Prerequisite materialHeat, Temperature, and Heat EnginesPressure Is Derived from Molecular Motion

The Number of States and Entropy

Initially, A contains 1 oscillator and B contains 3 oscillators, all with the same frequency. They share a total energy of 6ε.

熱平衡で、Aのエネルギーが0になる確率は？

The equations explain the phenomenon

  1. 1Counting the energy arrangementsWtotal=⁠q+N−1Cq……
  2. 2Counting the arrangements for each quantum number of AP(k)=⁠WA(k)×⁠WB(6 − k)/⁠Wtotal……
  3. 3Counting the arrangements for 2 solids with many oscillatorsW(qA)=⁠WA(qA) × WB(q − qA)……
  4. 4Taking the logarithm defines entropy.S=⁠kB ln W……
  5. 5Relating the increase in entropy with energy to temperature1T=⁠∂S∂E……
  6. Further development: deriving the direction of heat flow and the distribution of molecules
  7. 6The direction of heat flow follows from the entropy change.dS=⁠(1TB − 1TA) dE＞ 0……
  8. 7Calculating probabilities for a small systempi∝e−Ei/kB T……
  9. 8Deriving the distribution of molecular speedsf(v)∝v² e−m v²/2kB T……
  10. 9Normalize the probabilities and calculate the mean energy.Z = Σi e−Ei/kB T⁠，⟨E⟩ = −∂ ln Z∂β……