Temperature gradient → Electrical work

Matrix context

D = disequilibrium row · C = coupling family column. A cell reads the state of the atlas for that pair.

D.01 × C.08 · Temperature gradient × Heat engine (thermomechanical)family: Heat engine (thermomechanical)D.01 × C.15 · Temperature gradient × Magnetostrictivefamily: MagnetostrictiveD.01 × C.14 · Temperature gradient × Electromagnetic inductionfamily: Electromagnetic induction

Typed bounds on this route

Every limit the atlas holds for this route, as constraint records: reached through bounded_by claims on the route's own entities. Hard bounds (an upper bound or a formula) decide the thermodynamic-bound check above; benchmarks, constitutive relations and resource limits are listed and never decisive. No prose limit exists anywhere (pass 42).

• Carnot limit · formula-bound · hard, applies to this route
  η ≤ 1 − T_c/T_h for any heat engine between reservoirs at T_h and T_c
• Onsager–Casimir reciprocity · constitutive-relation · listed, never decisive
  L_ij(B) = ε_i ε_j L_ji(−B): the cross-coefficients of coupled linear transport are equal up to the time-reversal parities ε of the variables and reversal of the magnetic field; L_ij = L_ji is the zero-field, even-parity case

Steps

1. 01
   Temperature gradient —drives→ Thermal expansionestablishedclaim record

   thermal → mechanical

   refs [1] [2]
2. 02
   Thermal expansion —produces→ Mechanical stress or strainestablishedclaim record

   mechanical → mechanical

   • non-uniform thermal expansion is mechanically incompatible across adjacent regions of a solid, or is constrained by an adjoining material or structure, producing a strain or stress gradient; the constraining region may be the cooler part of the same continuous specimen rather than a separate stiffer body

   solid

   refs [3] [2]
3. 03
   Mechanical stress or strain —drives→ Villari effectestablishedclaim record

   mechanical → magnetic · loss to thermal

   • a magnetostrictive material under a bias field and cyclic stress

   solid · ferromagnet required

   refs [4] [5]
4. 04
   Villari effect —produces→ Changing magnetic fluxestablishedclaim record

   magnetic → magnetic

   refs [5] [6]
5. 05
   Changing magnetic flux —drives→ Electromagnetic inductionestablishedclaim record

   magnetic → electrical · loss to thermal

   ℰ = −dΦ/dt · rate of change [1/s]

   • the changing flux threads a conducting loop

   conductor required

   refs [7] [6]
6. 06
   Electromagnetic induction —produces→ Charge carriersestablishedclaim record

   electrical → electrical · loss to thermal

   refs [7] [6]
7. 07
   Charge carriers —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • carriers driven through an external load at the potential difference the source maintains

   refs [8]

Evidence for the complete composition · 0

No source on record for this exact composition. The sources below support the individual steps, not the route as a whole.

Evidence for the constituent relations · 8

1. [1]
   A. Bejan. Advanced Engineering Thermodynamics (4th ed.) (2016)

   Wiley · doi:10.1002/9781119245964 · crossref ✓ 2026-09-19 · record
2. [2]
   S. Iwakiri, Y. Miyata, T. Mori. Observation of universal thermopolarization effect in insulators (2026)

   arXiv:2605.17224 (17 May 2026), National Institute for Materials Science, Tsukuba · arxiv.org/abs/2605.17224 · record
3. [3]
   J. F. Nye. Physical Properties of Crystals: Their Representation by Tensors and Matrices (1985)

   Oxford University Press · book, no DOI · record
4. [4]
   E. Villari. Ueber die Aenderungen des magnetischen Moments, welche der Zug und das Hindurchleiten eines galvanischen Stroms in einem Stabe von Stahl oder Eisen hervorbringen (1865)

   Annalen der Physik 202, 87–122 · doi:10.1002/andp.18652020906 · crossref ✓ 2026-09-19 · record
5. [5]
   L. Wang, F. G. Yuan. Vibration energy harvesting by magnetostrictive material (2008)

   Smart Materials and Structures 17, 045009 · doi:10.1088/0964-1726/17/4/045009 · crossref ✓ 2026-09-19 · record
6. [6]
   L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii. Electrodynamics of Continuous Media (Course of Theoretical Physics, Vol. 8, 2nd ed.) (1984)

   Pergamon · book, no DOI · record
7. [7]
   M. Faraday. Experimental researches in electricity (1832)

   Philosophical Transactions of the Royal Society 122, 125–162 · doi:10.1098/rstl.1832.0006 · crossref ✓ 2026-09-19 · record
8. [8]
   N. W. Ashcroft, N. D. Mermin. Solid State Physics (1976)

   Holt, Rinehart and Winston · book, no DOI · record