# Administrative Information - **Class Meeting**: Monday, October 19, 2026, 1:00–1:50pm — **Coolbaugh 212** - Previous meeting: [[MATH310F26(Day 21) - The expected value that isn't there]] - 📄 [[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 21 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121823)** — 4 questions drawn from Friday's material, unlimited attempts, highest kept — due **Monday 10/19, 11:59pm** > - 🎯 **[LR] [Day 21 Reflection Statement](https://elearning.mines.edu/courses/81632/quizzes/121824)** — one free response, no wrong answers — due **Monday 10/19, 11:59pm** > - 🎯 **[LR] [Day 22 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121959)** — 4 questions drawn from today'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** > - 🎯 **[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**. **Applet trouble?** Use the `scottstrong.org` address (not the `obsidian.md` one), try off the campus network or in an incognito window, and email if it still won't load — due dates can move. > - 📝 **[MP3 — Initial Model and Test (first results)](https://elearning.mines.edu/courses/81632/assignments/590872)** — a start-to-finish statement about your context: simple math, reinterpreted by the context; compare against a known simple quantity where you can — due **Monday 10/26, 11:59pm** (moved from Friday 10/23) > - 📝 **[MA2 — Source and Background](https://elearning.mines.edu/courses/81632/assignments/590873)** — where the information comes from, and *what is coming to class* (forward thinking, not the presentation itself) — due **Monday 10/26, 11:59pm** ([[MATH310F26 - Project Work|Project Work]]; moved from Friday 10/23; > - 🕒 **Office block for 310:** **Friday 10/23, 3pm** is held for this class; more blocks next week. Tuesday 10/20 afternoon (before ~4pm) is also open by email. > - 🎯 **[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 22 Reflection Questions](https://elearning.mines.edu/courses/81632/quizzes/121959)** — 4 questions drawn from today'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** - 🎯 **[MR] [Media Reflection](https://elearning.mines.edu/courses/81632/quizzes/121701)** — due **Friday 10/23, 11:59pm**; report applet problems early - 📝 **[MP3 — Initial Model and Test](https://elearning.mines.edu/courses/81632/assignments/590872)** and **[MA2 — Source and Background](https://elearning.mines.edu/courses/81632/assignments/590873)** — both due **Monday 10/26, 11:59pm** (moved from Friday 10/23) Questions and statement are due Friday 10/23 at 11:59 pm. --- # Lecture Boards + Transcript + GenAI **Twenty-second meeting, October 19 — random walks and continuum limits (i.e., why the CLT), diffusion-transport model.** Board header: *22 | 10/19/26: RW and continuum limits (i.e., why CLT), Diffusion-transport Model.* The question underneath: the [central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem) says that walkers pulling steps from a population with a well-defined first and second moment pile up, for many steps and many walkers, into something normal — *why?* Today and Wednesday answer it by connecting the walk to **[diffusion and transport](https://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation)**, "to get us out of the stochastic fuzz of it all." Two movements: a finite-difference formula the class has never seen, built from the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) it has; then the balance equation from Friday pushed, puzzle-style, into a form where those finite differences appear — and, as the [Plinko](https://en.wikipedia.org/wiki/Plinko) board shrinks, become derivatives. The hour ends on a [partial differential equation](https://en.wikipedia.org/wiki/Partial_differential_equation) with a name. ## Housekeeping The **Media Reflection** has been out about a week: go to the website, work the applet, answer questions. One or two responses are in, and the applets have been finicky — Andrew's hadn't loaded; Rachel's had trouble reaching the site at all. Known issues: the URL sometimes resolves to the Obsidian host, where the applets won't run — be at `scottstrong.org`; the campus network has produced long load times where home connections are clean; a cached page from before the switch to "Applet Land" can misbehave, so try an [incognito window](https://en.wikipedia.org/wiki/Private_browsing) or another browser. "Part of the reason I assigned that for the media reflection is to see how these applets behave out in the not-my-own-computer." Keep the reports coming by email; due dates can move — "if you've got, I don't know, 18 quizzes, two tests this week… don't let it freak you out." **MP3, first results, at the end of the week.** Not a big lift: start talking about your context, do some mathematics associated with it, have the mathematics reinterpreted by the context. Take the simplest pathway through so the loop closes — the Plinko version is "what happens if I flip a coin, go right and go left," carried all the way to the bins on the histogram; once that jives with intuition it becomes a test pad for the next question ("what if I don't have a fair coin?"). By Friday or into Monday: something start-to-finish that makes statements about the thing, however small. **MA2** is the shorter one — background and sources, "the bag of goodies that's leading you on this model analysis quest" — and its last part, *what is coming to class*, asks for forward thinking rather than a presentation: keep the instructor in the loop so the project doesn't "manifest out of the ether at the end," which is both inadvisable and impossible to measure as learning. **Office time:** Friday at 3pm is cordoned off for 310; Thursday afternoon is taken by a thesis proposal; tomorrow afternoon before about four is open by email, and a stack of differential-equations exams makes any 310 conversation a welcome interruption. Last week was short and seemed a little tired; this week is full, it's Monday, and it seems a little tired. Stickers next time. (Or coffee — "you'd like to see me lecture on 50 minutes of straight coffee.") ## Recall, then add instead of subtract "Stop me if you've heard this one before" — the room did — the Taylor series, $(1)\qquad f(x)=f(x_{0})+f'(x_{0})(x-x_{0})+\frac{f''(x_{0})}{2!}(x-x_{0})^{2}+\cdots$ ("You say you hate Taylor series, but they're so lovable. Look at all the curvy parts. If you don't want to touch the pokey parts out here, you [linearize](https://en.wikipedia.org/wiki/Linearization).") Put $x=x_{0}\pm\Delta x$ in every $x$-slot — $x_{0}$ is a number and stays; the $x_{0} s cancel inside each parenthesis; the even power squares the sign away: $f(x_{0}\pm\Delta x)=f(x_{0})\pm f'(x_{0})\,\Delta x+\frac{f''(x_{0})}{2!}\Delta x^{2}\pm\frac{f'''(x_{0})}{3!}\Delta x^{3}+\mathcal{O}(\Delta x^{4}).$ What were these for, "a long time ago now"? Isolating a derivative of $f$ at $x_{0}$ in terms of *values* of $f$ — data — rather than derivatives: **[finite-difference approximations](https://en.wikipedia.org/wiki/Finite_difference)**, each with an accuracy set by where the series was chopped. [[MATH310F26(Day 4) - Differencing, the regression run, and a name for the pipeline|Day 4]] and [[MATH310F26(Workbook 2 Companion) - Taylor series and the art of cancellation|Workbook 2]] *subtracted* the two rows and got $f'(x_{0})=\frac{f(x_{0}+\Delta x)-f(x_{0}-\Delta x)}{2\,\Delta x}+\mathcal{O}(\Delta x^{2}),$ equality up to second order, approximation the moment the $\mathcal{O}$ is dropped — but a *known* approximation, since the growth of what was chopped is known. (Lily's question, on the crossed-out $\mathcal{O}(\Delta x^{2})$: yes, it still gets deleted; "[second-order accurate](https://en.wikipedia.org/wiki/Order_of_accuracy)" *means* truncated at $\Delta x^{2}$.) **Now add them.** Carry both rows at once — top row all pluses, second row all minuses — and add: $f(x_{0}+\Delta x)+f(x_{0}-\Delta x)=2f(x_{0})+0\cdot f'(x_{0})+f''(x_{0})\,\Delta x^{2}+\mathcal{O}(\Delta x^{4}).$ The $f(x_{0})$ term stays and doubles; the first-derivative term, plus on one row and minus on the other, is "toast"; the $f''$ term doubles but came with a half, so a whole survives; the triple-prime goes the way of the prime; and two unknown constants times $\Delta x^{4}$ are still some constant times $\Delta x^{4}$ — "this [big O](https://en.wikipedia.org/wiki/Big_O_notation) is like, I don't want to keep track of those constants." The only derivative left in the game is $f''$, and everything else is the original function. Scoot the $2f(x_{0})$ over, scoot the $\mathcal{O}$ over (sign untracked, by design), divide through by $\Delta x^{2}$: $f''(x_{0})\approx\frac{f(x_{0}+\Delta x)-2f(x_{0})+f(x_{0}-\Delta x)}{\Delta x^{2}},\qquad\text{error }\ \frac{\mathcal{O}(\Delta x^{4})}{\Delta x^{2}}=\mathcal{O}(\Delta x^{2}).$ Why offload the big-O at all? Because with only data — points somebody handed over, no formula — you can compute everything on the right and nothing hidden in the rest of the series; the offloaded piece is $\Delta x^{4}/\Delta x^{2}$, so the smaller $\Delta x$, the smaller its square, the less the remainder matters, "up until I can take the limit, and then it would be exact." **A second-order accurate finite-difference approximation to $f''$ at $x_{0}$.** And why now, mid–random walk? Because inside the continuum limit about to be finished, formulas exactly like this one will appear — and the more the Plinko board shrinks, horizontally and vertically, the more like derivatives they become. ## The balance, and a three-minute puzzle Dry-erase, so the chalk stays up. Friday's picture: a strip at $t$, a strip at $t+\Delta t$ above it, and the proportion $u(x,t+\Delta t)$ of walkers at $x$ on the upper strip, all of whom came from the lower one by one of three moves — from $x-\Delta x$ going right with probability $P_{r}$, from $x$ staying with $P_{0}$, from $x+\Delta x$ going left with $P_{l}$ ($u$ is a function of two variables, one spatial, one temporal): $u(x,t+\Delta t)=P_{r}\,u(x-\Delta x,t)+P_{0}\,u(x,t)+P_{l}\,u(x+\Delta x,t).$ Three moves, probabilities summing to one, so $P_{0}=1-P_{r}-P_{l}$, and the orange term $1\cdot u(x,t)$ it produces goes to the left-hand side: $u(x,t+\Delta t)-u(x,t)=\ \cdots$ Read it: the first slot doesn't change between the two functions, so this is *at* $x$ — the change in the proportion of walkers at $x$ as time takes its step. What's left on the right is four terms — two with $P_{r}$, two with $P_{l}$ — and here the derivation changed hands. "Every time I've ever tried to derive or explain this stuff on the fly, it's never really impacted people all that much," because there is no deep process in it, only knowing the target formulas and steering toward them. So the target was written with the insides left blank — the right-hand side *should* look like $\tfrac12\,(P_{r}+P_{l})\Big[\ \cdots\ \Big]-\tfrac12\,(P_{r}-P_{l})\Big[\ \cdots\ \Big]$ — and the room got three minutes ("five might be too much"; talk to one another; "or daydream — who doesn't like a good daydream about math?") to puzzle out the brackets from the four remaining terms. Tony loaded the first one: the $-2u(x,t)$, whose $-2$ cancels the $\tfrac12$ and returns $-P_{r}u-P_{l}u$, exactly the red leftovers. Then the blue $P_{r}$- and $P_{l}$-terms split across both brackets: with $u_{-}$ short for $u(x-\Delta x,t)$, a $\tfrac12 P_{r}u_{-}$ from the first bracket needs another $\tfrac12 P_{r}u_{-}$ from the second to make a whole $P_{r}u_{-}$ — so the second bracket must hold $-u(x-\Delta x,t)$, negative times negative; and then the $P_{l}$ bookkeeping comes out on its own: $u(x,t+\Delta t)-u(x,t)=\tfrac12(P_{r}+P_{l})\Big[u(x-\Delta x,t)-2u(x,t)+u(x+\Delta x,t)\Big]-\tfrac12(P_{r}-P_{l})\Big[u(x+\Delta x,t)-u(x-\Delta x,t)\Big].$ "Does anyone almost see it? Because it's almost there." The second bracket: only the $x$-slot changes, plus $\Delta x$ minus minus $\Delta x$ — the **central difference for the first derivative** from early in the semester. The first bracket: function at minus, function with a $-2$, function at plus — the formula from twenty minutes ago. What's missing is the denominators, the corresponding changes in $x$. "I don't have them, so I'm just going to put them in. That's what you like to see in a math class, right? That denominator doesn't look right. Make it look right, then." Legally, from the old [Laplace](https://en.wikipedia.org/wiki/Laplace_transform) days: a $\Delta x^{2}$ downstairs demands a $\Delta x^{2}$ upstairs; the second bracket needs a $2\Delta x$ downstairs — the $2$ is borrowed from the $\tfrac12$ already out front — and a $\Delta x$ upstairs to pay for it. The right side now counts changes of $u$ in $x$; the left counts a change of $u$ in $t$, so divide everything by $\Delta t$: $\underbrace{\frac{u(x,t+\Delta t)-u(x,t)}{\Delta t}}_{\to\ \partial u/\partial t}=\tfrac12(P_{r}+P_{l})\frac{\Delta x^{2}}{\Delta t}\underbrace{\left[\frac{u(x+\Delta x,t)-2u(x,t)+u(x-\Delta x,t)}{\Delta x^{2}}\right]}_{\to\ \partial^{2}u/\partial x^{2}}-(P_{r}-P_{l})\frac{\Delta x}{\Delta t}\underbrace{\left[\frac{u(x+\Delta x,t)-u(x-\Delta x,t)}{2\,\Delta x}\right]}_{\to\ \partial u/\partial x}.$ ## As $\Delta x,\Delta t\ll 1$: the continuum limit No limit sign on the board, deliberately — "the limiting process tends to be a snag point in terms of understanding the meaning of this derivation" — just the red instruction to make $\Delta x$ and $\Delta t$ real small. With $\Delta t$ small, the left side — $u$ at $t+\Delta t$ minus $u$ without it, over $\Delta t$, $x$ held fixed — is a [statement of derivative](https://en.wikipedia.org/wiki/Derivative): $\partial u/\partial t$. With $\Delta x$ small, the first bracket looks more and more like the **[second derivative](https://en.wikipedia.org/wiki/Second_derivative) in $x$**, and the second bracket (its corner erased, but it was there under the red, green, and orange) like the **first derivative in $x$**. Take the Plinko board, let the walkers go, and *squint* so the walkers blur and the gridding fogs — shrink the horizontal width and the vertical height between pins — and the differences turn continuous. **This is the continuum limit, and it is a partial differential equation: the diffusion–transport equation,** $\frac{\partial u}{\partial t}=D\,\frac{\partial^{2}u}{\partial x^{2}}-c\,\frac{\partial u}{\partial x},\qquad D=\tfrac12(P_{r}+P_{l})\frac{\Delta x^{2}}{\Delta t},\quad c=(P_{r}-P_{l})\frac{\Delta x}{\Delta t},$ which gives the big-scale, "squinty-scale" picture of what the walkers do without attending to every single one. Next time: the first-derivative term is responsible for **purely moving** the walkers through space, and the second-derivative term for **letting them [spread out](https://en.wikipedia.org/wiki/Diffusion)**; working together they produce [normal distributions](https://en.wikipedia.org/wiki/Normal_distribution) that move and spread — the central limit theorem, arrived at from the other side. (Tony's question: the first-derivative term is the **[transport](https://en.wikipedia.org/wiki/Advection)** term — a *spatial* derivative that purely moves stuff; a first derivative in *time*, a velocity term, would instead act like [damping](https://en.wikipedia.org/wiki/Damping).) > [!example]- The squint, re-run here > ![[day22_continuum_limit.png]] > Left: 200,000 walkers with $P_{r}=0.3$, $P_{l}=0.2$, $P_{0}=0.5$ ($\Delta x=\Delta t=1$), histogrammed at three times, against the PDE's answer — a [Gaussian](https://en.wikipedia.org/wiki/Gaussian_function) with mean $ct$ and variance $2Dt$: it moves at speed $c=0.1$ and spreads like $\sqrt{2Dt}$. Right: the new formula's error on $f=\sin$ at $x_{0}=1$ falls as $\Delta x^{2}$ — halve the step, quarter the error — along the predicted line $\frac{|f^{(4)}(x_{0})|}{12}\Delta x^{2}$. A meeting at 1:30 kept the instructor from the office until 2:30; "next time we break down the orange and green." > [!success]- Numbers checked in one sandbox run > Adding the two Taylor rows symbolically gives $2f+f''\Delta x^{2}+\frac{f^{(4)}}{12}\Delta x^{4}+\frac{f^{(6)}}{360}\Delta x^{6}$, so the centered second difference equals $f''(x_{0})+\frac{f^{(4)}(x_{0})}{12}\Delta x^{2}+\mathcal{O}(\Delta x^{4})$ — the board's $\mathcal{O}(\Delta x^{2})$ error, with its constant named. On $\sin$ at $x_{0}=1$, halving $\Delta x$ from $0.4$ down to $0.025$ divided the error by $3.984,\ 3.996,\ 3.999,\ 4.000$, and error$/\Delta x^{2}\to 0.0701=\sin(1)/12$. The bracket decomposition is an identity: $\tfrac12(P_{r}+P_{l})[u_{-}-2u_{0}+u_{+}]-\tfrac12(P_{r}-P_{l})[u_{+}-u_{-}]$ expands to exactly $P_{r}u_{-}+P_{l}u_{+}-(P_{r}+P_{l})u_{0}$. Walkers vs. PDE ($P_{r}=0.3$, $P_{l}=0.2$): at $t=50/150/400$ the sample means were $5.00/14.99/39.96$ against $ct=5/15/40$, and the variances $24.4/73.3/195.7$ against $2Dt=25/75/200$ — the small shortfall is the exact step variance $n[(P_{r}+P_{l})-(P_{r}-P_{l})^{2}]=24.5/73.5/196$, the $(P_{r}-P_{l})^{2}$ correction being the piece the continuum limit drops as $\Delta x\to 0$. Sign check on the transport term: $P_{r}>P_{l}$ gives $c>0$, and $u_{t}=-c\,u_{x}$ carries a profile to the *right*, as the walkers do. > [!example]- Board scans — October 19, 2026 > ![[MATH310F26-Day22-Board-1.jpg]] > ![[MATH310F26-Day22-Board-2.jpg]] > ![[MATH310F26-Day22-Board-3.jpg]] > ![[MATH310F26-Day22-Board-4.jpg]] > ![[MATH310F26-Day22-Board-5.jpg]] > > 📄 [[MATH310F26-Day22-Boards-2026-10-19.pdf|Full board capture (5 pages, PDF)]]