The DOI matches its Crossref record.
Stochastic Liouville Equations
Abstract
When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cited by
Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature
Ishizaki, A., & Fleming, G. R. (2009) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0908989106 · Cited by 891
Colored Noise in Dynamical Systems
Häunggi, P., & Jung, P. (1994) · Advances in Chemical Physics · doi:10.1002/9780470141489.ch4 · Cited by 522
A Stochastic Theory of Line Shape
Kubo, R. (1969) · Advances in Chemical Physics · doi:10.1002/9780470143605.ch6 · Cited by 441
Self-citationQuantum and classical relaxation rates from classical simulations
Bader, J. S., & Berne, B. J. (1994) · The Journal of Chemical Physics · doi:10.1063/1.466780 · Cited by 417
On the Cyto‐Toxicity Caused by Quantum Dots
Shiohara, A., Hoshino, A., Hanaki, K., Suzuki, K., & Yamamoto, K. (2004) · Microbiology and Immunology · doi:10.1111/j.1348-0421.2004.tb03478.x · Cited by 385
Metadata
- Authors
- Kubo, R.
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1963-02-01
- Volume
- 4
- Issue
- 2
- Pages
- 174-183
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W1969211920
- Cited by
- 1,013
- References
- 0
- FWCI
- 2.93
- Type
- journal article
- Language
- en
- Open access
- closed
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Stochastic Liouville Equations. https://app.cochat.ai/papers/10.1063/1.1703941
- The paper
- Kubo, R. (1963). Stochastic Liouville Equations. Journal of Mathematical Physics. https://doi.org/10.1063/1.1703941
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
