= Onsager reciprocal relation
{c}
An <Onsager reciprocal relation> connects cross-couplings between near-equilibrium transport processes. Choose thermodynamic forces $X_j$ and fluxes $J_i$ so that their products give the <entropy production>, and write $J_i=\sum_jL_{ij}X_j$. With <microscopic reversibility>, no imposed time-reversal-breaking field and variables with the appropriate matching time-reversal parity, the coefficients obey $L_{ij}=L_{ji}$. The choice of conjugate force and flux matters: arbitrary rescaling of just one coefficient need not preserve its numerical symmetry.
For coupled thermal and electrical transport, a thermal driving force can induce charge flow while an electrical driving force induces heat flow. Reciprocity relates those cross effects, reducing the number of independent transport measurements. <Lars Onsager>'s 1931 paper develops this reasoning from <microscopic reversibility> and distinguishes it from thermodynamic positivity alone.<Reciprocal Relations in Irreversible Processes. I. by Lars Onsager>[Abstract and §§1–3: coupled irreversible processes; reciprocal relation supplies information beyond thermodynamics]{ref} External <magnetic fields> and odd time-reversal variables require a generalized parity-sensitive relation rather than an unconditional symmetric matrix.
Back to article page
