Math 2.0

Statistical mechanics, structure of matter Definition

Entropy production

Builds on EntropyHeatSecond law of thermodynamicsLocal thermodynamic equilibriumDivergence

Definition

Let a closed system with entropy S(t)S(t) exchange heat with its surroundings through boundary parts jj at temperatures TjT_j, at rates Q˙j\dot Q_j. Split the rate of change of entropy as dSdt=∑jQ˙jTj+Σ.\frac{dS}{dt} = \sum_j \frac{\dot Q_j}{T_j} + \Sigma. The first term is the entropy exchange rate, and Σ\Sigma is the rate of entropy production of the system.

Now let a continuous system be in local thermodynamic equilibrium, with mass density ρ\rho and specific entropy ss. The local form is the balance law ρdsdt=−∇⋅Js+σ.\rho \frac{ds}{dt} = -\nabla \cdot J_s + \sigma. Here ddt\frac{d}{dt} is the material derivative, JsJ_s is the entropy flux and ∇⋅\nabla \cdot is the divergence. The term σ\sigma is the local entropy production per unit volume and time.

Remarks

The second law states Σ≥0\Sigma \ge 0 and σ≥0\sigma \ge 0, with equality exactly for reversible processes. Example: in a solid with heat flux JqJ_q and no other process, Js=Jq/TJ_s = J_q / T and σ=Jq⋅∇1T.\sigma = J_q \cdot \nabla \frac{1}{T}. With Fourier's law Jq=−k∇TJ_q = -k \nabla T and k>0k > 0, this gives σ=k∣∇T∣2/T2≥0\sigma = k \abs{\nabla T}^2 / T^2 \ge 0, as the second law requires. In general σ=∑kJk⋅Xk\sigma = \sum_k J_k \cdot X_k, a sum of products of fluxes JkJ_k and conjugate forces XkX_k, such as ∇(1/T)\nabla(1/T) for heat or −∇(μk/T)-\nabla(\mu_k/T) for diffusion. Non-example: the full rate dS/dtdS/dt is not the entropy production, because it also contains the exchange term, and it can be negative when a body cools.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Entropy, concept, no evidence levelEntropyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat, 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 processconceptLocal thermodynamic equilibrium, concept, no evidence levelLocal thermodynamic e…conceptReversible 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 workconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptClausius statement of the second law, statement, no evidence levelClausius statement of…statementPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptKelvin–Planck statement of the second law, statement, no evidence levelKelvin–Planck stateme…statementCarnot's theorem, statement, no evidence levelCarnot's theoremstatementDivergence, concept, no evidence levelDivergenceconceptCarnot cycle, example, no evidence levelCarnot cycleexampleClausius theorem, statement, no evidence levelClausius theoremstatementSecond law of thermodynamics, statement, no evidence levelSecond law of thermod…statementEntropy production, definition, no evidence levelEntropy productiondefinition
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. Euclidean space object
  3. Function concept
  4. Heat concept
  5. Heat capacity ratio concept
  6. Heat engine concept
  7. Ideal gas concept
  8. Isentropic process concept
  9. Isothermal process concept
  10. Local thermodynamic equilibrium concept
  11. Reversible process concept
  12. Temperature concept
  13. Thermal efficiency concept
  14. Thermodynamic cycle concept
  15. Thermodynamic system concept
  16. Thermodynamic temperature concept
  17. Thermodynamic work concept
  18. Function of several variables concept
  19. Vector-valued function concept
  20. Clausius statement of the second law statement
  21. Partial derivative concept
  22. Vector field concept
  23. Kelvin–Planck statement of the second law statement
  24. Carnot's theorem statement
  25. Divergence concept
  26. Carnot cycle example
  27. Clausius theorem statement
  28. Second law of thermodynamics statement
  29. Entropy production definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Entropy_production
About this page
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URIm2://8GVBZB9TAM
Kinddefinition
WritingW0 stub
MSC82C35, 80A17
DefinesEntropy production
UsesEntropy, Heat, Second law of thermodynamics, Local thermodynamic equilibrium, Divergence
Changed3 November 2026

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