- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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 and the grand canonical ensemble

Benedikt M. Reible* and Luigi Delle Site†

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

Phys. Rev. E 112, 024130 – Published 22 August, 2025

DOI: https://doi.org/10.1103/631r-2y71

Abstract

The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (44)

  1. D. Manzano, A short introduction to the Lindblad master equation, AIP Adv. 10, 025106 (2020).
  2. 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).
  3. 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).
  4. D. Manzano and P. I. Hurtado, Symmetry and the thermodynamics of currents in open quantum systems, Phys. Rev. B 90, 125138 (2014).
  5. 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).
  6. N. N. Bogoliubov, On a new method in the theory of superconductivity, Il Nuovo Cimento 7, 794 (1958).
  7. D. P. Sankovich, Bogolyubov's theory of superfluidity, Phys. Part. Nuclei 41, 1068 (2010).
  8. V. A. Zagrebnov and J.-B. Bru, The Bogoliubov model of weakly imperfect Bose gas, Phys. Rep. 350, 291 (2001).
  9. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2: Theory of the Condensed State, Landau and Lifshitz Course of Theoretical Physics Vol. 9 (Pergamon Press, Oxford, UK, 1980)
  10. H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics (Oxford University Press, Oxford, UK, 2004).
  11. P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, Cambridge, UK, 2015).
  12. G. Schaller, Quantum equilibration under constraints and transport balance, Phys. Rev. E 83, 031111 (2011).
  13. P. H. Guimarães, G. T. Landi, and M. J. de Oliveira, Nonequilibrium quantum chains under multisite Lindblad baths, Phys. Rev. E 94, 032139 (2016).
  14. G. Bulnes Cuetara, M. Esposito, and G. Schaller, Quantum thermodynamics with degenerate eigenstate coherences, Entropy 18, 447 (2016).
  15. H. Hauptmann and W. T. Strunz, Canonical c-field approach to interacting Bose gases: Stochastic interference of matter waves, arXiv:1911.10102.
  16. L. Delle Site and M. Praprotnik, Molecular systems with open boundaries: Theory and simulation, Phys. Rep. 693, 1 (2017).
  17. L. Delle Site, Grand canonical adaptive resolution simulation for molecules with electrons: A theoretical framework based on physical consistency, Comput. Phys. Commun. 222, 94 (2018).
  18. L. Delle Site, Simulation of many-electron systems that exchange matter with the environment, Adv. Theory Simul. 1, 1800056 (2018).
  19. R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics No. 717 (Springer, Berlin, Heidelberg, 2007).
  20. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, UK, 2007).
  21. A. Rivas and S. F. Huelga, Open Quantum Systems, SpringerBriefs in Physics (Springer, Berlin, Heidelberg, 2012).
  22. B. Vacchini, Open Quantum Systems, Graduate Texts in Physics (Springer, Cham, 2024).
  23. G. G. Emch and G. L. Sewell, Nonequilibrium statistical mechanics of open systems, J. Math. Phys. 9, 946 (1968).
  24. M. Takesaki, Theory of Operator Algebras i (Springer, New York, NY, 1979).
  25. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  26. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  27. D. A. Lidar, Z. Bihary, and K. B. Whaley, From completely positive maps to the quantum Markovian semigroup master equation, Chem. Phys. 268, 35 (2001).
  28. G. Schaller and T. Brandes, Preservation of positivity by dynamical coarse graining, Phys. Rev. A 78, 022106 (2008).
  29. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer, Berlin, Heidelberg, 1981).
  30. K. Huang, Statistical Mechanics, 2nd ed. (John Wiley & Sons, New York, NY, 1987).
  31. F. Schwabl, Statistical Mechanics, 2nd ed. (Springer, Berlin, Heidelberg, 2006).
  32. 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).
  33. L. Delle Site and R. Klein, Liouville-type equation for the n-particle distribution function of an open system, J. Math. Phys. 61, 083102 (2020).
  34. R. Klein and L. Delle Site, Derivation of Liouville-like equations for the n-state probability density of an open system with thermalized particle reservoirs and its link to molecular simulation, J. Phys. A: Math. Theor. 55, 155002 (2022).
  35. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Landau and Lifshitz Course of Theoretical Physics Vol. 5 (Pergamon Press, Oxford UK/New York, NY, 1980).
  36. L. Delle Site, G. Ciccotti, and C. Hartmann, Partitioning a macroscopic system into independent subsystems, J. Stat. Mech. (2017) 083201.
  37. B. M. Reible, C. Hartmann, and L. Delle Site, Two-sided Bogoliubov inequality to estimate finite size effects in quantum molecular simulations, Lett. Math. Phys. 112, 97 (2022).
  38. B. M. Reible, J. F. Hille, C. Hartmann, and L. Delle Site, Finite-size effects and thermodynamic accuracy in many-particle systems, Phys. Rev. Res. 5, 023156 (2023).
  39. L. Delle Site and C. Hartmann, Scaling law for the size dependence of a finite-range quantum gas, Phys. Rev. A 109, 022209 (2024).
  40. 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 (2024).
  41. B. Widom, Some topics in the theory of fluids, J. Chem. Phys. 39, 2808 (1963).
  42. In addition, from a mathematical point of view the choice −μNIdS⊗IdB is also very natural because the mapping B(HS)→B(HS⊗HB), T↦T⊗IdB, extending a bounded operator on HS to one on HS⊗HB, is the dual, with respect to the Hilbert-Schmidt inner product, of the partial trace trB:B1(HS⊗HB)→B1(HS), S⊗T↦Str(T), mapping a trace-class operator from HS⊗HB to one on HS; see, for example, Ref. [44], p. 122].
  43. M. E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation, 2nd ed. (Oxford University Press, Oxford, UK, 2023).
  44. D. Petz, Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics (Springer, Berlin, Heidelberg, 2008).

Outline

Information

Phys. Rev. E 112, 024130– Published 22 August, 2025
  • Received 9 May 2025
  • Revised 31 July 2025
  • Accepted 7 August 2025
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