Nonequilibrium effects for reactions with activation energy: Convergence of the expansions of solutions of the Boltzmann and Lorentz Fokker Planck equations with Sonine and Maxwell polynomials as basis functions
Bibliographic record
Abstract
In this paper, we consider the kinetic theory of a reactive species of mass m dilutely dispersed in a second reactive component of mass M, at equilibrium with a Boltzmann distribution. The primary objective of these studies is the correction, η, to the equilibrium rate coefficient, keq, where the nonequilibrium rate coefficient is expressed as k=keq(1−η). The Chapman–Enskog method of solution together with the Sonine (Laguerre) polynomials as basis functions for the solution of the Boltzmann equation with reaction is the method of choice of most researchers. The main objective of this paper is the study of the convergence of the Sonine polynomials in the solution of the reactive Boltzmann equation, for m/M→0, that is for the Lorentz limit. We report the solution of the Lorentz Fokker–Planck equation for m/M→0 with the expansion of the distribution function in the Maxwell polynomials as basis functions orthogonal on x∈[0, ∞] with weight function x2exp(−x2). The variable x=mv2/2kBTB is the reduced speed of the particle of mass m with TB the background gas temperature and kB the Boltzmann constant. We demonstrate that the nonclassical Maxwell polynomials provide a rapidly convergent solution of the chemical kinetic Fokker–Planck equation in the Lorentz limit. The reactive line-of-centers cross section is employed as representative of reactions with activation energy.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".