Bibliographic record
Abstract
The present study aims to find connections between two types of entropy production rates for continuous dynamical systems, one defined in the general dissipation context and the other one defined in a thermodynamics context. On one hand, the entropy production rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Subscript p"> <mml:semantics> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">e_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced by Andrey and Ruelle in the general dynamics context is based on Shannon’s entropy in the phase space that characterizes volume contraction relating to dissipation. On the other hand, the Helmholtz-Boltzmann entropy production rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper Q With dot slash upper T"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {Q}/T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced in the context of mechanical theory of thermodynamics, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper Q With dot"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the rate of heat generations and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the temperature, captures the thermodynamic entropy production relating to the mechanical energy dissipation. For certain cases of energy dissipative systems, we show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Subscript p"> <mml:semantics> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">e_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is indeed related to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper Q With dot slash upper T"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\dot {Q}/T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This consistency between the general mathematical notion of dissipation and the thermodynamic entropy production in dissipative systems suggests a unifying representation of nonequilibrium phenomenon in deterministic systems based on the theory of nonlinear dynamical systems.
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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".