Diffusion in channeled structures. III. Quantum corrections induced by lattice vibrations
Bibliographic record
Abstract
For large energy barrier systems the lattice vibrations can play an important role in the diffusion of a guest component inside channeled structures. Here, quantum corrections to the guest diffusion are studied within a semiclassical framework in which the guest behaves classically and the lattice quantum mechanically. The permeability is expressed in terms of correlation functions that are calculated using path integrals. Forward-backward path integrals for propagators are combined, and, using the Martin-Siggia-Rose formalism [Phys. Rev. A. 8, 423 (1973)], are transformed into a set of generalized Langevin equations that reduce to the classical equation of motion at high temperatures. The random initial conditions to these equations are specified by the density matrix from which an approximate expression for the quantum mechanical potential of mean force is derived. The quantum potential of mean force and activation energy obtained for the guest inside the lattice is slightly higher than the one for the fully classical system, and the diffusion of the guest in the quantum lattice is slower compared to its fully classical counterpart. The macroscopic intrinsic permeability of $\ensuremath{\alpha}$-quartz towards neon is reported, ${P}_{\mathrm{qtm}}^{\ensuremath{'}}(300\phantom{\rule{0.3em}{0ex}}\mathrm{K})=9.91\ifmmode\times\else\texttimes\fi{}{10}^{8}\phantom{\rule{0.3em}{0ex}}\mathrm{s}∕(\mathrm{mkg})$ within the semiclassical approximation and is lower than ${P}_{\mathrm{class}}^{\ensuremath{'}}(300\phantom{\rule{0.3em}{0ex}}\mathrm{K})=1.26\ifmmode\times\else\texttimes\fi{}{10}^{9}\phantom{\rule{0.3em}{0ex}}\mathrm{s}∕(\mathrm{mkg})$ obtained in the classical case at $300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ for the same potential model.
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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".