- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Open quantum systems beyond equilibrium: Lindblad equation  and path integral molecular dynamics

Benedikt M. Reible*, Somayeh Ahmadkhani†, and Luigi Delle Site‡

  • *Contact author: benedikt.reible@fu-berlin.de
  • †Contact author: s.ahmadkhani@fu-berlin.de
  • ‡Contact author: luigi.dellesite@fu-berlin.de
PDF

Phys. Rev. A 113, 042205 – Published 3 April, 2026

DOI: https://doi.org/10.1103/rrhf-h7xp

Abstract

The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype and surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multiatomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need for explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely, the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (88)

  1. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002).
  2. A. Rivas and S. F. Huelga, Open Quantum Systems (Springer, Berlin, 2012).
  3. B. Vacchini, Open Quantum Systems (Springer, Cham, 2024).
  4. R. M. Goyal, S. Maharaj, P. Kumar, and M. Chandrasekhar, Exploring quantum materials and applications: A review, J Mater. Sci.: Mater. Eng. 20, 4 (2025).
  5. G. Xavier, J. A. Larsson, P. Villoresi, G. Vallone, and A. Cabello, Energy-time and time-bin entanglement: Past, present and future, npj Quantum Inf. 11, 129 (2025).
  6. A. Acín, I. Bloch, H. Buhrman, et al., The quantum technologies roadmap: A European community view, New J. Phys. 20, 080201 (2018).
  7. J. Prior, A. W. Chin, S. F. Huelga, and M. B. Plenio, Efficient simulation of strong system-environment interactions, Phys. Rev. Lett. 105, 050404 (2010).
  8. G. McCauley, B. Cruikshank, D. I. Bondar, and K. Jacobs, Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes, npj Quantum Inf. 6, 74 (2020).
  9. E. Braaten, H.-W. Hammer, and G. P. Lepage, Lindblad equation for the inelastic loss of ultracold atoms, Phys. Rev. A 95, 012708 (2017).
  10. P. H. Guimarães, G. T. Landi, and M. J. de Oliveira, Nonequilibrium quantum chains under multisite Lindblad baths, Phys. Rev. E 94, 032139 (2016).
  11. A. Bose, Incorporation of empirical gain and loss mechanism in open quantum systems through path integral Lindblad dynamics, J. Phys. Chem. Lett. 15, 3363 (2024).
  12. B. M. Reible, A. Djurdjevac, and L. Delle Site, Chemical potential and variable number of particles control the quantum state: Quantum oscillators as a showcase, APL Quantum 2, 016124 (2025).
  13. S. Morais, Advances and applications of carbon nanotubes, Nanomaterials 13, 2674 (2023).
  14. A. Singh, L. Maisenbacher, Z. Lin, J. J. Axelrod, C. D. Panda, and H. Muller, Dynamics of a buffer-gas-loaded, deep optical trap for molecules, Phys. Rev. Res. 5, 033008 (2023).
  15. S. Chattaraj and G. Galli, Energy transfer between localized emitters in photonic cavities from first principles, Phys. Rev. Res. 7, 033229 (2025).
  16. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  17. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N‐level systems, J. Math. Phys. 17, 821 (1976).
  18. D. Manzano, A short introduction to the Lindblad master equation, AIP Adv. 10, 025106 (2020).
  19. R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20, 367 (1948).
  20. M. E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation, 2nd ed. (Oxford University Press, Oxford, 2023).
  21. N. Liang, A. Fiorentino, and B. Song, Thermal transport in amorphous carbon nanotubes, Phys. Rev. B 112, 094205 (2025).
  22. D. Donadio and G. Galli, Thermal conductivity of isolated and interacting carbon nanotubes: Comparing results from molecular dynamics and the Boltzmann transport equation, Phys. Rev. Lett. 99, 255502 (2007).
  23. G. Barbalinardo, Z. Chen, H. Dong, Z. Fan, and D. Donadio, Ultrahigh convergent thermal conductivity of carbon nanotubes from comprehensive atomistic modeling, Phys. Rev. Lett. 127, 025902 (2021).
  24. G. Benenti, D. Donadio, S. Lepri, and R. Livi, Non-Fourier heat transport in nanosystems, Riv. Nuovo Cimento 46, 105 (2023).
  25. R. Scipioni, D. Donadio, L. M. Ghiringhelli, and L. Delle Site, Proton wires via one-dimensional water chains adsorbed on metallic steps, J. Chem. Theory Comput. 7, 2681 (2011).
  26. D. Donadio and G. Galli, Shedding light on water wires, Physics 18, 54 (2025).
  27. J. Paulino, M. Yi, I. Hung, and T. A. Cross, Functional stability of water wire–carbonyl interactions in an ion channel, Proc. Natl. Acad. Sci. USA 117, 11908 (2020).
  28. S. Jang and G. A. Voth, Path integral centroid variables and the formulation of their exact real time dynamics, J. Chem. Phys. 111, 2357 (1999).
  29. S. Habershon, D. Manolopoulos, T. E. Markland, and T. F. Miller III, Ring-polymer molecular dynamics: Quantum effects in chemical dynamics from classical trajectories in an extended phase space, Annu. Rev. Phys. Chem. 64, 387 (2013).
  30. R. P. Feynman, A. R. Hibbs, and D. F. Styer, Quantum Mechanics and Path Integrals, emended ed. (Dover Publications, Mineola, NY, 2010).
  31. S. Mazzucchi, Mathematical Feynman Path Integrals and Their Applications, 2nd ed. (World Scientific, Singapore, 2022).
  32. H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10, 545 (1959).
  33. E. Nelson, Feynman integrals and the Schrödinger equation, J. Math. Phys. 5, 332 (1964).
  34. P. R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2, 238 (1968).
  35. M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed. (Academic Press, San Diego, 1980).
  36. K. Huang, Statistical Mechanics, 2nd ed. (John Wiley & Sons, New York, 1987).
  37. S. Blair, G. Zicari, A. Belenchia, A. Ferraro, and M. Paternostro, Nonequilibrium quantum probing through linear response, Phys. Rev. Res. 6, 013152 (2024).
  38. G. Ciccotti, R. Kapral, and A. Sergi, Non-equilibrium molecular dynamics, in Handbook of Materials Modeling, edited by S. Yip (Springer, Dordrecht, 2005), pp. 745–761.
  39. G. Ciccotti and G. Jacucci, Direct computation of dynamical response by molecular dynamics: The mobility of a charged Lennard-Jones particle, Phys. Rev. Lett. 35, 789 (1975).
  40. S. Bonella, M. Ferrario, and G. Ciccotti, Thermal diffusion in binary mixtures: Transient behavior and transport coefficients from equilibrium and nonequilibrium molecular dynamics, Langmuir 33, 11281 (2017).
  41. R. Ebrahimi Viand, F. Höfling, R. Klein, and L. Delle Site, Theory and simulation of open systems out of equilibrium, J. Chem. Phys. 153, 101102 (2020).
  42. H. Wang, C. Schütte, G. Ciccotti, and L. Delle Site, Exploring the conformational dynamics of alanine dipeptide in solution subjected to an external electric field: A nonequilibrium molecular dynamics simulation, J. Chem. Theory Comput. 10, 1376 (2014).
  43. M. Ferrario, G. Ciccotti, D. Mansutti, and A. DiCarlo, A NEMD approach to the melt-front evolution under gravity, J. Chem. Phys. 162, 074503 (2025).
  44. G. Ciccotti, G. Jacucci, and I. R. McDonald, “Thought-experiments” by molecular dynamics, J. Stat. Phys. 21, 1 (1979).
  45. C.-A. Pillet, Quantum dynamical systems, in Open Quantum Systems I. The Hamiltonian Approach, Lecture Notes in Mathematics, Vol. 1880, edited by S. Attal, A. Joye, and C.-A. Pillet (Springer, Berlin, 2006), pp. 107–182.
  46. From a structural point of view, the quantum-mechanical argument is analogous to the classical argument of (10) because the trace gives rise to an inner product on the space of Hilbert-Schmidt operators; see, e.g., Ref. [ [88], Sec. II, esp. Eq. (II.13)].
  47. E. B. Davies, Markovian master equations, Commun. Math. Phys. 39, 91 (1974).
  48. E. B. Davies, Markovian master equations. II, Math. Ann. 219, 147 (1976).
  49. R. Dümcke and H. Spohn, The proper form of the generator in the weak coupling limit, Z. Physik B 34, 419 (1979).
  50. A. Suárez, R. Silbey, and I. Oppenheim, Memory effects in the relaxation of quantum open systems, J. Chem. Phys. 97, 5101 (1992).
  51. F. Benatti and R. Floreanini, Open quantum dynamics: Complete positivity and entanglement, Int. J. Mod. Phys. B 19, 3063 (2005).
  52. R. S. Whitney, Staying positive: Going beyond Lindblad with perturbative master equations, J. Phys. A: Math. Theor. 41, 175304 (2008).
  53. D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, Searching for Lindbladians obeying local conservation laws and showing thermalization, Phys. Rev. A 107, 062216 (2023).
  54. F. Campaioli, J. H. Cole, and H. Hapuarachchi, Quantum master equations: Tips and tricks for quantum optics, quantum computing, and beyond, PRX Quantum 5, 020202 (2024).
  55. P. Gaspard and M. Nagaoka, Slippage of initial conditions for the Redfield master equation, J. Chem. Phys. 111, 5668 (1999).
  56. G. Schaller and T. Brandes, Preservation of positivity by dynamical coarse graining, Phys. Rev. A 78, 022106 (2008).
  57. D. Taj and F. Rossi, Completely positive Markovian quantum dynamics in the weak-coupling limit, Phys. Rev. A 78, 052113 (2008).
  58. F. Benatti, R. Floreanini, and U. Marzolino, Entangling two unequal atoms through a common bath, Phys. Rev. A 81, 012105 (2010).
  59. C. Majenz, T. Albash, H.-P. Breuer, and D. A. Lidar, Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations, Phys. Rev. A 88, 012103 (2013).
  60. J. Jeske, D. J. Ing, M. B. Plenio, S. F. Huelga, and J. H. Cole, Bloch-Redfield equations for modeling light-harvesting complexes, J. Chem. Phys. 142, 064104 (2015).
  61. D. Farina and V. Giovannetti, Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation, Phys. Rev. A 100, 012107 (2019).
  62. R. Hartmann and W. T. Strunz, Accuracy assessment of perturbative master equations: Embracing nonpositivity, Phys. Rev. A 101, 012103 (2020).
  63. A. Trushechkin, Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation, Phys. Rev. A 103, 062226 (2021).
  64. D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths, Phys. Rev. A 105, 032208 (2022).
  65. L. Diosi, On high-temperature Markovian equation for quantum Brownian motion, Europhys. Lett. 22, 1 (1993).
  66. A. Das, A. Dhar, I. Santra, U. Satpathi, and S. Sinha, Quantum Brownian motion: Drude and Ohmic baths as continuum limits of the Rubin model, Phys. Rev. E 102, 062130 (2020).
  67. Z. Chen, Y. Lu, H. Wang, Y. Liu, and T. Li, Quantum Langevin dynamics for optimization, Commun. Math. Phys. 406, 52 (2025).
  68. B. M. Reible and L. Delle Site, Open quantum systems and the grand canonical ensemble, Phys. Rev. E 112, 024130 (2025).
  69. L. Delle Site and A. Djurdjevac, An effective Hamiltonian for the simulation of open quantum molecular systems, J. Phys. A: Math. Theor. 57, 255002 (2024).
  70. L. Delle Site and C. Hartmann, Computationally feasible bounds for the free energy of nonequilibrium steady states, applied to simple models of heat conduction, Mol. Phys. 123, e2391484 (2025).
  71. A. Agarwal and L. Delle Site, Path integral molecular dynamics within the grand canonical-like adaptive resolution technique: Simulation of liquid water, J. Chem. Phys. 143, 094102 (2015).
  72. S. Panahian Jand, Z. Nourbakhsh, and L. Delle Site, Nuclear quantum effects in fullerene–fullerene aggregation in water, Front. Chem. 10, 1072665 (2022).
  73. M. Kryvohuz, Semiclassical instanton approach to calculation of reaction rate constants in multidimensional chemical systems, J. Chem. Phys. 134, 114103 (2011).
  74. K. A. Jung, P. E. Videla, and V. S. Batista, Multi-time formulation of Matsubara dynamics, J. Chem. Phys. 151, 034108 (2019).
  75. A. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio, Markovian master equations: A critical study, New J. Phys. 12, 113032 (2010).
  76. S. Nakajima, On quantum theory of transport phenomena: Steady diffusion, Prog. Theor. Phys. 20, 948 (1958).
  77. R. Zwanzig, Ensemble method in the theory of irreversibility, J. Chem. Phys. 33, 1338 (1960).
  78. A. G. Redfield, On the theory of relaxation processes, IBM J. Res. Dev. 1, 19 (1957).
  79. S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti, Path integral based calculations of symmetrized time correlation functions. I., J. Chem. Phys. 133, 164104 (2010).
  80. A. Agarwal and L. Delle Site, Grand-canonical adaptive resolution centroid molecular dynamics: Implementation and application, Comput. Phys. Commun. 206, 26 (2016).
  81. A. Agarwal, C. Clementi, and L. Delle Site, Path integral-GC-AdResS simulation of a large hydrophobic solute in water: a tool to investigate the interplay between local microscopic structures and quantum delocalization of atoms in space, Phys. Chem. Chem. Phys. 19, 13030 (2017).
  82. S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117, 1 (1995).
  83. A. P. Thompson, H. M. Aktulga, R. Berger, et al., LAMMPS—A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).
  84. S. Habershon, T. E. Markland, and D. E. Manolopoulos, Competing quantum effects in the dynamics of a flexible water model, J. Chem. Phys. 131, 024501 (2009).
  85. M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, Efficient stochastic thermostatting of path integral molecular dynamics, J. Chem. Phys. 133, 124104 (2010).
  86. B. Fernando, Non-equilibrium computer simulations of coupling effects under thermal gradients, CMST 23, 165 (2017).
  87. D. Surblys, H. Matsubara, G. Kikugawa, and T. Ohara, Methodology and meaning of computing heat flux via atomic stress in systems with constraint dynamics, J. Appl. Phys. 130, 215104 (2021).
  88. V. Bach, J. Fröhlich, and I. M. Sigal, Return to equilibrium, J. Math. Phys. 41, 3985 (2000).

Outline

Information

Phys. Rev. A 113, 042205– Published 3 April, 2026
  • Received 10 December 2025
  • Accepted 9 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation