When geometry is irrelevant for heat diffusion: the transition from lumped-element to field formulations
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Abstract Thermal systems are an attractive setting for exploring the connections between the lumped-element approximations of elementary circuit theory and the partial-differential field equations of mathematical physics. In a calculation suitable for an undergraduate course in mathematical physics, we show that the response function between an oscillating heater and temperature probe has a smooth crossover between a low-frequency, ‘lumped-element’ regime where the system behaves as a whole and a high-frequency regime dominated by the spatial dependence of the temperature field. Undergraduates can also easily (and cheaply) explore these ideas experimentally in a typical advanced laboratory course. Because the characteristic frequencies are low, ≈0.03 Hz, measuring the response frequency by frequency is slow and challenging; to speed up the measurements, we introduce a useful, if underappreciated, experimental technique based on a multisine power signal that sums carefully chosen harmonic components with random phases. Strikingly, a simple model assuming a one-dimensional, finite rod predicts a temperature response in the frequency domain that closely approximates experimental measurements from an irregular, blob-shaped object. A spherical model gives similar results. Thus, the frequency response of this irregular thermal system has surprisingly little sensitivity to its geometry, an example of—and justification for—the toy models so beloved of physicists.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it