- 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

Statistical theory of electronic degrees of freedom in wave packet molecular dynamics

Daniel Plummer1,*, Pontus Svensson1,2, Wiktor Jasniak1, Patrick Hollebon3, Sam M. Vinko1, and Gianluca Gregori1

  • *Contact author: daniel.plummer@physics.ox.ac.uk
PDF

Phys. Rev. E 114, 015219 – Published 24 July, 2026

DOI: https://doi.org/10.1103/3w8y-wj1r

Abstract

We derive statistical distributions for the degrees of freedom in wave packet molecular dynamics models. Specifically, a theory is developed for the width distributions of Gaussian wave packets in both isotropic and anisotropic formulations. The resulting distribution functions show good agreement with molecular dynamics data under warm dense matter conditions, providing practical guidance for constraining the confining potential, an empirical parameter in the model. We also discuss how these distributions influence the resulting effective Coulomb interactions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. J. Vorberger, F. Graziani, D. Riley, A. D. Baczewski, I. Baraffe, M. Bethkenhagen, S. Blouin, M. P. Böhme, M. Bonitz, M. Bussmann, A. Casner, W. Cayzac, P. Celliers, G. Chabrier, N. Chamel, D. Chapman, M. Chen, J. Clérouin, G. Collins, F. Coppari, et al., Roadmap for warm dense matter physics, Plasma Phys. Control. Fusion 68, 073501 (2026).
  2. D. Saumon, S. Blouin, and P.-E. Tremblay, Current challenges in the physics of white dwarf stars, Phys. Rep. 988, 1 (2022).
  3. D. Saumon, W. B. Hubbard, G. Chabrier, and H. M. van Horn, The role of the molecular-metallic transition of hydrogen in the evolution of Jupiter, Saturn, and brown dwarfs, Astrophys. J. 391, 827 (1992).
  4. T. Guillot, Interiors of giant planets inside and outside the solar system, Science 286, 72 (1999).
  5. S. X. Hu, L. A. Collins, T. R. Boehly, Y. H. Ding, P. B. Radha, V. N. Goncharov, V. V. Karasiev, G. W. Collins, S. P. Regan, and E. M. Campbell, A review on ab initio studies of static, transport, and optical properties of polystyrene under extreme conditions for inertial confinement fusion applications, Phys. Plasmas 25, 056306 (2018).
  6. T. Dornheim, Z. A. Moldabekov, K. Ramakrishna, P. Tolias, A. D. Baczewski, D. Kraus, T. R. Preston, D. A. Chapman, M. P. Böhme, T. Döppner, F. Graziani, M. Bonitz, A. Cangi, and J. Vorberger, Electronic density response of warm dense matter, Phys. Plasmas 30, 032705 (2023).
  7. M. Bonitz, J. Vorberger, M. Bethkenhagen, M. P. Böhme, D. M. Ceperley, A. Filinov, T. Gawne, F. Graziani, G. Gregori, P. Hamann, S. B. Hansen, M. Holzmann, S. X. Hu, H. Kählert, V. V. Karasiev, U. Kleinschmidt, L. Kordts, C. Makait, B. Militzer, Z. A. Moldabekov, et al., Toward first principles-based simulations of dense hydrogen, Phys. Plasmas 31, 110501 (2024).
  8. B. Militzer, F. González-Cataldo, S. Zhang, K. P. Driver, and F. Soubiran, First-principles equation of state database for warm dense matter computation, Phys. Rev. E 103, 013203 (2021).
  9. Z. Moldabekov, S. Schwalbe, M. P. Böhme, J. Vorberger, X. Shao, M. Pavanello, F. R. Graziani, and T. Dornheim, Bound-state breaking and the importance of thermal exchange–correlation effects in warm dense hydrogen, J. Chem. Theory Comput. 20, 68 (2024).
  10. A. Kononov, C.-W. Lee, T. P. dos Santos, B. Robinson, Y. Yao, Y. Yao, X. Andrade, A. D. Baczewski, E. Constantinescu, A. A. Correa, Y. Kanai, N. Modine, and A. Schleife, Electron dynamics in extended systems within real-time time-dependent density-functional theory, MRS Commun. 12, 1002 (2022).
  11. A. V. Filinov, V. O. Golubnychiy, M. Bonitz, W. Ebeling, and J. W. Dufty, Temperature-dependent quantum pair potentials and their application to dense partially ionized hydrogen plasmas, Phys. Rev. E 70, 046411 (2004).
  12. J. N. Glosli, F. R. Graziani, R. M. More, M. S. Murillo, F. H. Streitz, M. P. Surh, L. X. Benedict, S. Hau-Riege, A. B. Langdon, and R. A. London, Molecular dynamics simulations of temperature equilibration in dense hydrogen, Phys. Rev. E 78, 025401(R)(2008).
  13. G. S. Demyanov and P. R. Levashov, Molecular dynamics of nondegenerate hydrogen plasma using improved Kelbg pseudopotential with electron thermal de Broglie wavelength correction, Phys. Plasmas 32, 123901 (2025).
  14. H. Feldmeier and J. Schnack, Molecular dynamics for fermions, Rev. Mod. Phys. 72, 655 (2000).
  15. P. E. Grabowski, in Frontiers and Challenges in Warm Dense Matter, edited by F. Graziani, M. P. Desjarlais, R. Redmer, and S. B. Trickey (Springer, Cham, 2014), pp. 265–282.
  16. D. Klakow, C. Toepffer, and P.-G. Reinhard, Hydrogen under extreme conditions, Phys. Lett. A 192, 55 (1994).
  17. B. Jakob, P.-G. Reinhard, C. Toepffer, and G. Zwicknagel, Wave packet simulation of dense hydrogen, Phys. Rev. E 76, 036406 (2007).
  18. Y. Lavrinenko, P. R. Levashov, D. V. Minakov, I. V. Morozov, and I. A. Valuev, Equilibrium properties of warm dense deuterium calculated by the wave packet molecular dynamics and density functional theory method, Phys. Rev. E 104, 045304 (2021).
  19. P. Svensson, Y. Aziz, T. Dornheim, S. Azadi, P. Hollebon, A. Skelt, S. M. Vinko, and G. Gregori, Modeling of warm dense hydrogen via explicit real-time electron dynamics: Dynamic structure factors, Phys. Rev. E 110, 055205 (2024).
  20. I. V. Morozov and I. A. Valuev, Localization constraints in Gaussian wave packet molecular dynamics of nonideal plasmas, J. Phys. A: Math. Theor. 42, 214044 (2009).
  21. P. Svensson, P. Hollebon, D. Plummer, S. M. Vinko, and G. Gregori, Modeling of warm dense hydrogen via explicit real-time electron dynamics: Electron transport properties, Phys. Rev. E 111, 045208 (2025).
  22. R. A. Davis, W. A. Angermeier, R. K. T. Hermsmeier, and T. G. White, Ion modes in dense ionized plasmas through nonadiabatic molecular dynamics, Phys. Rev. Res. 2, 043139 (2020).
  23. Y. Yao, Q. Zeng, K. Chen, D. Kang, Y. Hou, Q. Ma, and J. Dai, Reduced ionic diffusion by the dynamic electron-ion collisions in warm dense hydrogen, Phys. Plasmas 28, 012704 (2021).
  24. W. A. Angermeier and T. G. White, An investigation into the approximations used in wave packet molecular dynamics for the study of warm dense matter, Plasma 4, 294 (2021).
  25. W. Ebeling and B. Militzer, Quantum molecular dynamics of partially ionized plasmas, Phys. Lett. A 226, 298 (1997).
  26. D. Plummer, P. Svensson, W. Jasniak, P. Hollebon, S. M. Vinko, and G. Gregori, Modeling partially ionized dense plasma using wavepacket molecular dynamics, Phys. Rev. E 113, 045206 (2026).
  27. D. Plummer, P. Svensson, D. O. Gericke, P. Hollebon, S. M. Vinko, and G. Gregori, Ionization calculations using classical molecular dynamics, Phys. Rev. E 111, 015204 (2025).
  28. Q. Ma, J. Dai, D. Kang, M. S. Murillo, Y. Hou, Z. Zhao, and J. Yuan, Extremely low electron-ion temperature relaxation rates in warm dense hydrogen: Interplay between quantum electrons and coupled ions, Phys. Rev. Lett. 122, 015001 (2019).
  29. W. Ebeling, A. Filinov, M. Bonitz, V. Filinov, and T. Pohl, The method of effective potentials in the quantum-statistical theory of plasmas, J. Phys. A: Math. Gen. 39, 4309 (2006).
  30. P. Svensson, T. Campbell, F. Graziani, Z. Moldabekov, N. Lyu, V. S. Batista, S. Richardson, S. M. Vinko, and G. Gregori, Development of a new quantum trajectory molecular dynamics framework, Philos. Trans. R. Soc. A 381, 20220325 (2023).
  31. D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods (Cambridge University Press, Cambridge, 2009).
  32. J. T. Su and W. A. Goddard, Excited electron dynamics modeling of warm dense matter, Phys. Rev. Lett. 99, 185003 (2007).
  33. Y. S. Lavrinenko, I. V. Morozov, and I. A. Valuev, Wave packet molecular dynamics–density functional theory method for non-ideal plasma and warm dense matter simulations, Contrib. Plasma Phys. 59, e201800179 (2019).
  34. Y. S. Lavrinenko, I. Morozov, and I. A. Valuev, Reflecting boundary conditions for classical and quantum molecular dynamics simulations of nonideal plasmas, Contrib. Plasma Phys. 56, 448 (2016).
  35. D. Klakow, C. Toepffer, and P. Reinhard, Semiclassical molecular dynamics for strongly coupled Coulomb systems, J. Chem. Phys. 101, 10766 (1994).
  36. Y. S. Lavrinenko, I. V. Morozov, and I. A. Valuev, High performance wave packet molecular dynamics with density functional exchange-correlation term for non-ideal plasma simulations, J. Phys.: Conf. Ser. 1787, 012043 (2021).
  37. M. L. Mehta, Random Matrices, 3rd ed. (Elsevier, Amsterdam, 2004).
  38. M. Knaup, P.-G. Reinhard, and C. Toepffer, Wave packet molecular dynamics simulations of hydrogen near the transition to a metallic fluid, Contrib. Plasma Phys. 39, 57 (1999).
  39. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21, 1087 (1953).
  40. W. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57, 97 (1970).
  41. C. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed. (Springer, Berlin, 2004).
  42. A. Gelman, J. Carlin, H. Stern, D. Dunson, A. Vehtari, and D. Rubin, in Bayesian Data Analysis, 3rd ed. (CRC, Boca Raton, 2013), Chap. 11, pp. 275–292.
  43. A. Gelman and D. B. Rubin, Inference from iterative simulation using multiple sequences, Stat. Sci. 7, 457 (1992).
  44. A. Ono and H. Horiuchi, Antisymmetrized molecular dynamics for heavy ion collisions, Prog. Part. Nucl. Phys. 53, 501 (2004).
  45. D. Plummer and G. Gregori, Figure data: A statistical theory of electronic degrees of freedom in wave packet molecular dynamics, University of Oxford, ORA, 2026, https://ora.ox.ac.uk/objects/uuid:ba82ae3a-6af8-490f-b16c-6b341e5c875a.
  46. I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series, and Products, 6th ed., edited by A. Jeffrey and D. Zwillinger (Academic, San Diego, 2000).

Outline

Information

Phys. Rev. E 114, 015219– Published 24 July, 2026
  • Received 17 February 2026
  • Accepted 9 June 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