Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids
Bibliographic record
Abstract
Analytical expressions for radial distribution function (RDF) are of critical importance for various applications, such as development of the perturbation theories. Theoretically, RDF expressions for odd-dimensional fluids can be obtained by solving the Percus-Yevick integral equations. But for even-dimensional cases, such as the hard disk (2D) fluid, analytical expressions are infeasible. The only 2D RDF is a heuristic expression which provides acceptable estimations for an intermediate and low density range. In this work, we employ a simple and empirical expression for the 2D RDF and the 3D RDF based on an approach proposed for the 3D RDF. The parameters are determined in such a way that the final RDF expressions are thermodynamically consistent, namely the pressure and the isothermal compressibility constraints are both satisfied. The new RDFs for the 2D and 3D hard spheres are highly accurate for the entire density range up to the first-order phase transition points. The predictions of the first coordination numbers are consistent with simulation results for the 3D fluid. Finally, by using the 2D RDF with a primitive second-order perturbation theory, the pressure-volume-temperature relation and vapor-liquid equilibrium are calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data show promising results.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".