Ballistic Behaviour in Bounded Velocity Transport

  • F. DebbaschEmail author
  • D. Espaze
  • V. Foulonneau
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 247)


Stochastic models of bounded velocity transport are revisited. It is proven that these models exhibit short-time propagative (as opposed to diffusive) behavior for a large class of initial conditions. Numerical simulations also show that this propagative effect is different from the damped propagation predicted by common hyperbolic models. A fit of the density profiles is finally presented and a geometrical generalization of Fick’s law is also proposed.


Diffusions Fick’s law Geometric flows Hyperbolic diffusion Relativistic Ornstein-Uhlenbeck process  Stochastic processes 



Part of this work was funded by the ANR Grant 09-BLAN-0364-01.


  1. 1.
    Itina TE, Mamatkulov M, Sentis M (2005) Nonlinear fluence dependencies in femtosecond laser ablation of metals and dielectrics materials. Opt Eng 44(5):051109–051116CrossRefGoogle Scholar
  2. 2.
    Klossika JJ, Gratzke U, Vicanek M, Simon G (1996) Importance of a finite speed of heat propagation in metal irradiated by femtosecond laser pulses. Phys Rev B 54(15):10277–10279CrossRefGoogle Scholar
  3. 3.
    Chen HT, Song JP, Liu KC (2004) Study of hyperbolic heat conduction problem in IC Chip. Japanese, J Appl Phys 43(7A):4404–4410CrossRefGoogle Scholar
  4. 4.
    Jaunich MK et al (2006) Bio-heat transfer analysis during short pulse laser irradiation of tissues. Intl J Heat Mass Transf 51:5511–5521CrossRefGoogle Scholar
  5. 5.
    Kim K, Guo Z (2007) Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulse irradiations. Comput Methods Programs Biomed 86(2):112–123CrossRefGoogle Scholar
  6. 6.
    Freidberg J (2007) Plasma physics and fusion energy. CambridgeGoogle Scholar
  7. 7.
    Martin-Solis JR et al (2006) Enhanced production of runaway electrons during a disruptive termination of discharges heated with lower hybrid power in the frascati tokamak upgrade. Phys Rev Lett 97:165002CrossRefGoogle Scholar
  8. 8.
    Cattaneo C (1948) Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena, 3.Google Scholar
  9. 9.
    Chevalier C, Debbasch F, Rivet JP (2008) A review of finite speed transport models. In: Proceedings of the second international forum on heat transfer (IFHT08), 17–19 Sept (2008), Tokyo, Japan, Heat Transfer Society of JapanGoogle Scholar
  10. 10.
    Herrera L, Pavon D (2001) Why hyperbolic theories of dissipation cannot be ignored: comments on a paper by Kostadt and Liu. Phys Rev D 64:088503CrossRefGoogle Scholar
  11. 11.
    Israel W (1987) Covariant fluid mechanics and thermodynamics: an introduction. In: Anile A, Choquet-Bruhat Y (eds) Relativistic fluid dynamics. Lecture notes in mathematics, vol 1385. Springer, Berlin.Google Scholar
  12. 12.
    Müller I, Ruggeri T (1993) Extended thermodynamics. Springer Tracts in Natural Philosophy, vol 37. Springer, New-YorkGoogle Scholar
  13. 13.
    Debbasch F, Espaze D, Foulonneau V (2012) Novel aspects of bounded veloity transport. In: Lecture notes in engineering and computer science: proceedings of the world congress on engineering and computer science, WCECS 2012, USA, San Francisco, 24–26 Oct 2012, pp 1198–1201Google Scholar
  14. 14.
    Debbasch F, Mallick K, Rivet JP (1997) Relativistic Ornstein-Uhlenbeck process. J Stat Phys 88:945MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Debbasch F, Chevalier C (2006) Relativistic stochastic processes. In: Proceedings of XV conference on non-equilibrium statistical mechanics and nonlinear physics, Mar del Plata, Argentina, 4–8 Dec 2006, A.I.P. Conference Proceedings, 2007Google Scholar
  16. 16.
    Debbasch F (2008) Equilibrium distribution function of a relativistic dilute perfect gas. Process Phys A 387:2443–2454MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jüttner F (1911) Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie. Ann Phys (Leipzig) 34:856CrossRefzbMATHGoogle Scholar
  18. 18.
    Angst J, Franchi J (2007) Central limit theorem for a class of relativistic diffusions. J Math Phys 48(8)Google Scholar
  19. 19.
    Debbasch F, Rivet JP (1998) A diffusion equation from the relativistic Ornstein-Uhlenbeck process. J Stat Phys 90:1179MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chevalier C, Debbasch F (2007) Multi-scale diffusion on an interface. Eur Phys Lett 77:20005–20009MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chevalier C, Debbasch F (2008) Is brownian motion sensitive to geometry fluctuations? J Stat Phys 131:717–731MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Itô K (1950) On stochastic differential equations on a differentiable manifold. Nagoy Math J 1:35–47zbMATHGoogle Scholar
  23. 23.
    Itô K (1953) On stochastic differential equations on a differentiable manifold ii. M.K. 28:82–85Google Scholar
  24. 24.
    Øksendal B (1998) Stochastic differential equations, 5th edn. Universitext, Springer, BerlinGoogle Scholar
  25. 25.
    Chevalier C, Debbasch F (2010) Lateral diffusions: the influence of geometry fluctuations. Eur Phys Lett 89(3):38001CrossRefGoogle Scholar
  26. 26.
    Debbasch F, Di Molfetta G, Espaze D, Foulonneau V (2012) Propagation in quantum walks and relativistic diffusions. Accepted for publication in Physica ScriptaGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.LERMAParisFrance

Personalised recommendations