Problem set — orientation

Problem set: orientation¶

Exam-style problems. Work them with a pen before touching a computer; the last one is meant to be finished numerically. Solutions and marking rubrics live with the instructor material and are deliberately not on this site.

Take $\kB = 1.381 \times 10^{-23}\ \mathrm{J\,K^{-1}}$ throughout.

Problem 1 — the honest die¶

(a) From the definitions, compute the mean $\mu_1$ and variance $\sigma_1^2$ of one fair six-sided die. Show the sums; do not quote the closed form.

(b) A die is loaded so that a six comes up with probability $1/3$ and each of the other five faces with probability $2/15$. Compute its mean and variance, and say in one sentence why the variance moved in the direction it did.

(c) Two fair dice are rolled independently. By enumerating all 36 equally likely outcomes, compute $\langle X + Y \rangle$ and $\operatorname{Var}(X+Y)$ directly, and check both against the addition rules derived on the module page.

(d) Now let $Y = X$ — the “second die” is a copy of the first, read twice. Compute $\operatorname{Var}(X + Y)$ for this case and compare with (c). Which hypothesis of the addition rule has been violated, and by how much does the answer change?

Problem 2 — scatter bookkeeping¶

(a) You average 100 fair dice. Give the mean and the standard deviation of the result.

(b) How many dice must you average for the standard deviation of the answer to fall to 0.001? Comment on whether this is a practical experiment.

(c) For the same $N$ as in (b), give the standard deviation of the sum of those dice, and state how it changed relative to $N = 100$. Explain in one sentence how the sum can become less predictable while the average becomes more predictable.

(d) A laboratory measurement is repeated $M$ times and the results averaged. The precision is said to improve as $\sqrt{M}$. State precisely what must be true of the individual measurements for this to hold, and give one realistic way an experiment can violate it — in which case averaging more data buys you much less than $\sqrt{M}$.

Problem 3 — dimensional detective¶

(a) Three candidate expressions are offered for a molecular speed: $\sqrt{3\kB T/m}$, $\quad 3\kB T/m$, $\quad \sqrt{3\kB T m}$. Reject the ones that cannot be a speed, showing the dimensions of each.

(b) From $PV = N\kB T$, with $N$ a pure number, deduce the SI units of $\kB$.

(c) Show that the pascal and the joule per cubic metre are the same unit. Given that, may we conclude that the pressure of a gas equals its energy density? Explain what dimensional analysis does and does not license here.

(d) A calculation returns an internal energy of $6.214 \times 10^{-21}\ \mathrm{J}$ where theory predicts $6.207 \times 10^{-21}\ \mathrm{J}$. A student checks agreement by testing whether the two differ by less than $10^{-12}\ \mathrm{J}$ and reports a perfect match. State what is wrong with the test, propose a better one, and apply it.

Problem 4 — partial derivatives at work¶

An ideal gas has $P(V,T) = N \kB T / V$ with $N$ fixed.

(a) Compute $(\partial P/\partial T)_V$ and $(\partial P/\partial V)_T$, and write the total differential $\mathrm{d}P$.

(b) Take $N = 1.00\times10^{22}$, $V = 1.00\times10^{-3}\ \mathrm{m^3}$ and $T = 300\ \mathrm{K}$. Use the total differential to estimate the change in pressure when the temperature rises by $3\ \mathrm{K}$ while the volume falls by $1.0\times10^{-5}\ \mathrm{m^3}$. Then compute the change exactly and give the fractional error of the estimate.

(c) Repeat (b) with the volume rising by $1.0\times10^{-5}\ \mathrm{m^3}$ instead. Explain the result without calculation, using only the form of $P(V,T)$.

(d) For $f(x,y) = x^2 y$, evaluate $(\partial f/\partial x)_y$ at $(1,1)$ and $\mathrm{d}f/\mathrm{d}x$ at $x=1$ along the line $y=x$. The two differ. Say exactly which question each of them answers, and identify the term in the total differential that accounts for the difference.

Problem 5 — exact or not¶

Consider the two differential forms

(a) Apply the mixed-partials criterion to each and state which is exact.

(b) For the exact one, find its potential $f$ explicitly.

(c) Integrate both forms from $(0,0)$ to $(1,1)$ along (i) the straight line $y=x$ and (ii) the two-leg route across to $(1,0)$ and then up. Present the four numbers in a table and comment on the pattern.

(d) Compute $\oint \omega_2$ counter-clockwise around the unit square with corners $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. Explain why the answer had to be non-zero.

(e) In module 5 the plane becomes the $P$–$V$ plane and one of the forms you integrate is inexact. In one sentence, say what physical consequence follows from that inexactness — no calculation.

Problem 6 — computational¶

Using the module’s laboratory notebook and the thermolab.sampling module:

(a) For $N \in \{4, 16, 64, 256, 1024\}$, measure the relative spread of the average of $N$ dice over at least 500 repetitions. Plot it against $N$ on log-log axes and fit the exponent with validation.scaling_exponent. Report the fitted value.

(b) Check the coefficient as well as the exponent: multiply each measured spread by $\sqrt{N}$ and compare with $\sigma_1/\mu_1$. Explain why a test of the exponent alone would be weak evidence.

(c) Design and run the falsifying experiment for the claim “the average settles because later rolls compensate earlier ones”: generate many runs, keep only those whose first ten rolls averaged above 4.5, and measure the mean of the subsequent rolls in those runs. Report the result with an uncertainty and state what it rules out.

(d) Repeat (a) with a loaded die of your choice. State which of the two numbers — the exponent or the coefficient — changed, and why the derivation on the module page predicts exactly that.

(e) State one thing your numerical results do not prove.