Math 2.0

Classical thermodynamics, heat transfer Laws of thermodynamics Fact

Principle of minimum energy

Builds on Thermodynamic systemThermodynamic equilibriumEntropyInternal energyPrinciple of maximum entropy in thermodynamics

Statement

Let a composite thermodynamic system have fixed total entropy SS, total volume VV and total amounts of substance NN. Let xx denote the internal variables left free after some internal constraints are removed. Then in equilibrium the values of xx minimise the total internal energy U(S,V,N;x)U(S, V, N; x). The minimum is over all values of xx compatible with the fixed totals.

Remarks

The principle is equivalent to the principle of maximum entropy, given that temperature is positive, that is, SS increases with UU. Suppose the energy were not minimal at an equilibrium state. Then one could withdraw some energy as work at constant entropy, and give it back as heat. The energy would return to its old value while the entropy would increase, so the entropy was not maximal, a contradiction. Example: for two subsystems exchanging volume at fixed entropies, minimising U1(S1,V1)+U2(S2,V−V1)U_1(S_1, V_1) + U_2(S_2, V - V_1) gives p1=p2p_1 = p_2, where pi=−∂Ui/∂Vip_i = -\partial U_i / \partial V_i is the pressure.

Going deeper

For a student

The minimum energy principle is the version of the maximum entropy principle in the energy representation. Its Legendre transforms give the minimum principles for the other potentials. For example, at fixed temperature and volume the Helmholtz free energy is minimal in equilibrium, and at fixed temperature and pressure the Gibbs free energy is.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Entropy, concept, no evidence levelEntropyconceptHeat, concept, no evidence levelHeatconceptHeat capacity ratio, concept, no evidence levelHeat capacity ratioconceptHeat engine, concept, no evidence levelHeat engineconceptIdeal gas, concept, no evidence levelIdeal gasconceptInternal energy, concept, no evidence levelInternal energyconceptIsentropic process, concept, no evidence levelIsentropic processconceptIsothermal process, concept, no evidence levelIsothermal processconceptReversible process, concept, no evidence levelReversible processconceptTemperature, concept, no evidence levelTemperatureconceptThermal efficiency, concept, no evidence levelThermal efficiencyconceptThermodynamic cycle, concept, no evidence levelThermodynamic cycleconceptThermodynamic equilibrium, concept, no evidence levelThermodynamic equilib…conceptThermodynamic system, concept, no evidence levelThermodynamic systemconceptThermodynamic temperature, concept, no evidence levelThermodynamic tempera…conceptThermodynamic work, concept, no evidence levelThermodynamic workconceptClausius statement of the second law, statement, no evidence levelClausius statement of…statementKelvin–Planck statement of the second law, statement, no evidence levelKelvin–Planck stateme…statementCarnot's theorem, statement, no evidence levelCarnot's theoremstatementCarnot cycle, example, no evidence levelCarnot cycleexampleClausius theorem, statement, no evidence levelClausius theoremstatementSecond law of thermodynamics, statement, no evidence levelSecond law of thermod…statementPrinciple of maximum entropy in thermodynamics, statement, no evidence levelPrinciple of maximum …statementPrinciple of minimum energy, statement, no evidence levelPrinciple of minimum …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Entropy concept
  2. Heat concept
  3. Heat capacity ratio concept
  4. Heat engine concept
  5. Ideal gas concept
  6. Internal energy concept
  7. Isentropic process concept
  8. Isothermal process concept
  9. Reversible process concept
  10. Temperature concept
  11. Thermal efficiency concept
  12. Thermodynamic cycle concept
  13. Thermodynamic equilibrium concept
  14. Thermodynamic system concept
  15. Thermodynamic temperature concept
  16. Thermodynamic work concept
  17. Clausius statement of the second law statement
  18. Kelvin–Planck statement of the second law statement
  19. Carnot's theorem statement
  20. Carnot cycle example
  21. Clausius theorem statement
  22. Second law of thermodynamics statement
  23. Principle of maximum entropy in thermodynamics statement
  24. Principle of minimum energy statement

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Principle_of_minimum_energy
About this page
TagC2QNWE50TW
URIm2://C2QNWE50TW
Kindstatement, fact
WritingW0 stub
MSC80A05, 80A10
StatusThis copy of the site has no build data. Run m2 status C2QNWE50TW.
AboutInternal energy, Thermodynamic equilibrium
UsesThermodynamic system, Thermodynamic equilibrium, Entropy, Internal energy, Principle of maximum entropy in thermodynamics
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Changed3 November 2026

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