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Fast kinetic simulator for relativistic matter

Nature Computational Science volume 2, pages 641–654 (2022)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Relativistic kinetic theory is ubiquitous to several fields of modern physics, finding application at large scales in systems in astrophysical contexts, all of the way down to subnuclear scales and into the realm of quark–gluon plasmas. This motivates the quest for powerful and efficient computational methods that are able to accurately study fluid dynamics in the relativistic regime as well as the transition to beyond hydrodynamics—in principle all of the way down to ballistic regimes. We present a family of relativistic lattice kinetic schemes for the efficient simulation of relativistic flows in both strongly (fluid) and weakly (rarefied gas) interacting regimes. The method can deal with both massless and massive particles, thereby encompassing ultra- and mildly relativistic regimes alike. The computational performance of the method for the simulation of relativistic flows across the aforementioned regimes is discussed in detail, along with prospects of future applications.

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Fig. 1: Riemann problem for an ultra-relativistic gas of particles for various viscous regimes.
Fig. 2: Riemann problem for a relativistic gas of particles in the free-streaming regime at different values of m corresponding to ζ.
Fig. 3: Discretization error versus the radial/angular quadrature at different Kn and for different values of the relativistic coldness.
Fig. 4: Evolution of the pressure anisotropy χ with respect to the scaling variable at initial conditions τ0 = 0.2 fm/c and T0 = 0.5 GeV, and various values of η/s.
Fig. 5: Simulation of a vortical flow using set initial conditions and a cubic domain of side ℓ = 20 fm.
Fig. 6: Performance scaling with respect to the number of discrete velocities.

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Data availability

Source Data are provided with this paper.

Code availability

The code—as well as examples running the Riemann problem, data and scripts, to reproduce Figs. 1, 2 and 4—has been deposited to Code Ocean92.

References

  1. Rezzolla, L. & Zanotti, O. Relativistic Hydrodynamics (Oxford Univ. Press, 2013).

  2. Florkowski, W., Heller, M. P. & Spaliński, M. New theories of relativistic hydrodynamics in the LHC era. Rep. Prog. Phys. 81, 046001 (2018).

    Article  MathSciNet  Google Scholar 

  3. Lucas, A. & Fong, K. C. Hydrodynamics of electrons in graphene. J. Phy. Condens. Matter 30, 053001 (2018).

    Article  Google Scholar 

  4. Maldacena, J. The large-N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999).

  5. Romatschke, P. & Romatschke, U. Relativistic Fluid Dynamics In and Out of Equilibrium: And Applications to Relativistic Nuclear Collisions (Cambridge Univ. Press, 2019).

  6. Lucas, A., Davison, R. A. & Sachdev, S. Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals. Proc. Natl Acad. Sci. USA 113, 9463–9468 (2016).

    Article  Google Scholar 

  7. Succi, S. Lattice Boltzmann 2038. Europhys. Lett. 109, 50001 (2015).

    Article  Google Scholar 

  8. Boltzmann, L. Lectures on Gas Theory (Univ. California Press, 2020).

  9. Rischke, D. H., Bernard, S. & Maruhn, J. A. Relativistic hydrodynamics for heavy ion collisions. 1. General aspects and expansion into vacuum. Nucl. Phys. A 595, 346–382 (1995).

    Article  Google Scholar 

  10. Huovinen, P., Kolb, P. F., Heinz, U., Ruuskanen, P. V. & Voloshin, S. A. Radial and elliptic flow at RHIC: further predictions. Phys. Lett. B 503, 58–64 (2001).

    Article  Google Scholar 

  11. Aguiar, C. E., Kodama, T., Osada, T. & Hama, Y. Smoothed particle hydrodynamics for relativistic heavy-ion collisions. J. Phys. G 27, 75–94 (2000).

    Article  Google Scholar 

  12. Schenke, B., Jeon, S. & Gale, C. (3+1)D Hydrodynamic simulation of relativistic heavy-ion collisions. Phys. Rev. C 82, 014903 (2010).

    Article  Google Scholar 

  13. Molnar, E., Niemi, H. & Rischke, D. H. Numerical tests of causal relativistic dissipative fluid dynamics. Eur. Phys. J. C 65, 615–635 (2010).

    Article  Google Scholar 

  14. Gerhard, J., Lindenstruth, V. & Bleicher, M. Relativistic hydrodynamics on graphic cards. Comput. Phys. Commun. 184, 311–319 (2013).

    Article  MathSciNet  Google Scholar 

  15. Del Zanna, L. et al. Relativistic viscous hydrodynamics for heavy-ion collisions with ECHO-QGP. Eur. Phys. J. C 73, 2524 (2013).

    Article  Google Scholar 

  16. Karpenko, I., Huovinen, P. & Bleicher, M. A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions. Comput. Phys. Commun. 185, 3016–3027 (2014).

    Article  MATH  Google Scholar 

  17. Pandya, A., Most, E. R. & Pretorius, F. Conservative finite volume scheme for first-order viscous relativistic hydrodynamics. Phys. Rev. D 105, 123001 (2022).

    Article  MathSciNet  Google Scholar 

  18. Nonaka, C., Honda, E. & Muroya, S. (3+1)-Dimensional relativistic hydrodynamical expansion of hot and dense matter in ultra-relativistic nuclear collision. Eur. Phys. J. C 17, 663–673 (2000).

    Article  Google Scholar 

  19. Xu, Z. & Greiner, C. Transport rates and momentum isotropization of gluon matter in ultrarelativistic heavy-ion collisions. Phys. Rev. C 76, 024911 (2007).

    Article  Google Scholar 

  20. Petersen, H., Steinheimer, J., Burau, G., Bleicher, M. & Stöcker, H. A fully integrated transport approach to heavy ion reactions with an intermediate hydrodynamic stage. Phys. Rev. C 78, 044901 (2008).

    Article  Google Scholar 

  21. Plumari, S., Puglisi, A., Scardina, F. & Greco, V. Shear viscosity of a strongly interacting system: Green–Kubo correlator versus Chapman–Enskog and relaxation-time approximations. Phys. Rev. C 86, 054902 (2012).

    Article  Google Scholar 

  22. Weil, J. et al. Particle production and equilibrium properties within a new hadron transport approach for heavy-ion collisions. Phys. Rev. C 94, 054905 (2016).

    Article  Google Scholar 

  23. Gallmeister, K., Niemi, H., Greiner, C. & Rischke, D. H. Exploring the applicability of dissipative fluid dynamics to small systems by comparison to the Boltzmann equation. Phys. Rev. C 98, 024912 (2018).

    Article  Google Scholar 

  24. Benzi, R., Succi, S. & Vergassola, M. The lattice Boltzmann equation: theory and application. Phys. Rep. 222, 145–197 (1992).

    Article  Google Scholar 

  25. Mendoza, M., Boghosian, B. M., Herrmann, H. J. & Succi, S. Fast lattice Boltzmann solver for relativistic hydrodynamics. Phys. Rev. Lett. 105, 014502 (2010).

    Article  Google Scholar 

  26. Mendoza, M., Boghosian, B. M., Herrmann, H. J. & Succi, S. Derivation of the lattice Boltzmann model for relativistic hydrodynamics. Phys. Rev. D 82, 105008 (2010).

    Article  Google Scholar 

  27. Gabbana, A., Simeoni, D., Succi, S. & Tripiccione, R. Relativistic lattice Boltzmann methods: theory and applications. Phys. Rep. 863, 1–63 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  28. Romatschke, P., Mendoza, M. & Succi, S. A fully relativistic lattice Boltzmann algorithm. Phys. Rev. C 84, 034903 (2011).

    Article  Google Scholar 

  29. Romatschke, P. Relativistic (lattice) Boltzmann equation with nonideal equation of state. Phys. Rev. D 85, 065012 (2012).

    Article  Google Scholar 

  30. Gabbana, A., Mendoza, M., Succi, S. & Tripiccione, R. Numerical evidence of electron hydrodynamic whirlpools in graphene samples. Comput. Fluids 172, 644–650 (2018).

  31. Weih, L. R. et al. Beyond moments: relativistic lattice Boltzmann methods for radiative transport in computational astrophysics. Mon. Notices R. Astron. Soc. 498, 3374–3394 (2020).

    Article  Google Scholar 

  32. Ambruş, V. E. & Blaga, R. High-order quadrature-based lattice Boltzmann models for the flow of ultrarelativistic rarefied gases. Phys. Rev. C 98, 035201 (2018).

    Article  Google Scholar 

  33. Coelho, R. C. V., Mendoza, M., Doria, M. M. & Herrmann, H. J. Fully dissipative relativistic lattice Boltzmann method in two dimensions. Comput. Fluids 172, 318–331 (2018).

  34. Bazzanini, L., Gabbana, A., Simeoni, D., Succi, S. & Tripiccione, R. A lattice Boltzmann method for relativistic rarefied flows in (2+1) dimensions. J. Comput. Sci. 51, 101320 (2021).

    Article  MathSciNet  Google Scholar 

  35. Ahrens, C. & Beylkin, G. Rotationally invariant quadratures for the sphere. Proc. R. Soc. A 465, 3103–3125 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  36. Gabbana, A. et al. Dissipative hydrodynamics of relativistic shock waves in a quark gluon plasma: comparing and benchmarking alternate numerical methods. Phys. Rev. C 101, 064904 (2020).

    Article  Google Scholar 

  37. Cercignani, C. & Kremer, G. M. The Relativistic Boltzmann Equation: Theory and Applications (Birkhäuser Basel, 2002).

  38. Ambruş, V. E. & Guga-Roşian, C. Lattice Boltzmann study of the one-dimensional boost-invariant expansion with anisotropic initial conditions. AIP Conference Proc. 2071, 020014 (2019).

  39. Bjorken, J. D. Highly relativistic nucleus–nucleus collisions: the central rapidity region. Phys. Rev. D 27, 140–151 (1983).

    Article  Google Scholar 

  40. Heller, M. P. & Spalinski, M. Hydrodynamics beyond the gradient expansion: resurgence and resummation. Phys. Rev. Lett. 115, 072501 (2015).

    Article  Google Scholar 

  41. Soloviev, A. Hydrodynamic attractors in heavy ion collisions: a review. Eur. Phys. J. C 82, 319 (2022).

    Article  Google Scholar 

  42. Kurkela, A., Wiedemann, U. A. & Wu, B. Flow in AA and pA as an interplay of fluid-like and non-fluid like excitations. Eur. Phys. J. C 79, 965 (2019).

    Article  Google Scholar 

  43. Giacalone, G., Mazeliauskas, A. & Schlichting, S. Hydrodynamic attractors, initial state energy and particle production in relativistic nuclear collisions. Phys. Rev. Lett. 123, 262301 (2019).

    Article  Google Scholar 

  44. Ambruş, V. E., Schlichting, S. & Werthmann, C. Development of transverse flow at small and large opacities in conformal kinetic theory. Phys. Rev. D 105, 014031 (2022).

    Article  MathSciNet  Google Scholar 

  45. Strickland, M. & Tantary, U. Exact solution for the non-equilibrium attractor in number-conserving relaxation time approximation. JHEP 10, 069 (2019).

    Article  Google Scholar 

  46. Kamata, S., Martinez, M., Plaschke, P., Ochsenfeld, S. & Schlichting, S. Hydrodynamization and nonequilibrium Green’s functions in kinetic theory. Phys. Rev. D 102, 056003 (2020).

    Article  MathSciNet  Google Scholar 

  47. Blaizot, J. P. & Yan, L. Attractor and fixed points in Bjorken flows. Phys. Rev. C 104, 055201 (2021).

    Article  Google Scholar 

  48. Ambruş, V. E., Busuioc, S., Fotakis, J. A., Gallmeister, K. & Greiner, C. Bjorken flow attractors with transverse dynamics. Phys. Rev. D 104, 094022 (2021).

    Article  MathSciNet  Google Scholar 

  49. Romatschke, P. Relativistic fluid dynamics far from local equilibrium. Phys. Rev. Lett. 120, 012301 (2018).

    Article  Google Scholar 

  50. Adamczyk, L., STAR Collaboration. Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid. Nature 548, 62–65 (2017).

    Article  Google Scholar 

  51. Adam, J., STAR Collaboration. Global polarization of Λ hyperons in Au + Au collisions at \(\sqrt{s_{NN}}\)= 200 GeV. Phys. Rev. C 98, 014910 (2018).

  52. Fukushima, K., Kharzeev, D. E. & Warringa, H. J. The chiral magnetic effect. Phys. Rev. D 78, 074033 (2008).

    Article  Google Scholar 

  53. Kharzeev, D. E., Liao, J., Voloshin, S. A. & Wang, G. Chiral magnetic and vortical effects in high-energy nuclear collisions—a status report. Prog. Part. Nucl. Phys. 88, 1–28 (2016).

    Article  Google Scholar 

  54. Ambruş, V. E. & Chernodub, M. N. Hyperon-anti-hyperon polarization asymmetry in relativistic heavy-ion collisions as an interplay between chiral and helical vortical effects. Eur. Phys. J. C 82, 61 (2022).

    Article  Google Scholar 

  55. Ambruş, V. E. & Chernodub, M. N. Vortical effects in Dirac fluids with vector, chiral and helical charges. Preprint at https://arxiv.org/abs/1912.11034 (2019).

  56. Becattini, F. et al. A study of vorticity formation in high energy nuclear collisions. Eur. Phys. J. C 75, 406 (2015).

    Article  Google Scholar 

  57. Becattini, F., Chandra, V., Del Zanna, L. & Grossi, E. Relativistic distribution function for particles with spin at local thermodynamical equilibrium. Ann. Phys. 338, 32–49 (2013).

  58. Karpenko, I. & Becattini, F. Study of Λ polarization in relativistic nuclear collisions at \(\sqrt{{s}_{{{{\rm{NN}}}}}}=7.7\)-200 GeV. Eur. Phys. J. C 77, 213 (2017).

  59. Friman, B., Florkowski, W., Jaiswal, A., Ryblewski, R. & Speranza, E. Relativistic fluid dynamics of spin-polarized systems of particles. In Proc. XIII Quark Confinement and the Hadron Spectrum – PoS(Confinement2018) Vol. 336, 158 (PoS, 2019).

  60. Gabbana, A., Simeoni, D., Succi, S. & Tripiccione, R. Probing bulk viscosity in relativistic flows. Philos. Trans. R. Soc. A 378, 20190409 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  61. Kovtun, P. K., Son, D. T. & Starinets, A. O. Viscosity in strongly interacting quantum field theories from black hole physics. Phys. Rev. Lett. 94, 111601 (2005).

    Article  Google Scholar 

  62. Zhang, Y. et al. Temperature-dependent shear viscosity in a multi-phase transport model for ultrarelativistic heavy-ion collisions at RHIC and LHC. J. Phys. G 46, 055101 (2019).

    Article  Google Scholar 

  63. Niemi, H., Denicol, G. S., Huovinen, P., Molnár, E. & Rischke, D. H. Influence of shear viscosity of quark–gluon plasma on elliptic flow in ultrarelativistic heavy-ion collisions. Phys. Rev. Lett. 106, 212302 (2011).

    Article  Google Scholar 

  64. Calore, E. et al. Massively parallel lattice–Boltzmann codes on large GPU clusters. Parallel Comput. 58, 1–24 (2016).

  65. Calore, E., Gabbana, A., Kraus, J., Schifano, S. F. & Tripiccione, R. Performance and portability of accelerated lattice Boltzmann applications with OpenACC. Concurr. Comput. 28, 3485–3502 (2016).

    Article  Google Scholar 

  66. Di Staso, G., Clercx, H. J. H., Succi, S. & Toschi, F. Lattice Boltzmann accelerated direct simulation Monte Carlo for dilute gas flow simulations. Philos. Trans. R. Soc. A 374, 20160226 (2016).

    Article  MathSciNet  Google Scholar 

  67. Fries, R. J., Müller, B., Nonaka, C. & Bass, S. A. Hadronization in heavy-ion collisions: recombination and fragmentation of partons. Phys. Rev. Lett. 90, 202303 (2003).

    Article  Google Scholar 

  68. Molnár, D. & Voloshin, S. A. Elliptic flow at large transverse momenta from quark coalescence. Phys. Rev. Lett. 91, 092301 (2003).

    Article  Google Scholar 

  69. Greco, V., Ko, C. M. & Lévai, P. Parton coalescence and the antiproton/pion anomaly at RHIC. Phys. Rev. Lett. 90, 202302 (2003).

    Article  Google Scholar 

  70. Bouras, I. et al. Relativistic shock waves in viscous gluon matter. Phys. Rev. Lett. 103, 032301 (2009).

    Article  Google Scholar 

  71. Bonaccorso, F. et al. LBcuda: a high-performance CUDA port of lbsoft for simulation of colloidal systems, 2022. Comput. Phys. Commun. 277, 108380 (2022).

    Article  MathSciNet  Google Scholar 

  72. Succi, S. et al. Towards exascale lattice Boltzmann computing. Comput. Fluids 181, 107–115 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  73. Romatschke, P. Azimuthal anisotropies at high momentum from purely non-hydrodynamic transport. Eur. Phys. J. C 78, 636 (2018).

    Article  Google Scholar 

  74. Adamczyk, L., STAR Collaboration. Energy dependence of moments of net-proton multiplicity distributions at RHIC. Phys. Rev. Lett. 112, 032302 (2014).

    Article  Google Scholar 

  75. Habich, M. & Romatschke, P. Onset of cavitation in the quark–gluon plasma. JHEP 12, 054 (2014).

    Article  Google Scholar 

  76. Nahrgang, M., Bluhm, M., Schaefer, T., Thomas, B. & Steffen, A. Diffusive dynamics of critical fluctuations near the QCD critical point. Phys. Rev. D 99, 116015 (2019).

    Article  MathSciNet  Google Scholar 

  77. Denicol, G. S., Molnár, E., Niemi, H. & Rischke, D. H. Resistive dissipative magnetohydrodynamics from the Boltzmann–Vlasov equation. Phys. Rev. D 99, 056017 (2019).

    Article  Google Scholar 

  78. Bacchini, F. et al. Fully kinetic shearing-box simulations of magnetorotational turbulence in 2D and 3D. I. Pair plasmas. Preprint at https://arxiv.org/abs/2206.07061 (2022).

  79. Parise, G. et al. Lattice Boltzmann simulations of plasma wakefield acceleration. Phys. Plasmas 29, 043903 (2022).

    Article  Google Scholar 

  80. Anderson, J. L. & Witting, H. R. Relativistic quantum transport coefficients. Physica 74, 489–495 (1974).

  81. Anderson, J. L. & Witting, H. R. A relativistic relaxation-time model for the Boltzmann equation. Physica 74, 466–488 (1974).

  82. Landau, L. D. & Lifshitz, E. M. Fluid Mechanics (Elsevier, 1987).

  83. Abramowitz, M., Stegun, I. A. & Miller, D. Handbook of mathematical functions with formulas, graphs and mathematical tables (National Bureau of Standards Applied Mathematics Series no. 55). J. Appl. Mech. 32, 239 (1965).

  84. Delsarte, P., Goethals, J. M. & Seidel, J. J. Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977).

  85. Womersley, R. S. Efficient Spherical Designs with Good Geometric Properties (Springer, 2018).

  86. Thompson, K. W. The special relativistic shock tube. J. Fluid Mech. 171, 365–375 (1986).

  87. Israel, W. Nonstationary irreversible thermodynamics: a causal relativistic theory. Ann. Phys. 100, 310–331 (1976).

  88. Israel, W. & Stewart, J. M. Thermodynamics of nonstationary and transient effects in a relativistic gas. Phys. Lett. A 58, 213–215 (1976).

    Article  Google Scholar 

  89. Jaiswal, A. Relativistic dissipative hydrodynamics from kinetic theory with relaxation time approximation. Phys. Rev. C 87, 051901 (2013).

    Article  Google Scholar 

  90. Romatschke, P. & Strickland, M. Collective modes of an anisotropic quark gluon plasma. Phys. Rev. D 68, 036004 (2003).

    Article  Google Scholar 

  91. Florkowski, W., Ryblewski, R. & Strickland, M. Testing viscous and anisotropic hydrodynamics in an exactly solvable case. Phys. Rev. C 88, 024903 (2013).

    Article  Google Scholar 

  92. Gabbana, A. & Ambruş, V. E. Relativistic Lattice Boltzmann Method (Code Ocean, 2022); https://doi.org/10.24433/CO.5625382.v2

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Acknowledgements

D.S. was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant (agreement no. 765048). S.S. acknowledges funding from the European Research Council under the European Union’s Horizon 2020 framework programme (grant no. P/2014-2020)/ERC (grant agreement no. 739964) (COPMAT). V.E.A. gratefully acknowledges the support of the Alexander von Humboldt Foundation through a Research Fellowship for post-doctoral researchers. All numerical work was performed on the COKA computing cluster at Università di Ferrara. The funders had no role in study design, data collection and analysis, the decision to publish, nor the preparation of the manuscript. The authors thank Dr. Kai Gallmeister for kindly providing the BAMPS data for the Bjorken flow simulations. This paper is dedicated to the memory of Raffaele Tripiccione, our dear friend, colleague and mentor.

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Frankfurt, Germany

    V. E. Ambruş

  2. Department of Physics, West University of Timişoara, Timisoara, Romania

    V. E. Ambruş

  3. Università di Ferrara and INFN-Ferrara, Ferrara, Italy

    L. Bazzanini, D. Simeoni & R. Tripiccione

  4. Eindhoven University of Technology, Eindhoven, Netherlands

    A. Gabbana

  5. Bergische Universität Wuppertal, Wuppertal, Germany

    D. Simeoni

  6. University of Cyprus, Physics department, Nicosia, Cyprus

    D. Simeoni

  7. Center for Life, Nano & Neuro Science @ La Sapienza, Italian Institute of Technology, Roma, Italy

    S. Succi

  8. Department of Physics, Harvard University, Cambridge, MA, USA

    S. Succi

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  1. V. E. Ambruş
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Contributions

A.G. R.T and S.S. conceived the research, L.B., A.G. and D.S. performed the numerical work related to the Riemann problem and the anisotropic flow, whereas V.E.A. performed the numerical work for the Bjorken flow. All authors contributed to the discussion of the results, the editing and revision of the paper.

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Correspondence to D. Simeoni.

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Nature Computational Science thanks Paul Romatschke and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Handling editor: Jie Pan, in collaboration with the Nature Computational Science team. Peer reviewer reports are available.

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Source Data Fig. 1

Simulation results for all different curves presented in the plots of Fig. 1a,b.

Source Data Fig. 2

Simulation results for all different curves presented in the plots of Fig. 2a,b.

Source Data Fig. 3

Simulation results for all different curves presented in the plots of Fig. 3a,b.

Source Data Fig. 4

Simulation results for all different curves presented in the plots in Fig. 4.

Source Data Fig. 5

Simulation results of the color plots presented in Fig. 5b,c.

Source Data Fig. 6

Simulation results of the color plots presented in Fig. 6b,c.

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Ambruş, V.E., Bazzanini, L., Gabbana, A. et al. Fast kinetic simulator for relativistic matter. Nat Comput Sci 2, 641–654 (2022). https://doi.org/10.1038/s43588-022-00333-x

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