Onsager Reciprocal Relations — Thermoelectric Transport
Linear irreversible thermodynamics: L_ij = L_ji enforced by time-reversal symmetry
Temperature & Voltage Profile
Heat & Charge Fluxes
Seebeck Efficiency η(ZT)
Peltier Effect (cooling)
Onsager matrix [L_ij]
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L₁₂ = L₂₁ ✓
ZT =
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η_max/η_Carnot =
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Figure of merit S²σ/κ
Seebeck coeff S (μV/K)
200
Electrical conductivity σ (S/m)
50000
Thermal conductivity κ (W/m·K)
1.50
ΔT hot–cold (K)
100
T_cold (K)
300
Onsager formalism:
Fluxes
J_e = L₁₁·(-∇μ/T) + L₁₂·(-∇T/T²)
and
J_Q = L₂₁·(-∇μ/T) + L₂₂·(-∇T/T²)
. Onsager proved
L₁₂ = L₂₁
from microscopic time-reversal invariance. Seebeck S=L₁₂/(T·L₁₁), Peltier Π=S·T. ZT=S²σT/κ governs efficiency.
Open circuit EMF:
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| Max power:
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| Entropy production:
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