Entropy and microstates – counting Ω, S = k ln Ω and the direction of change
PhysicsThermal Physics & GasesAges 17–18
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Next in Thermal Physics & Gases
Two Einstein solids A and B swap energy quanta at random: the simulation counts the microstates Ω of every macrostate, plots the probability and the entropy S = k ln Ω, and shows that heat flows from hot to cold because the shared state has by far the most microstates. The Gas in a box screen removes a partition so a gas expands or two gases mix; the reverse becomes vanishingly unlikely as the number of particles grows.
Lesson: Entropy and the statistical meaning of the second law of thermodynamics: microstates, macrostates and S = k ln Ω
What it shows
Entropy counts possibilities. A macrostate, such as how many energy quanta sit in solid A, can be made in Ω different ways, the microstates, and S = k ln Ω. When two solids exchange quanta at random, every microstate is equally likely, so the system drifts to the macrostate with the most microstates: equal temperatures. For a gas, spread-out arrangements vastly outnumber gathered ones. With small numbers the reverse sometimes happens; with 10²³ particles it never does. That is the second law of thermodynamics.
How to use
On Two solids, set Oscillators in A, Oscillators in B and Energy quanta in total, choose Start, then press Run and watch q in A move to the most probable value; use Step for single exchanges and read the table of macrostates. On Gas in a box, choose Gas expands or Two gases mix, set Particles N and press Run.
Parameters you can change
- Screen Two solids exchanging energy, Gas in a box
- Oscillators in A 1–200
- Oscillators in B 1–200
- Energy quanta in total 1–200
- Starting state All in A, All in B, Half each
- Exchange steps per second 1–5000
- Process Gas expands, Two gases mix
- Particles N (each colour when mixing) 2–200
Questions to explore
- Why does energy flow from the hotter solid to the colder one if every single exchange is random?
- How does the chance of all the gas returning to the left half change as N grows from 4 to 100?
- Why does entropy use the logarithm of Ω instead of Ω itself?
