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Path integral approach to quantum Brownian motion

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  • 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
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    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. 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.
    5. 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).
    6. 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).
    7. 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.
    8. 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.
    9. Kristina Giesel & Michael Kobler, 2022. "An Open Scattering Model in Polymerized Quantum Mechanics," Mathematics, MDPI, vol. 10(22), pages 1-32, November.
    10. Zarezadeh Zakarya & Costantini Giovanni, 2019. "Particle diffusion Monte Carlo (PDMC)," Monte Carlo Methods and Applications, De Gruyter, vol. 25(2), pages 121-130, June.
    11. 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.
    12. 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).
    13. 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).
    14. 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.

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