Conventions used everywhere

Thermal physics has more arbitrary choices in it than most subjects: signs, symbols, which variables are held fixed, what counts as established. None of them changes the physics, and all of them change the formulas. Textbooks differ, and a student comparing two sources silently switching between conventions can lose a week to a stray minus sign.

So this course fixes its choices once, here, and never varies them. This page is the place to come back to when a formula elsewhere does not match one of ours.

The sign of work¶

The first law says that the internal energy of a system changes by whatever energy crosses its boundary. The arbitrary part is which direction counts as positive for work. Both common choices appear in the literature:

	This course	The other common choice
First law	$\mathrm{d}U = \dbar Q + \dbar \Won$	$\mathrm{d}U = \dbar Q - \dbar W_{\mathrm{by}}$
Positive work means	energy put into the system	energy taken out by the system
Quasistatic compression, $\mathrm{d}V<0$	$\dbar \Won = -P\,\mathrm{d}V > 0$	$\dbar W_{\mathrm{by}} = P\,\mathrm{d}V < 0$
Converting between them	$\Won = -W_{\mathrm{by}}$	$W_{\mathrm{by}} = -\Won$

This course always uses the first column. Heat added to the system is positive; work done on the system is positive; both add to $U$ , with no minus sign to remember. It is the convention of most modern statistical-physics texts, and it has the pedagogical advantage that the first law reads as plain energy bookkeeping: everything that goes in, adds.

The other convention appears symbolically on this page and nowhere else in the course, with a single exception: the engines and refrigerators of module 7 are conventionally analysed in terms of the work delivered by the machine, so that one module converts explicitly, in one clearly marked block, and converts back. Everywhere else — including problem sets, quizzes and the laboratories — the first column is the only one in use. A project lint enforces this.

$\mathrm{d}$ and $\dbar$ : state functions and path functions¶

Symbol	Meaning	Examples
$\mathrm{d}X$	an exact differential: $X$ is a property of the state	$\mathrm{d}U$ , $\mathrm{d}S$ , $\mathrm{d}V$ , $\mathrm{d}T$
$\dbar Y$	an inexact differential: $Y$ describes a transfer, not a property	$\dbar Q$ , $\dbar \Won$

A system has an internal energy and has an entropy; it does not have a heat or a work. Heat and work are names for energy in transit, and how much of each crosses the boundary depends on the route the system took, not merely on where it started and finished. Writing $\mathrm{d}Q$ would assert that some function $Q$ of the state exists to be differentiated. None does.

The mathematics behind the distinction — what makes a differential form exact, and how to test — is in the refresher; the physics arrives in module 5.

Units and constants¶

SI throughout. Metres, kilograms, seconds, kelvin, and their combinations. Where a problem quotes litres or atmospheres, convert before computing.
$\kB$ is always written explicitly. Many advanced texts set $\kB = 1$ and measure temperature in energy units. That is elegant and it hides exactly the conversion a student is trying to learn, so the core modules never do it. Entropy therefore carries units of $\mathrm{J\,K^{-1}}$ , not “nats”.
$R = N_A \kB$ . Where chemistry writes $PV = nRT$ with $n$ in moles, we write $PV = N\kB T$ with $N$ a plain count of particles. The two are the same statement.
Temperatures are absolute. Kelvin, always, unless a difference is being quoted.
The library works in plain SI floats. thermolab deliberately does not wrap quantities in a units package, so that a student can read P = N * K_B * T / V and recognise it. Dimensional correctness is enforced in the test suite instead, where every formula is re-evaluated with pint quantities.

Epistemic labels: what kind of claim is this?¶

Thermal physics mixes exact mathematics, empirical laws, modelling choices and numerical evidence, often within one paragraph. Blurring them is how students end up believing that a simulation “proved” an identity. Every boxed claim in this course therefore carries a label saying what kind of statement it is, and the boxes below are one live example of each.

Model specifications¶

Every simulation in the course — on a page or in a notebook — carries a seven-field block saying exactly what is being modelled. The fields never change and never reorder, so you can scan them. This is the one from the orientation module, which is the simplest in the course:

Model specification

System: $N$ dice, each showing one of six equally likely faces; the observable is their sum or their average.
Dynamics: none — each draw is independent of every other, with no memory between them and nothing evolving in time.
Boundary: closed trivially; $N$ is fixed for one experiment and nothing is exchanged with anything.
Ensemble: every face equally probable, by assumption rather than by proof.
Ignored: all physics of real dice — the tumble, the bounce, pip asymmetry, air drag, and the deterministic chaos that is the actual reason a throw looks random.
Valid when: the randomness is well described by independent uniform draws; a seeded pseudo-random generator is, by construction.
Failure modes: correlated draws, loaded dice, and any question about a single roll beyond its distribution.

The two fields students skip are the two worth reading. Ignored tells you what the model is blind to, which is where its answers will be wrong. Failure modes names the regimes where it stops being a useful description at all — and an exam question that asks “when does this break down?” is asking you to reproduce that field.

The shape of a module¶

Every module page climbs in the same four steps, and you can enter and leave at the step that suits you.

Opening — a physical puzzle, and predictions you commit to before any equation. No prerequisites at all.
Core — the laboratory, the derivation, and the computational check. This is the exam-relevant spine, and it is the part to work through with a pen.
Consolidation — transfer to other settings, quizzes, exam-style problems, and reflection questions.
Advanced — deeper material, always marked with the box below and always last.

Simulation never proves¶

A simulation is a numerical experiment on a model. It can show you that a result is plausible, reveal where your intuition is wrong, and catch algebra errors. It cannot establish an identity, because it only ever examined finitely many cases of one particular model.

The division of labour this course keeps to:

Derivations establish. An identity is true because it follows from stated premises.
Simulations illustrate and check. They make a result visible and they catch mistakes.
Experiments decide. Whether a model describes the world is not a question mathematics or computation can answer.

When a page says “the fitted exponent is -0.5”, that is a numerical observation about a model. When it says “the relative scatter of an average of $N$ independent quantities is $\sigma_1/(\mu_1\sqrt{N})$ ”, that is a theorem. The first is evidence for having implemented the second correctly. It is not why the second is true.