IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v121y1983i3p587-616.html
   My bibliography  Save this article

Path integral approach to quantum Brownian motion

Author

Listed:
  • Caldeira, A.O.
  • Leggett, A.J.

Abstract

We apply the influence-functional method of Feynman and Vernon to the study of Brownian motion at arbitrary temperature. By choosing a specific model for the dissipative interaction of the system of interest with its environment, we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient. We show that in the limit h→0 the results obtained from the influence functional formalism reduce to the classical Fokker-Planck equation. In the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature, we obtain an explicit expression for the time evolution of the complete density matrix ϱ(x, x′, t) when the system starts in a particular kind of pure state. We compare our results with those of other approaches to the problem of dissipation in quantum mechanics.

Suggested Citation

  • Caldeira, A.O. & Leggett, A.J., 1983. "Path integral approach to quantum Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 121(3), pages 587-616.
    • Handle: RePEc:eee:phsmap:v:121:y:1983:i:3:p:587-616
    DOI: 10.1016/0378-4371(83)90013-4
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0378437183900134
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/0378-4371(83)90013-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Iche, G. & Nozieres, P., 1978. "Quantum Brownian motion of a heavy particle: An adiabatic expansion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 91(3), pages 485-506.
    2. Micheline Thomine-Desmazures, 1965. "L'investigation psychosociologique en milieu rural," Économie rurale, Programme National Persée, vol. 64(1), pages 19-24.
    3. Cortázar, René & Moreno, Ernesto & Pizarro, Crisóstomo, 1977. "Condicionantes culturales y sociales de las políticas de erradicación de la pobreza," Sede de la CEPAL en Santiago (Estudios e Investigaciones) 34558, Naciones Unidas Comisión Económica para América Latina y el Caribe (CEPAL).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dorofeyev, Illarion, 2016. "Dynamics of two externally driven coupled quantum oscillators interacting with separate baths based on path integrals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 200-220.
    2. Satpathi, Urbashi & Sinha, Supurna, 2018. "Quantum Brownian motion in a magnetic field: Transition from monotonic to oscillatory behaviour," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 692-701.
    3. Tay, B.A., 2017. "Symmetry of bilinear master equations for a quantum oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 578-589.
    4. Ghosh, Aritra & Kaur, Jasleen & Bandyopadhyay, Malay, 2024. "Energetics of the dissipative quantum oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 643(C).
    5. Jasmina Jekni'c-Dugi'c & Sonja Radi' c & Igor Petrovi'c & Momir Arsenijevi'c & Miroljub Dugi'c, 2018. "Quantum Brownian oscillator for the stock market," Papers 1901.10544, arXiv.org.
    6. Tanaka, Shigenori & Umegaki, Toshihito & Nishiyama, Akihiro & Kitoh-Nishioka, Hirotaka, 2022. "Dynamical free energy based model for quantum decision making," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 605(C).
    7. Tay, B.A., 2020. "Eigenvalues of the Liouvillians of quantum master equation for a harmonic oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).
    8. Ramírez, J.E. & Herrera, J.N. & Martínez, M.I., 2016. "Getting a stochastic process from a conservative Lagrangian: A first approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 1-9.
    9. Tay, B.A., 2017. "Solutions of generic bilinear master equations for a quantum oscillator—Positive and factorized conditions on stationary states," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 42-64.
    10. Kristina Giesel & Michael Kobler, 2022. "An Open Scattering Model in Polymerized Quantum Mechanics," Mathematics, MDPI, vol. 10(22), pages 1-32, November.
    11. Zarezadeh Zakarya & Costantini Giovanni, 2019. "Particle diffusion Monte Carlo (PDMC)," Monte Carlo Methods and Applications, De Gruyter, vol. 25(2), pages 121-130, June.
    12. Saša Nježić & Jasna Radulović & Fatima Živić & Ana Mirić & Živana Jovanović Pešić & Mina Vasković Jovanović & Nenad Grujović, 2022. "Chaotic Model of Brownian Motion in Relation to Drug Delivery Systems Using Ferromagnetic Particles," Mathematics, MDPI, vol. 10(24), pages 1-19, December.
    13. Ali, Md. Manirul & Dinakaran, Rohith & Radhakrishnan, Chandrashekar, 2023. "Coherence crossover dynamics in the strong coupling regime," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 614(C).
    14. Bhattacharjee, Suraka & Satpathi, Urbashi & Sinha, Supurna, 2022. "Quantum Brownian motion of a charged oscillator in a magnetic field coupled to a heat bath through momentum variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 605(C).
    15. Meng, Xiangyi & Zhang, Jian-Wei & Guo, Hong, 2016. "Quantum Brownian motion model for the stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 281-288.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cáceres, Manuel O. & Chattah, Ana K., 1996. "On the Schrödinger-Langevin picture and the master equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 234(1), pages 322-340.
    2. Remaud, B. & Hernández, E.S., 1981. "On the phase-space dynamics of a time-dependent harmonic oscillator at finite temperature," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 107(3), pages 553-566.
    3. Braun, E., 1980. "A model of Brownian dynamics for colloidal suspensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 103(1), pages 325-342.
    4. Tegmark, Max & Yeh, Leehwa, 1994. "Steady states of harmonic oscillator chains and shortcomings of harmonic heat baths," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 202(1), pages 342-362.
    5. Bafaluy, J. & Rubí, J.M., 1988. "The “harmonic liquid” away from equilibrium," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 153(1), pages 129-146.
    6. Dekker, H., 1987. "Quantum mechanical barrier problems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 146(3), pages 387-395.
    7. Dekker, H., 1988. "Incoherent tunneling at weak bias: Strong quantum damping," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 154(1), pages 61-88.
    8. Gorini, Vittorio & Verri, Maurizio & Frigerio, Alberto, 1989. "Non-markovian behavior in low-temperature damping," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 161(2), pages 357-384.
    9. Dekker, H., 1991. "Multisite spin hopping analysis of multilevel dissipative quantum tunneling and coherence at finite temperatures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 179(1), pages 81-102.
    10. Lindenberg, Katja & Mohanty, Udayan & Seshadri, V., 1983. "Hamiltonian model for the Brownian motion of a rigid rotor," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 1-16.
    11. Kagermann, H., 1982. "Stochastic equations arising from test particle problems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(1), pages 199-206.
    12. Eckhardt, W., 1987. "On the consistent description of quantum relaxation processes: The harmonic oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 141(1), pages 81-110.
    13. Kagermann, H., 1981. "Stochastic equations arising from test particle problems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 105(3), pages 365-379.
    14. Yannouleas, C., 1985. "Quantal Brownian motion from RPA dynamics: The master and Fokker-Planck equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 132(1), pages 1-27.
    15. Dekker, H., 1985. "A contribution to the dynamical theory of radiating electrons," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 133(1), pages 1-34.
    16. Hernández, E.S. & Mizrahi, S.S., 1983. "Parametrization of open systems with effective quadratic Hamiltonians plus stochastic force," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 159-176.
    17. Yannouleas, C., 1984. "The RPA description of the quantum damped harmonic oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 125(2), pages 549-576.
    18. Brustein, Ram & Marianer, Shlomo & Schwartz, Moshe, 1991. "Langevin memory kernel and noise from Lagrangian dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 175(1), pages 47-58.
    19. Dekker, H., 1987. "Dissipative quantum mechanics: A proof of dynamical consistency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 144(2), pages 445-452.
    20. Gu, Jian-zhong & Wu, Xi-zhen & Zhuo, Yi-zhong, 1995. "The full solution of the independent oscillator model and the properties of the environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 221(4), pages 539-553.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:121:y:1983:i:3:p:587-616. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.