Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach
Bibliographic record
Abstract
The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 0"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">C_{0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -semigroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper V left-parenthesis t right-parenthesis right-parenthesis Subscript t greater-than-or-slanted-equals 0"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mo> ⩾ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\left (\mathcal {V}(t)\right )_{t \geqslant 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1 Baseline left-parenthesis double-struck upper T Superscript d Baseline times double-struck upper R Superscript d Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo> × </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> governing conservative linear kinetic equations on the torus with general scattering kernel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic k left-parenthesis v comma v prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="bold-italic">k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>v</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\boldsymbol {k}(v,v’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and degenerate (i.e. not bounded away from zero) collision frequency <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis v right-parenthesis equals integral Underscript double-struck upper R Superscript d Baseline Endscripts bold-italic k left-parenthesis v prime comma v right-parenthesis bold-italic m left-parenthesis normal d v prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> σ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo> ∫ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mi mathvariant="bold-italic">k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>v</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi mathvariant="bold-italic">m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:msup> <mml:mi>v</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v’,v)\boldsymbol {m}(\mathrm {d}v’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , (with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic m left-parenthesis normal d v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="bold-italic">m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\boldsymbol {m}(\mathrm {d}v)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> being absolutely continuous with respect to the Lebesgue measure). We show in particular that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".