Math 2.0

Classical thermodynamics, heat transfer Definition

Local thermodynamic equilibrium

Builds on Thermodynamic equilibriumIntensive and extensive propertiesFundamental thermodynamic relationTemperature

Definition

Let a continuous system occupy a region Ω\Omega during a time interval II. The system is in local thermodynamic equilibrium if these conditions hold.

  1. Around each point x∈Ωx \in \Omega there are cells that are small against the lengths over which macroscopic quantities vary. Yet they contain enough particles for thermodynamics to apply.
  2. At each time t∈It \in I, each cell has the variables of an equilibrium state. These are the specific internal energy u(x,t)u(x,t), specific volume v(x,t)v(x,t) and mass fractions wk(x,t)w_k(x,t). They also include the intensive variables: temperature T(x,t)T(x,t), pressure p(x,t)p(x,t) and chemical potentials μk(x,t)\mu_k(x,t).
  3. These fields obey the equilibrium equations of state. The specific entropy s(x,t)s(x,t) obeys the fundamental relation in the Gibbs form T ds=du+p dv−∑kμk dwkT\,ds = du + p\,dv - \sum_k \mu_k\,dw_k along the motion of each material element.

Remarks

Example: heat conduction in a copper bar with ends at 300300 kelvin and 400400 kelvin. The temperature changes over centimetres, while a cell of side 1 μm1\,\mu\mathrm{m} holds about 101110^{11} atoms, so clauses 1 to 3 hold to high accuracy. Non-example: a shock wave in a dilute gas, a few mean free paths thick, fails clause 1. No cell is both small against the shock and large against the mean free path. Here μk\mu_k is the chemical potential per unit mass of component kk.

Reading path

To understand this page from the ground up, read these 27 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 gasconceptIntensive and extensive properties, concept, no evidence levelIntensive and extensi…conceptInternal energy, concept, no evidence levelInternal energyconceptIsentropic process, concept, no evidence levelIsentropic processconceptIsothermal process, concept, no evidence levelIsothermal processconceptReversible process, concept, no evidence levelReversible processconceptState function, concept, no evidence levelState functionconceptTemperature, 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…statementFirst law of thermodynamics, statement, no evidence levelFirst law of thermody…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…statementFundamental thermodynamic relation, statement, no evidence levelFundamental thermodyn…statementLocal thermodynamic equilibrium, definition, no evidence levelLocal thermodynamic e…definition
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. Intensive and extensive properties concept
  7. Internal energy concept
  8. Isentropic process concept
  9. Isothermal process concept
  10. Reversible process concept
  11. State function concept
  12. Temperature concept
  13. Thermal efficiency concept
  14. Thermodynamic cycle concept
  15. Thermodynamic equilibrium concept
  16. Thermodynamic system concept
  17. Thermodynamic temperature concept
  18. Thermodynamic work concept
  19. Clausius statement of the second law statement
  20. First law of thermodynamics statement
  21. Kelvin–Planck statement of the second law statement
  22. Carnot's theorem statement
  23. Carnot cycle example
  24. Clausius theorem statement
  25. Second law of thermodynamics statement
  26. Fundamental thermodynamic relation statement
  27. Local thermodynamic equilibrium definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics
About this page
TagB5YGBYPDBM
URIm2://B5YGBYPDBM
Kinddefinition
WritingW0 stub
MSC80A17, 82C35
DefinesLocal thermodynamic equilibrium
UsesThermodynamic equilibrium, Intensive and extensive properties, Fundamental thermodynamic relation, Temperature
Changed3 November 2026

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