Analytical expression for the first-order density matrix of a<i>d</i>-dimensional harmonically confined Fermi gas at finite temperature
Bibliographic record
Abstract
We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-$N$ limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.
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.
How this classification was reachedexpand
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) | 0.001 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".