there to spread out, normal-distribution style.
> [!example]- The convolution, drawn
> ![[day24_convolution.png]]
> Left: the three-delta start — three normals that merge into one as they widen (and drift, $c=0.3$). Right: a "mountain range" drawn by hand, under (1): the pointy peaks are mollified first, then the whole range smooths toward one bump; the area under the curve is conserved.
## Two applets, and a 3Blue1Brown clip
First, the news: **IT has concluded the website is not spyware**, so it should be accessible on campus — "big achievement, yes." (And now that the trust is built up, obviously the time to put spyware on it: something that secretly makes your computers do mathematical calculations. On the record, so probably not a good way to start a life of crime.)
**Applet one** shows three things at once: the histogram bins of literally simulated walkers; the lattice equation with its $u