- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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
  • Letter
  • Open Access

Quantum counterpart of equipartition theorem: A Möbius inversion approach

Xin-Hai Tong1,* and Yao Wang2

  • *Contact author: xinhai@iis.u-tokyo.ac.jp
PDF

Phys. Rev. Research 6, L032064 – Published 13 September, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L032064

Abstract

The equipartition theorem is crucial in classical statistical physics, and recent studies have revealed its quantum counterpart for specific systems. This raises the question: does a quantum counterpart of the equipartition theorem exist for any given system, and if so, what is its concrete form? In this Letter, we employ the Möbius inversion approach to address these questions, providing a criterion to determine whether a system adheres to the quantum counterpart of the equipartition theorem. If it does, the corresponding distribution function can be readily derived. Furthermore, we construct the fermionic version of the criterion in a manner analogous to the bosonic case.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (40)

  1. E. Schrödinger, Statistical Thermodynamics (Dover Publications Inc., New York, 1989).
  2. F. Reif, Fundamentals of Statistical and Thermal Physics (Waveland Press Inc., Long Grove, IL, 1985).
  3. K. Huang, Statistical Mechanics (Wiley, Hoboken, NJ, 2008).
  4. S. Dattagupta, J. Kumar, S. Sinha, and P. A. Sreeram, Dissipative quantum systems and the heat capacity, Phys. Rev. E 81, 031136 (2010).
  5. M. Campisi, On the limiting cases of nonextensive thermostatistics, Phys. Lett. A 366, 335 (2007).
  6. T. Koide and T. Kodama, Thermodynamic laws and equipartition theorem in relativistic Brownian motion, Phys. Rev. E 83, 061111 (2011).
  7. E. M. C. Abreu, Jr., J. A. Neto, E. M. Barboza, A. C. R. Mendes, and B. B. Soares, On the equipartition theorem and black holes non-Gaussian entropies, Mod. Phys. Lett. A 35, 2050266 (2020).
  8. E. M. C. Abreu, Jr., J. A. Neto, and E. M. Barboza, Barrow's black hole entropy and the equipartition theorem, Europhys. Lett. 130, 40005 (2020).
  9. E. M. Barboza, Jr., R. da C. Nunes, E. M. C. Abreu, and J. A. Neto, Dark energy models through nonextensive Tsallis' statistics, Physica A 436, 301 (2015).
  10. A. van der Ziel, Equivalent circuit and equipartition theorem in ideal dielectric and ferroelectric capacitors, J. Appl. Phys. 44, 1400 (1973).
  11. R. Sarpeshkar, T. Delbruck, and C. A. Mead, White noise in MOS transistors and resistors, IEEE Circuits Devices Mag. 9, 23 (1993).
  12. M. H. Matheny, L. G. Villanueva, R. B. Karabalin, J. E. Sader, and M. L. Roukes, Nonlinear mode-coupling in nanomechanical systems, Nano Lett. 13, 1622 (2013).
  13. J. Spiechowicz, P. Bialas, and J. Łuczka, Quantum partition of energy for a free Brownian particle: Impact of dissipation, Phys. Rev. A 98, 052107 (2018).
  14. P. Bialas, J. Spiechowicz, and J. Łuczka, Partition of energy for a dissipative quantum oscillator, Sci. Rep. 8, 16080 (2018).
  15. J. Spiechowicz and J. Łuczka, On superstatistics of energy for a free quantum Brownian particle, J. Stat. Mech.: Theory Exp. (2019) 064002.
  16. P. Bialas, J. Spiechowicz, and J. Łuczka, Quantum analogue of energy equipartition theorem, J. Phys. A: Math. Theor. 52, 15LT01 (2019).
  17. J. Kaur, A. Ghosh, and M. Bandyopadhyay, Quantum counterpart of energy equipartition theorem for fermionic systems, J. Stat. Mech.: Theory Exp. (2022) 053105 .
  18. J. Spiechowicz and J. Łuczka, Energy of a free Brownian particle coupled to thermal vacuum, Sci. Rep. 11, 4088 (2021).
  19. J. Kaur, A. Ghosh, and M. Bandyopadhyay, Partition of free energy for a Brownian quantum oscillator: Effect of dissipation and magnetic field, Physica A 599, 127466 (2022).
  20. A. Ghosh, Generalised energy equipartition in electrical circuits, Pramana 97, 82 (2023).
  21. J. Kaur, A. Ghosh, and M. Bandyopadhyay, Partition of kinetic energy and magnetic moment in dissipative diamagnetism, Physica A 625, 128993 (2023).
  22. X.-H. Tong, Quantum counterpart of equipartition theorem in quadratic systems, Phys. Rev. Res. 6, 023157 (2024).
  23. A. Ghosh, M. Bandyopadhyay, S. Dattagupta, and S. Gupta, Independent-oscillator model and the quantum Langevin equation for an oscillator: A review, J. Stat. Mech.: Theory Exp. (2024) 074002.
  24. J. Kaur, A. Ghosh, and M. Bandyopadhyay, Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field, Phys. Rev. E 104, 064112 (2021).
  25. J. Łuczka, Quantum counterpart of classical equipartition of energy, J. Stat. Phys. 179, 839 (2020).
  26. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, 1979).
  27. N.-x. Chen, Modified Möbius inverse formula and its applications in physics, Phys. Rev. Lett. 64, 1193 (1990).
  28. C. Nan-xian and R. Er-Qian, Unified solution of the inverse capacity problem, Phys. Rev. E 57, 1302 (1998).
  29. N.-x. Chen and G.-b. Ren, Carlsson-Gelatt-Ehrenreich technique and the Möbius inversion theorem, Phys. Rev. B 45, 8177 (1992).
  30. K. Huang, Introduction to Statistical Physics (CRC Press, 2009).
  31. L. D. Landau and E. M. Lifshitz, Statistical Physics: Volume 5 (Elsevier, 2013).
  32. D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics (Cambridge University Press, 2018).
  33. D. Schumayer and D. A. W. Hutchinson, Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83, 307 (2011).
  34. B. Julia, Statistical theory of numbers, in Number Theory and Physics: Proceedings of the Winter School, Les Houches, France, March 7–16, 1989 (Springer, 1990), pp. 276–293.
  35. S. G. Brush, History of the Lenz-Ising model, Rev. Mod. Phys. 39, 883 (1967).
  36. A. O. Caldeira and A. J. Leggett, Influence of dissipation on quantum tunneling in macroscopic systems, Phys. Rev. Lett. 46, 211 (1981).
  37. T. M. Apostol, Introduction to Analytic Number Theory (Springer Science & Business Media, 1998).
  38. N.-x. Chen and G.-b. Ren, Inverse problems on Fermi systems and ionic crystals, Phys. Lett. A 160, 319 (1991).
  39. F. Haake, Quantum Signatures of Chaos (Springer, 1991).
  40. H. Schomerus, Random matrix approaches to open quantum systems, in Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School: Volume 104, July 2015 (Oxford Academic, 2017), pp. 409–473.

Outline

Information

Phys. Rev. Research 6, L032064– Published 13 September, 2024
  • Received 24 April 2024
  • Accepted 28 August 2024
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