# Administrative Information - **Class Meeting**: Wednesday, October 21, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 22) - Shrinking the Plinko board]] - 📄 [[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)** — 4 questions drawn from Monday's material, unlimited attempts, highest kept — due **Friday 10/23, 11:59pm** > - 🎯 **[LR] [Day 22 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121960)** — one free response, no wrong answers — due **Friday 10/23, 11:59pm** > - 🎯 **[LR] [Day 23 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121971)** — 4 questions drawn from today'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** > - 🎯 **[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**, **to be extended as the campus-network issue is resolved** (see Housekeeping). The applets need the `scottastrong.org` address; off-campus access works, on-campus is being looked at by IT. If it fails for you *off* campus, email — that's the case still unexplained. > - 📝 **[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)** — forms reworked this morning; due **Monday 10/26, 11:59pm** on Canvas (go in Friday if you're ready; ask if you need past Monday). ([[MATH310F26 - Project Work|Project Work]]) > - 🕒 **Office block for 310:** **Friday 10/23, 3pm**; 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 --- # Deliverables - 🎯 **[LR] [Day 23 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121971)** — 4 questions drawn from today'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** - 🎯 **[MR] [Media Reflection](https://elearning.mines.edu/courses/81632/quizzes/121701)** — due **Friday 10/23**, extension pending the network fix - 📝 **[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 Friday 10/23; the pair went live late). --- # Lecture Boards + Transcript + GenAI **Twenty-third meeting, October 21 — the diffusion–[transport equation](https://en.wikipedia.org/wiki/Advection) and its properties.** Board header: *23 | 10/21/26: The diffusion-transport Equation and its properties.* Monday's continuum limit gets a name, a numbering, and a warning label, and then gets taken apart: set one coefficient to zero and the **[diffusion (heat) equation](https://en.wikipedia.org/wiki/Heat_equation)** is left, argued geometrically on an interval with doors at both ends; scale the other way and the **transport equation** is left, solved by a guess and a [chain rule](https://en.wikipedia.org/wiki/Chain_rule); put them back together on the whole line with every walker starting at the origin, and the answer is a bell curve that drifts, widens, and shrinks — the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem), read off a [partial differential equation](https://en.wikipedia.org/wiki/Partial_differential_equation). Three examples, a coffee cup, a perfume bottle, and a [Gaussian blur](https://en.wikipedia.org/wiki/Gaussian_blur). ([Dum Dums](https://en.wikipedia.org/wiki/Dum_Dums) are coming: "it feels like about the dum-dum time of the semester." Root beer has one taker; grape and the mystery flavor are the instructor's.) ## Housekeeping The website: IT confirmed that the School of Mines **[domain name server](https://en.wikipedia.org/wiki/Domain_Name_System) has been actively blocking `scottastrong.org`** — the word *[spyware](https://en.wikipedia.org/wiki/Spyware)* came up, which was news — and the security team is reviewing it; if they clear it, the campus network will let you through to the base URL the applets need. Off-campus networks and other DNS providers resolve it fine. A redirect toggled Monday night to force traffic to `scottastrong.org` made things *worse* on campus Tuesday (some hands went up), and has been undone: the Obsidian-hosted address now loads anywhere, but its applets don't run. **Due dates will be extended as IT reports back.** The one case still unexplained: anyone who can't reach the site or its applets *off* campus — "it's working for me, and I only have so many ways I can test it myself" — email. Also: the [[MATH310F26 - Project Work|project]] forms for **MP3** (first calculation out of the gates) and **MA2** (an update on the model you're breaking down) were reworked this morning and posted; the forms said Friday, Canvas says **Monday**, and more time is available on request. The point of a first calculation now, mid-to-late October, is a full month left to retune it if it isn't working — or, better, to push it in new directions if it is. (Outside of class: hundreds of differential-equations exams graded, with a 2 a.m. on the clock; "these old bones.") ## Equation (1), and a warning label Monday's achievement, recapped on the board that stayed up: an organism that in each step of time goes right a fixed distance, left the same distance, or nowhere, with total probability one; the balance for the proportion of walkers at $x$ at the next time; and, as $\Delta t$ and $\Delta x$ go to zero, the left side becoming the time derivative, the first bracket the centered second difference from Monday, the second bracket the centered first difference from [[MATH310F26(Workbook 2 Companion) - Taylor series and the art of cancellation|Workbook 2]]. So, *loosely* — "we're kind of living in a gray area here, and that's fine" — if $\Delta x,\Delta t$ get very small the walkers' equation looks more and more like $(1)\qquad \frac{\partial u}{\partial t}=\frac{D}{2}\,\frac{\partial^{2}u}{\partial x^{2}}-c\,\frac{\partial u}{\partial x},\qquad D=(P_{R}+P_{L})\frac{\Delta x^{2}}{\Delta t},\quad c=(P_{R}-P_{L})\frac{\Delta x}{\Delta t},$ a **partial differential equation** — two independent variables, location and time, so two kinds of derivative, which is why it never appeared in your ODE class — and **the continuum limit of the discrete-time random process.** (Bookkeeping: the $\tfrac12$ that Monday's page folded into $D$ now sits in the equation, so today's $D$ is twice Monday's; same equation.) In red, the question underneath all of it: *why the central limit theorem comes up.* A wake-up check, then the trouble. Who is getting small? $\Delta x$ and $\Delta t$. What is inside $c$? The ratio $\Delta x/\Delta t$ — numerator and denominator both shrinking — and inside $D$ that same ratio *times another* $\Delta x$. "So these coefficients seem fishy in this continuum limit." The standard phrase is: send $\Delta x,\Delta t\to 0$ but **keep $\Delta x/\Delta t$ finite** — a Calc 1 sort of limit — and then $D$ carries an extra $\Delta x\to 0$, so the two coefficients **clash**. Beyond the bookkeeping, what does an [infinitesimal](https://en.wikipedia.org/wiki/Infinitesimal) parcel of space or time even mean? Those are mathematical objects; microbes doing the walking "don't care about our calculus, our limits, any of that process." Deciding what *small* means for a given system is a philosophical question with hard mathematics behind it, and not today's goal. What is known: **equation (1) can be a proxy for the process if we squint well enough** — presumably, if the walk converges smoothly to this equation, there is some scaling at which it is the reasonable thing to look at. So, assuming a scaling where (1) applies, *what does it predict?* Which is the house emphasis: what is the analysis giving me? ## Ex 1 — the parabolic limit: diffusion, or heat First instinct with any equation: chill it out — which terms can an assumption kill? The $c$ term. What about the walkers' reality zeroes $c$? Andrew, over the air conditioning: **make the probabilities the same.** $P_{R}=P_{L}$ gives $c=0$ and the so-called **diffusion equation**, $\frac{\partial u}{\partial t}=\frac{D}{2}\,\frac{\partial^{2}u}{\partial x^{2}}.$ Now prod it, informally — "I'm investigating. I'm searching." — and geometrically. Horizontal axis space, restricted for now to $0\le x\le L$ (the infinite line comes later), and $u(x,0)=f(x)$: in ODE language, an **[initial condition](https://en.wikipedia.org/wiki/Initial_condition)**, except that here the initial condition is the *entire distribution* of walkers across the domain. Draw it as a big bow, more walkers in the interior than at the edges. Plucked guitar strings flop around; this is a different process. The endpoints are blue for a reason: think of them as **doorways** — a walker that reaches one leaves the system and is never thought about again. Walkers jiggle; "we don't know what we're doing, but eventually one of us is going to jiggle over here," out the door that never closes, while the rest keep jiggling; let it run and everything should flatten. That's the picture; now the argument. The first spatial derivative is slope; the second is **curvature, or [concavity](https://en.wikipedia.org/wiki/Concave_function)**. The red bow is concave down, so $\partial^{2}u/\partial x^{2}<0$ there. $D$ is a sum of probabilities (non-negative) times $\Delta x^{2}$ (positive) over $\Delta t$ (positive), so the whole right-hand side is negative, so $\partial u/\partial t<0$: **$u$ decreases in time** wherever it is concave down. The first-derivative-in-time term coupled with the second-derivative-in-space term "takes concave-down regions and decays them" — a string settling through [damping](https://en.wikipedia.org/wiki/Damping) is the right feel. Hence the names. **Heat:** paint the hump red for hot and the doors blue for cold; heat moves from hot to cold to equalize everything, and at the particle level the walkers bounce around until they even out. A hot coffee cup is a lot of jiggling; come back in two hundred hours ("I don't know a good time scale for this") and it is the background temperature. **Diffusion:** break a perfume bottle — more molecules here than out in the room, all jiggling, so they spread, and then we smell it. (The [olfactory](https://en.wikipedia.org/wiki/Olfaction) tangent, put out there and not pursued: two geometrically similar molecules can smell very different; somehow the nose reads energy states.) ## Ex 2 — the hyperbolic limit: transport Scale the other way: let $\Delta x/\Delta t\to$ a constant, so that $\Delta x\,(\Delta x/\Delta t)\to 0$. The first demand keeps $c$; the second takes $D$ away. What is left is $\frac{\partial u}{\partial t}=-c\,\frac{\partial u}{\partial x},$ the **transport equation** — called *[convection](https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation)* or *advection* in engineering, *transport* on the math side; one of the PDEs "that just come up on the regular." *Why transport?* In the great way of differential equations: guess something, shove it in, and see whether it satisfies the equation. The guess: cram the two variables into one function of a single variable, $u(x,t)=f(x-ct).$ "This guy spits out a lot of stuff all day long. I don't know if I should trust him. I should check." Take derivatives, chain-rule style, with the bothersome inside covered by a letter, $z=x-ct$ ("whenever it bothers me in mathematics, I cover that thing up with another letter"). Since $f$ depends on $z$ alone, $\partial f/\partial z$ is an ordinary derivative: $\frac{\partial u}{\partial x}=\frac{df}{dz}\frac{\partial z}{\partial x}=f'(z)\cdot 1,\qquad \frac{\partial u}{\partial t}=\frac{df}{dz}\frac{\partial z}{\partial t}=f'(z)\cdot(-c).$ ($x$ doesn't think about time differentiation; $-ct$ pulls out a $-c$.) Left side $-c\,f'(z)$, right side $-c\,f'(z)$: **check — a solution.** And the transportation: at $t=0$ it is $f(x)$; picture your favorite graph, a parabola $f(x)=x^{2}$, and replace $x$ by $x-ct$ to get $(x-ct)^{2}$. Subtracting inside the argument **shifts the graph right**, by more as $t$ grows, with no change of shape at all: **pure transport.** So the big equation is the *diffusion–transport* equation because it is built from two: the diffusion piece, the walk seen from a bird's-eye, macroscopic level, says the coffee's heat, the perfume, the walkers in the room *spread out*; the $-c\,\partial u/\partial x$ piece takes a shape and *moves it* without deforming it. Together, if the world is nice: spread it out and let it drift while spreading — exactly the two things the random walkers did, spreading, and drifting when the coin was biased. (For the coffee cup the transport would be a background flow carrying the heat along.) > [!example]- The three examples, drawn > ![[day23_three_limits.png]] > Left, Ex 1 on $[0,L]$ with doors at both ends: the concave-down bow decays in place. Middle, Ex 2: a shape carried right at speed $c$, unchanged. Right, Ex 3 on the whole line from $u(x,0)=\delta(x)$: the bell curve drifts at $c$, widens like $\sqrt{Dt}$, and shrinks to keep its area one; run backward toward $t=0$ it sharpens into the delta. ## Ex 3 — transport–diffusion on $\mathbb{R}$, by announcement Not a course that works the diffusion–transport equation in "gory, gory detail," so the result is stated to be stared at. Given (1) on the whole real line with $u(x,0)=\delta(x-0)=\delta(x),$ interpretation first: $\delta$ has shown up, so integrals must show up with it (a delta alone means little; integration gives it its [generalized-function](https://en.wikipedia.org/wiki/Generalized_function) context), and the integral tool of this course is Fourier — so, unsurprisingly, the tactic is to [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) the whole shebang, solve, and come back, much as with [Laplace](https://en.wikipedia.org/wiki/Laplace_transform); "not my game" here, a three-or-four-lecture derivation. Contextually (Andrew): **every walker starts at $x=0$**, a hundred percent of them, like the [Plinko](https://en.wikipedia.org/wiki/Plinko) board. From the Plinko discussions, expect spreading, and if the central limit theorem is really backing us, expect [normal distributions](https://en.wikipedia.org/wiki/Normal_distribution). Fourier delivers $u(x,t)=\frac{1}{\sqrt{2\pi D t}}\;e^{-\dfrac{(x-ct)^{2}}{2Dt}}$ (the board wrote the prefactor as $1/\sqrt{Dt}$; the $2\pi$ is the normalization that keeps the area one). Cover the $ct$ and it is the normal distribution with its $e^{-x^{2}}$; the only place $c$ appears is inside the argument, where subtracting $ct$ moves the bell without changing its shape — the instructor as a bell curve, top and all, sliding sideways. As $t$ grows the height goes down and the width grows: **drift, widen, shrink**, until everything equalizes. And where did everyone start? Dropping $t=0$ straight into the formula is troubling, so think of it as a limit: as $t\to 0^{+}$ the prefactor goes up — the height goes up — and the number multiplying the square in the exponent goes up, so the spatial decay is rapid: a very pointy, narrow bell, and in the limit, the [Dirac delta](https://en.wikipedia.org/wiki/Dirac_delta_function). "Everybody starts at the origin, time starts, the bell curve takes over, the bell curve widens, the height goes down and it drifts. That is the continuum approximation to what was going on at our microscopic walking level." One minute left, a walkers-aside: in a computer image, white is the big number, 255. Then this initial condition is one white pixel at the origin with black on either side, and the equation blurs it into a normal distribution — **every Gaussian blur you have ever applied is this**, acting as if a crowd of color-walkers underneath were spreading the information out. > [!success]- Numbers checked in one sandbox run > Symbolically, $u=f(x-ct)$ satisfies $u_{t}+c\,u_{x}=0$ for any $f$ (residual $0$). The Gaussian $\frac{1}{\sqrt{2\pi Dt}}e^{-(x-ct)^{2}/(2Dt)}$ satisfies $u_{t}=\frac{D}{2}u_{xx}-c\,u_{x}$ exactly (residual $0$), integrates to $1$ for all $t$, and has mean $ct$ and variance $Dt$ — the drift and the widening; with Monday's $\tfrac12$-included $D$ that variance is $2Dt$, the same number. The board's prefactor $1/\sqrt{Dt}$ gives area $\sqrt{2\pi}$, so the $2\pi$ was the only thing missing. For Ex 1 on $[0,L]$ with $u=0$ at both doors and $f(x)=x(L-x)$, the Fourier-sine series solution has $\partial u/\partial t<0$ at every interior point checked ($t=0.05$, max $-0.063$), as the concavity argument predicts, and each mode decays like $e^{-(D/2)(n\pi/L)^{2}t}$ — the lowest mode is the last to go, which is why the bow flattens rather than wrinkles. Bookkeeping between pages: Day 22 wrote $u_{t}=D'u_{xx}-c\,u_{x}$ with $D'=\tfrac12(P_{r}+P_{l})\Delta x^{2}/\Delta t$; today's $D=(P_{R}+P_{L})\Delta x^{2}/\Delta t=2D'$ with the half moved into the equation. Same $c$. > [!example]- Board scans — October 21, 2026 > ![[MATH310F26-Day23-Board-1.jpg]] > ![[MATH310F26-Day23-Board-2.jpg]] > ![[MATH310F26-Day23-Board-3.jpg]] > ![[MATH310F26-Day23-Board-4.jpg]] > ![[MATH310F26-Day23-Board-5.jpg]] > ![[MATH310F26-Day23-Board-6.jpg]] > ![[MATH310F26-Day23-Board-7.jpg]] > > 📄 [[MATH310F26-Day23-Boards-2026-10-21.pdf|Full board capture (7 pages, PDF)]]