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, k e q , where the nonequilibrium rate coefficient is expressed as k = k e q ( 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 x 2 exp ( − x 2 ) . The variable x = m v 2 / 2 k B T B is the reduced speed of the particle of mass m with T B the background gas temperature and k B 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. • Nonequilibrium effects for reactions with activation energy. • The reactant to heat bath mass ratio is an important variable. • Solutions of the Boltzmann equation and Lorentz Fokker Planck equation are considered. • The Chapman–Enskog method of solution is employed. • The solutions with Sonine polynomials and Maxwell polynomials are compared.
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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) | 0.000 | 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 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".