Math 2.0

Classical thermodynamics, heat transfer Non-equilibrium thermodynamics Example

Szilard engine

Builds on Landauer's principleIsothermal processIdeal gasSecond law of thermodynamics

Example

The Szilard engine is a thought experiment of Leo Szilard that makes Maxwell's demon concrete. A single gas molecule moves in a box of volume VV in contact with a heat bath at temperature TT.

  1. A partition is inserted in the middle, so the molecule is in the left or the right half, each with probability 1/21/2.
  2. A demon measures which half holds the molecule and records the result as one bit.
  3. Using the result, it attaches a load to the correct side. The molecule pushes the partition as a piston in an isothermal expansion from V/2V/2 to VV.
  4. The partition is removed, and the box is back in its initial state.

Treat the one molecule as an ideal gas with pV=kBTpV = k_B T. The work done in step 3 is W=∫V/2VkBTV′ dV′=kBTln⁡2,W = \int_{V/2}^{V} \frac{k_B T}{V'}\, dV' = k_B T \ln 2, and it equals the heat taken from the bath. So each cycle seems to turn heat from one bath fully into work, against the second law.

Remarks

The cycle is not closed until the demon's one-bit memory is reset. By Landauer's principle, the reset releases at least kBTln⁡2k_B T \ln 2 of heat to the bath, which cancels the gain of step 3. So the net work per full cycle is at most 00, and the second law holds.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
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 gasconceptIsentropic 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 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…statementLandauer's principle, statement, no evidence levelLandauer's principlestatementSzilard engine, example, no evidence levelSzilard engineexample
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. Isentropic process concept
  7. Isothermal process concept
  8. Reversible process concept
  9. Temperature concept
  10. Thermal efficiency concept
  11. Thermodynamic cycle concept
  12. Thermodynamic system concept
  13. Thermodynamic temperature concept
  14. Thermodynamic work concept
  15. Clausius statement of the second law statement
  16. Kelvin–Planck statement of the second law statement
  17. Carnot's theorem statement
  18. Carnot cycle example
  19. Clausius theorem statement
  20. Second law of thermodynamics statement
  21. Landauer's principle statement
  22. Szilard engine example

Further reading

  1. Web https://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_information_theory
About this page
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Kindexample
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MSC80A05, 82B30, 94A17
IllustratesMaxwell's demon
UsesLandauer's principle, Isothermal process, Ideal gas, Second law of thermodynamics
Changed3 November 2026

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