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.09 · Temperature gradient × Caloric (magneto/electro/elasto)family: Caloric (magneto/electro/elasto)D.01 × C.04 · Temperature gradient × Pyroelectricfamily: Pyroelectric

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

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→ Elastocaloric effectreplicatedclaim record

   mechanical → thermal

   • stress large enough to drive the martensitic transformation of a shape-memory alloy

   solid

   refs [4] [5]
4. 04
   Elastocaloric effect —produces→ Temporal temperature changereplicatedclaim record

   thermal → thermal

   refs [4] [5]
5. 05
   Temporal temperature change —drives→ Pyroelectric effectestablishedclaim record

   thermal → electrical · loss to thermal

   ΔP = p · ΔT · pyroelectric coefficient p [C/(m^2 K)]

   ΔT is the material's temperature change between two times, not a spatial temperature difference.

   • the temperature of the polar material must change in time (dT/dt ≠ 0); a static spatial temperature gradient alone produces no pyroelectric current
   • the material must be below its Curie temperature

   solid · ferroelectric required · transient only

   Re-spelled in loop-3 pass 30 from the temperature-gradient row, where it had sat as an indexing compromise, onto the temporal temperature change that is its physical driver.

   refs [6] [7] [8]
6. 06
   Pyroelectric effect —produces→ Separated surface chargeestablishedclaim record

   electrical → electrical

   refs [6]
7. 07
   Separated surface charge —converts into→ Electrical workestablishedclaim record

   electrical → electrical · loss to thermal

   • electrodes on the charged surfaces connected through a load

   refs [9]

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 · 9

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]
   J. Tušek et al.. A regenerative elastocaloric heat pump (2016)

   Nature Energy 1, 16134 · doi:10.1038/nenergy.2016.134 · crossref ✓ 2026-09-19 · record
5. [5]
   X. Moya, S. Kar-Narayan, N. D. Mathur. Caloric materials near ferroic phase transitions (2014)

   Nature Materials 13, 439–450 · doi:10.1038/nmat3951 · crossref ✓ 2026-09-19 · record
6. [6]
   S. B. Lang. Pyroelectricity: from ancient curiosity to modern imaging tool (2005)

   Physics Today 58(8), 31–36 · doi:10.1063/1.2062916 · crossref ✓ 2026-09-19 · record
7. [7]
   R. W. Whatmore. Pyroelectric devices and materials (1986)

   Reports on Progress in Physics 49, 1335 · doi:10.1088/0034-4885/49/12/002 · crossref ✓ 2026-09-19 · record
8. [8]
   C. R. Bowen et al.. Pyroelectric materials and devices for energy harvesting applications (2014)

   Energy & Environmental Science 7, 3836–3856 · doi:10.1039/C4EE01759E · crossref ✓ 2026-09-19 · record
9. [9]
   H. B. Radousky, H. Liang. Energy harvesting: an integrated view of materials, devices and applications (2012)

   Nanotechnology 23, 502001 · doi:10.1088/0957-4484/23/50/502001 · crossref ✓ 2026-09-19 · record