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  • Open Access

Lorentz-boosted diffusion: Initial value formulation and exact solutions

L. Gavassino

  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

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Phys. Rev. D 113, 114038 – Published 22 June, 2026

DOI: https://doi.org/10.1103/k646-mgz5

Abstract

It is well known that the diffusion equation, when treated as a stand-alone partial differential equation, exhibits exponential instabilities in boosted frames, which render the corresponding initial value problem ill posed. Recently, however, it was shown that Fick-type diffusion arises as the exact hydrodynamic sector of relativistic Fokker-Planck kinetic theory. In this work, we exploit this kinetic embedding to formulate a modified initial value problem for one-dimensional Lorentz-boosted diffusion. We show that the resulting dynamics are well posed both forward and backward in time, provided the boosted density profiles admit a kinetic-theory realization. Such profiles form a space of band-limited functions, within which the evolution can be expressed as a discrete superposition of spatially sampled initial data, weighted by a Shannon-Whittaker-type Green’s function defined on the full Minkowski plane. The Green’s function is obtained in closed analytic form.

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References (36)

  1. L. Peliti, Statistical Mechanics in a Nutshell, In a nutshell (Princeton University Press, Princeton, NJ, 2011).
  2. W. Hiscock and L. Lindblom, Phys. Rev. D 31, 725 (1985).
  3. P. Kostädt and M. Liu, Phys. Rev. D 62, 023003 (2000).
  4. G. Başar, J. Bhambure, R. Singh, and D. Teaney, Phys. Rev. C 110, 044903 (2024).
  5. V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Applied Mathematical Sciences Vol. 127 (Springer, New York, 2006).
  6. L. Gavassino, Phys. Rev. X 12, 041001 (2022).
  7. C. Cattaneo, Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, Comptes rendus hebdomadaires des séances de l’Académie des sciences (Gauthier-Villars, 1958), https://scispace.com/papers/a-form-of-heat-conduction-equations-which-eliminates-the-4gvmur1p0o.
  8. W. Israel and J. Stewart, Ann. Phys. (N.Y.) 118, 341 (1979).
  9. L. Gavassino and M. Antonelli, Front. Astron. Space Sci. 8, 686344 (2021).
  10. L. Gavassino and M. Antonelli, Phys. Rev. D 112, 104052 (2025).
  11. L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, edited by L. Rezzolla and O. Zanotti (Oxford University Press, New York, 2013), ISBN-10: 0198528906, ISBN-13: 978-0198528906.
  12. R. Geroch, J. Math. Phys. (N.Y.) 36, 4226 (1995).
  13. L. Lindblom, Ann. Phys. (N.Y.) 247, 1 (1996).
  14. R. P. Geroch, arXiv:gr-qc/0103112.
  15. L. Gavassino, Phys. Rev. D 112, 034026 (2025).
  16. L. Gavassino, arXiv:2601.19464.
  17. L. Gavassino, arXiv:2601.19474.
  18. L. Gavassino, arXiv:2601.03081.
  19. L. C. Evans, Partial Differential Equations (American Mathematical Society, Providence, RI, 2010).
  20. K. Huang, Statistical Mechanics, 2nd ed. (John Wiley & Sons, New York, 1987).
  21. F. Debbasch, K. Mallick, and J.-P. Rivet, J. Stat. Phys. 88, 945 (1997).
  22. J. Dunkel and P. Hänggi, Phys. Rep. 471, 1 (2009).
  23. S. R. de Groot, W. A. van Leeuwen, and C. G. van Weert, Relativistic Kinetic Theory: Principles and Applications (North-Holland Publishing Company, Amsterdam, 1980).
  24. L. Gavassino, arXiv:2604.07031.
  25. L. Gavassino, M. Antonelli, and B. Haskell, Phys. Rev. Lett. 128, 010606 (2022).
  26. M. P. Heller, A. Serantes, M. Spaliński, and B. Withers, Phys. Rev. Lett. 130, 261601 (2023).
  27. L. Gavassino, Phys. Lett. B 840, 137854 (2023).
  28. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications Vol. 19 (American Mathematical Society, Providence, RI, 1934).
  29. A. I. Zayed, Rev. Unión Mat. Argent. 49, 99 (2008), https://www.scielo.org.ar/pdf/ruma/v49n1/v49n1a09.pdf.
  30. J. Rauch, Partial Differential Equations, Graduate Texts in Mathematics (Springer, New York, 2012).
  31. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill Book Company, New York, 1953), Vol. 1.
  32. O. F. Brevig, A. Chirre, J. Ortega-Cerdà, and K. Seip, arXiv:2210.13922.
  33. A. G. García, Adv. Imaging Electron Phys. 124, 63 (2003).
  34. I. Ostrovskii and A. Ulanovskii, C. R. Acad. Sci. 333, 735 (2001).
  35. F. S. Bemfica, M. M. Disconzi, and J. Noronha, Phys. Rev. X 12, 021044 (2022).
  36. J. L. Anderson and H. R. Witting, Physica (Amsterdam) 74, 466 (1974).

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Phys. Rev. D 113, 114038– Published 22 June, 2026
  • Received 8 March 2026
  • Accepted 27 May 2026
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