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Onsager Reciprocal Relations

1930
  • Lars Onsager
Laboratory scene demonstrating thermoelectricity applications with Peltier cooler and generator.

A key theorem in non-equilibrium thermodynamics, these relations express the equality of certain cross-coefficients between coupled flows of energy and matter. They state that in the absence of a magnetic field, the matrix of transport coefficients is symmetric: [latex]L_{\alpha\beta} = L_{\beta\alpha}[/latex]. This links seemingly unrelated transport phenomena, such as the Seebeck and Peltier effects in thermoelectricity.

The Onsager reciprocal relations extend classical thermodynamics, which primarily deals with systems in equilibrium, to systems that are near equilibrium but experiencing irreversible processes. These processes are described by a set of linear equations relating termodinámica ‘fluxes’ (like heat flow, electric current, or mass diffusion) to thermodynamic ‘forces’ (like temperature gradient, electric potential gradient, or chemical potential gradient). For example, a heat flux ([latex]J_q[/latex]) can be caused by a temperature gradient ([latex]X_q[/latex]) and an electric potential gradient ([latex]X_e[/latex]), so [latex]J_q = L_{qq}X_q + L_{qe}X_e[/latex].

The novelty of Onsager’s work, for which he won the 1968 Nobel Prize in Chemistry, was to prove that the cross-coefficient linking the electrical force to heat flow ([latex]L_{qe}[/latex]) is equal to the coefficient linking the thermal force to electric current ([latex]L_{eq}[/latex]). This symmetry is not obvious from macroscopic observation but arises from the principle of microscopic reversibility—the idea that the equations of motion for individual particles are symmetric with respect to time reversal. These relations dramatically simplified the study of complex transport phenomena by reducing the number of independent coefficients that need to be measured experimentally.

UNESCO Nomenclature: 2212
– Thermodynamics, statistical physics, and condensed matter

Tipo

Sistema abstracto

Disrupción

Incremental

Utilización

Nicho/Especialización

Precursores

  • classical thermodynamics (First and Second Laws)
  • empirical laws like Fourier’s law of heat conduction and Fick’s law of diffusion
  • discovery of various cross-phenomena like the Seebeck, Peltier, and Thomson effects
  • statistical mecánica of Ludwig Boltzmann and J. Willard Gibbs
  • Einstein’s work on Brownian motion and fluctuation-dissipation theorems

Aplicaciones

  • thermoelectricity (peltier coolers, radioisotope thermoelectric generators)
  • electrokinetics (electro-osmosis for microfluidics, streaming potential)
  • analysis of transport phenomena across biological membranes
  • materials science for designing multi-functional materials
  • geophysics for modeling coupled heat and fluid flow in earth’s crust

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Related to: Onsager reciprocal relations, non-equilibrium thermodynamics, transport phenomena, fluxes, forces, thermoelectricity, seebeck effect, Peltier effect, microscopic reversibility, statistical mechanics.

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