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Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

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Tanimura, Yoshitaka·2006·Journal of the Physical Society of Japan
DOI 10.1143/jpsj.75.082001
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Abstract

Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker–Planck equation and a quantum master equation from a system–bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.

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200 of 217 shown

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Metadata

Authors
Tanimura, Y.
Venue
Journal of the Physical Society of Japan
Publisher
Physical Society of Japan
Published
2006-08-15
Volume
75
Issue
8
Pages
082001
ISSN
  • 0031-9015 (ISSN-L)
  • 1347-4073
DOI
10.1143/jpsj.75.082001
OpenAlex
W1975391438
Cited by
891
References
9
FWCI
17.80
Type
journal article
Language
en
Open access
hybrid

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 ›

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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, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems. https://app.cochat.ai/papers/10.1143/jpsj.75.082001
The paper
Tanimura, Y. (2006). Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems. Journal of the Physical Society of Japan. https://doi.org/10.1143/jpsj.75.082001

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.