The Lambert W function and quantum statistics
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
We present some applications of the Lambert W function (W function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the W function. We then present some applications of this formula including Bose–Einstein systems in d dimensions, Maxwell–Boltzmann systems, and black body radiation. We also show that for the appropriate parameter values, this formula reduces to an analytic expression in connection with Wien’s displacement law that was found in a previous study. In addition, we show that for Maxwell–Boltzmann and Bose–Einstein systems, the W function allows us to express the temperature of the system as a function of the thermodynamically relevant chemical potential, the particle density, and other parameters. Finally, we explore an indirect relationship of the W function to the polylogarithm function and to the Lambert transform.
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