Math 2.0

Classical thermodynamics, heat transfer Definition

Rational thermodynamics

Builds on Continuum mechanicsConstitutive equationClausius–Duhem inequalityEntropyTemperature

Definition

Rational thermodynamics is the theory of thermodynamic processes in continuous media built on these principles.

  1. Absolute temperature θ>0\theta > 0 and specific entropy η\eta are primitive fields, defined for all states and not only near equilibrium.
  2. Every process satisfies the balance laws of mass, linear momentum, angular momentum and energy.
  3. A material is given by constitutive equations for the stress, heat flux, specific internal energy and specific entropy. They are functions of the history of the deformation and the temperature.
  4. The second law is the Clausius–Duhem inequality, and the constitutive equations must be such that it holds in every process that satisfies the balance laws.

Remarks

Example: a thermoelastic solid with heat conduction, whose constitutive equations depend on the deformation gradient, the temperature and the temperature gradient, is a material in this sense. Clause 4 then implies that the free energy does not depend on the temperature gradient and that the stress and the entropy are derivatives of the free energy. Non-example: classical non-equilibrium thermodynamics based on local equilibrium defines entropy only through the equilibrium Gibbs relation, so it does not take clause 1 as an axiom. The approach is also called continuum thermodynamics; its critics question the use of entropy and temperature far from equilibrium.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Cauchy stress tensor, concept, no evidence levelCauchy stress tensorconceptConstitutive equation, concept, no evidence levelConstitutive equationconceptContinuum mechanics, concept, no evidence levelContinuum mechanicsconceptEntropy, concept, no evidence levelEntropyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTemperature, concept, no evidence levelTemperatureconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptDivergence, concept, no evidence levelDivergenceconceptClausius–Duhem inequality, statement, no evidence levelClausius–Duhem inequa…statementRational thermodynamics, definition, no evidence levelRational thermodynami…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. Cauchy stress tensor concept
  2. Constitutive equation concept
  3. Continuum mechanics concept
  4. Entropy concept
  5. Euclidean space object
  6. Function concept
  7. Temperature concept
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Partial derivative concept
  11. Vector field concept
  12. Divergence concept
  13. Clausius–Duhem inequality statement
  14. Rational thermodynamics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rational_thermodynamics
About this page
Tag2BB088TENN
URIm2://2BB088TENN
Kinddefinition
WritingW0 stub
MSC80A17, 74A15
DefinesRational thermodynamics
UsesContinuum mechanics, Constitutive equation, Clausius–Duhem inequality, Entropy, Temperature
Changed3 November 2026

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