A reading series for physics-loving high-schoolers and undergraduates
Temperature is the exchange rate between energy and bits ── and k_B is not a constant of nature at all but a unit conversion factor that vanishes the moment you measure temperature in energy. Only one dimensionless ratio matters, E/k_BT, and the single number 25 meV at room temperature lights up chemistry, semiconductors and life alike. This series adds a fourth axis, k_B, to the c · ℏ · G of "The Physics Cube."
In 2019 humanity stopped measuring k_B and fixed its value instead. Measure temperature in energy and k_B = 1 ── the kelvin is a historical leftover, and the only thing that matters is the dimensionless ratio E/k_BT. One number, 25 meV at room temperature, tells you what breaks and what stays frozen.k_BT(300K) ≈ 1/40 eV
Why e^(−E/k_BT)? Toss counters into boxes at random and an exponential distribution rises by itself ── its slope is the temperature. The real quantity is not T but β = 1/k_BT, which is why negative and infinite temperatures cause no trouble at all.β = 1/k_BT
Work and heat differ only in whether you are keeping the books (a handshake with "Renormalization That Clicks," Ep. 1). Equipartition gives ½k_BT per degree of freedom ── until ℏω exceeds k_BT, at which point that degree of freedom refuses delivery. Hence the staircase in the heat capacity of a diatomic molecule: quantum mechanics showing up on a thermometer.the stairs come from ℏω/k_BT
In small systems over short times, entropy really does decrease sometimes ── and the odds are given exactly: P(+σ)/P(−σ) = e^(σ/k_B). Verified with colloidal beads and RNA hairpins. The second law looks like a law only because N is large.P(+σ)/P(−σ) = e^(σ/k_B)
Where energy has a ceiling, T < 0 is realisable ── and it is not cold but hotter than any positive temperature. Obvious once you accept that the real quantity is 1/T, which slides smoothly from + through 0 to −. Achieved with nuclear spins (1951) and with ultracold atoms (2013).the real quantity is 1/T, not T
Rotate time 90 degrees onto the imaginary axis (a Wick rotation) and the quantum partition function appears ── and going once around imaginary time multiplies you by the Boltzmann factor. The period is ℏ/k_BT. Temperature was the inverse of a circumference. This settles the debt from "Cosmology That Clicks," Ep. 9.period of τ = ℏ/k_BT
Coast and it is vacuum; accelerate and the same vacuum becomes a bath at T = ℏa/2πck_B (the Unruh effect). The final form of Episode 1's "temperature is not a property of matter" ── temperature depends on the observer's state of motion too. One line about horizons hands the baton to Hawking temperature.T = ℏa / 2πck_B