Liouville equation in statistical mechanics is not applicable to gases composed of colliding molecules
Bibliographic record
Abstract
Liouville equation is a fundamental one in statistical mechanics. It is rooted in ensemble theory. By ensemble theory, the variation of the system’s microscopic state is indicated by the moving of the phase point, and the moving trajectory is believed continuous. Thus, the ensemble density is thought to be a smooth function, and it observes continuity equation. When the Hamiltonian canonical equations of the molecules are applied to the continuity equation, Liouville equation can be obtained. We carefully analyze a gas composed of a great number of molecules colliding with each other. The defects in deriving Liouville equation are found. Due to collision, molecules’ momenta changes discontinuously, so that the trajectories of the phase points are actually not continuous. In statistical mechanics, infinitesimals in physics and in mathematics should be distinguished. In continuity equation that the ensemble density satisfies, the derivatives with respect to space and time should be physical infinitesimals, while in Hamiltonian canonical equations that every molecule follows, the derivatives take infinitesimals in mathematics. In the course of deriving Liouville equation, the infinitesimals in physics are unknowingly replaced by those in mathematics. The conclusion is that Liouville equation is not applicable to gases.
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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 | 0.001 |
| 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".