##  [Onsager Reciprocal Relations](/onsager-reciprocal-relations-0) 

 Definition

Symmetry relations among linear transport coefficients that hold in the near‑equilibrium linear response regime when microscopic dynamics respect time‑reversal symmetry.

 

 

 

 

 

 





## Principle

Principle

Microscopic reversibility implies that cross‑coefficients coupling different thermodynamic fluxes and forces are equal when variables are paired according to their time‑reversal parity, reducing independent transport parameters.

 

 

 

 

 





## Demonstration

Demonstration

In coupled heat and charge transport, the coefficient relating a temperature gradient to an electrical current equals the coefficient relating an electric field to a heat current, when measured under conditions that preserve the relevant symmetry.

 

 

 

 

## Misapplication

Misapplication

Applying the reciprocity relations far from equilibrium, to nonlinear response, or when time‑reversal symmetry is broken (for example by a static magnetic field) leads to invalid equalities among coefficients.

 

 

 

 

 





## Consequence

Consequence

When valid, the relations constrain and reduce the number of independent linear transport coefficients, enabling consistency checks across different transport measurements and simplifying phenomenological descriptions.

 

 

 

 

## Reversal

Reversal

In media lacking time‑reversal invariance or in strongly driven regimes, reciprocal symmetry is lost and antisymmetric components or additional independent coefficients appear.

 

 

 

 

 





## Boundary

Boundary

Valid in the linear response (near‑equilibrium) regime for macroscopic averaged fluxes and forces and under microscopic time‑reversal invariance; not applicable to strongly nonlinear, far‑from‑equilibrium dynamics or in the presence of symmetry‑breaking fields or nonreciprocal materials.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Distinct from fluctuation–response statements that relate response magnitudes to equilibrium fluctuations; Onsager reciprocity concerns equality relations among distinct transport coefficients rather than fluctuation amplitudes.

 

 

 

 

 





## Synthesis

Synthesis

A set of symmetry constraints stating that, near equilibrium and under microscopic time‑reversal symmetry, the linear matrix of transport coefficients is symmetric in appropriately paired variables, constraining coupled transport phenomena.