- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

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

Expanded ensemble method for bubble nucleation

Jaakko Hällfors* and Kari Rummukainen†

  • *Contact author: jaakko.hallfors@helsinki.fi
  • †Contact author: kari.rummukainen@helsinki.fi
PDF

Phys. Rev. D 111, 114512 – Published 17 June, 2025

DOI: https://doi.org/10.1103/b61n-q1qq

Abstract

In the absence of impurities and boundary effects, first order phase transitions are initiated by the nucleation of critical bubbles. In thermally driven transitions many systems can remain metastable for an extended time, possibly tens of orders of magnitude longer than typical microscopic timescales. In standard Markov chain Monte Carlo simulations of these systems the probability of critical bubbles can be too suppressed for the transition to happen in any practical simulation time. The computation can be accelerated by using modified sampling methods, for example multicanonical or Wang-Landau sampling. However, even using these methods, there remains a condensation barrier which dramatically reduces the efficiency at large volumes. We present a novel sampling method, the method of expanded ensembles, which very effectively circumvents the condensation barrier and enables efficient simulations at large volumes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (31)

  1. K. Kajantie, M. Laine, K. Rummukainen, and M. E. Shaposhnikov, Is there a hot electroweak phase transition at mH≳mW?, Phys. Rev. Lett. 77, 2887 (1996).
  2. M. D’Onofrio and K. Rummukainen, Standard model cross-over on the lattice, Phys. Rev. D 93, 025003 (2016).
  3. V. A. Rubakov and M. E. Shaposhnikov, Electroweak baryon number non-conservation in the early Universe and in high-energy collisions, Phys. Usp. 39, 461 (1996).
  4. C. Caprini, M. Chala, G. C. Dorsch, M. Hindmarsh, S. J. Huber, T. Konstandin, J. Kozaczuk, G. Nardini, J. M. No, K. Rummukainen, P. Schwaller, G. Servant, A. Tranberg, and D. J. Weir, Detecting gravitational waves from cosmological phase transitions with LISA: An update, J. Cosmol. Astropart. Phys. 03 (2020) 024.
  5. M. Hindmarsh, M. Lüben, J. Lumma, and M. Pauly, Phase transitions in the early universe, SciPost Phys. Lecture Notes 024 (2021).
  6. Amaro-Seoane et al., Laser interferometer space antenna, arXiv:1702.00786.
  7. C. Caprini, M. Hindmarsh, S. Huber, T. Konstandin, J. Kozaczuk, G. Nardini, J. M. No, A. Petiteau, P. Schwaller, G. Servant, and D. J. Weir, Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, J. Cosmol. Astropart. Phys. 04 (2016) 001.
  8. J. S. Langer, Statistical theory of the decay of metastable states, Ann. Phys. (N.Y.) 54, 258 (1969).
  9. I. Affleck, Quantum-statistical metastability, Phys. Rev. Lett. 46, 388 (1981).
  10. S. Coleman, Fate of the false vacuum: Semiclassical theory, Phys. Rev. D 15, 2929 (1977).
  11. C. G. Callan and S. Coleman, Fate of the false vacuum. II. First quantum corrections, Phys. Rev. D 16, 1762 (1977).
  12. A. D. Linde, Decay of the false vacuum at finite temperature, Nucl. Phys. B216, 421 (1983).
  13. K. Farakos, K. Kajantie, K. Rummukainen, and M. Shaposhnikov, 3D physics and the electroweak phase transition: A framework for lattice Monte Carlo analysis, Nucl. Phys. B 442, 317 (1995).
  14. G. D. Moore and K. Rummukainen, Electroweak bubble nucleation, nonperturbatively, Phys. Rev. D 63, 045002 (2001).
  15. T. Neuhaus and J. S. Hager, 2D crystal shapes, droplet condensation, and exponential slowing down in simulations of first-order phase transitions, J. Stat. Phys. 113, 47 (2003).
  16. O. Gould, S. Güyer, and K. Rummukainen, First-order electroweak phase transitions: A nonperturbative update, Phys. Rev. D 106, 114507 (2022).
  17. O. Gould, A. Kormu, and D. J. Weir, A nonperturbative test of nucleation calculations for strong phase transitions, Phys. Rev. D 111, L051901 (2025).
  18. K. Rummukainen, R. Seppä, and D. J. Weir, Resolving the critical bubble in SU(8) deconfinement transition, Proc. Sci. LATTICE2024 (2025) 434.
  19. G. D. Moore, Measuring the broken phase sphaleron rate nonperturbatively, Phys. Rev. D 59, 014503 (1998).
  20. B. A. Berg and T. Neuhaus, Multicanonical algorithms for first order phase transitions, Phys. Lett. B 267, 249 (1991).
  21. F. Wang and D. P. Landau, Efficient, multiple-range random walk algorithm to calculate the density of states, Phys. Rev. Lett. 86, 2050 (2001).
  22. E. Bennett, B. Lucini, D. Mason, M. Piai, E. Rinaldi, and D. Vadacchino, The density of states method for symplectic gauge theories at finite temperature, arXiv:2409.19426.
  23. M. Caselle, G. Costagliola, A. Nada, M. Panero, and A. Toniato, Jarzynski’s theorem for lattice gauge theory, Phys. Rev. D 94, 034503 (2016).
  24. G. D. Moore, K. Rummukainen, and A. Tranberg, Non-perturbative computation of the bubble nucleation rate in the cubic anisotropy model, J. High Energy Phys. 04 (2001) 017.
  25. K. Binder, Theory of the evaporation/condensation transition of equilibrium droplets in finite volumes, Physica A (Amsterdam) 319, 99 (2003).
  26. K. Binder and M. H. Kalos, “Critical clusters” in a supersaturated vapor: Theory and Monte Carlo simulation, J. Stat. Phys. 22, 363 (1980).
  27. A. Nußbaumer, E. Bittner, and W. Janke, Free-energy barrier at droplet condensation, Prog. Theor. Phys. Suppl. 184, 400 (2010).
  28. A. Nußbaumer, E. Bittner, and W. Janke, Monte Carlo study of the droplet formation-dissolution transition on different two-dimensional lattices, Phys. Rev. E 77, 041109 (2008).
  29. A. P. Lyubartsev, A. A. Martsinovski, S. V. Shevkunov, and P. N. Vorontsov-Velyaminov, New approach to Monte Carlo calculation of the free energy: Method of expanded ensembles, J. Chem. Phys. 96, 1776 (1992).
  30. HILA lattice simulation framework, https://github.com/CFT-HY/HILA.
  31. J. Hällfors and K. Rummukainen, Data for paper “Expanded ensemble method for bubble nucleation”, Version v1 (2025), 10.5281/zenodo.15423156.

Outline

Information

Phys. Rev. D 111, 114512– Published 17 June, 2025
  • Received 3 April 2025
  • Accepted 13 May 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. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation