Shock structure in the 14 moment system of extended thermodynamics with high order closure based on the maximum entropy principle


An analysis of the shock structure in the 14 moment system of extended thermodynamics with first, second and third order closure based on the maximum entropy principle (MEP) is presented, as a follow up of a recent investigation of the shock structure in the 13 moment system with first and second MEP-based closure. It is seen that when adopting higher order closures, the strength of the subshock that appears in the shock structure profile for large enough Mach numbers is remarkably reduced with respect to what is found with the first order closure, and the overall profile of the shock structure solution is in better agreement with experimental results.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer, New York (1993)

    Google Scholar 

  2. 2.

    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)

    Google Scholar 

  3. 3.

    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271–292 (2012)

    Article  Google Scholar 

  4. 4.

    Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics. Phys. Lett. A 377, 2136–2140 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Arima, T., Mentrelli, A., Ruggeri, T.: Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments. Ann. Phys. 345, 111–140 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Basel (2015)

    Google Scholar 

  7. 7.

    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171–190 (1957)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kogan, M.N.: Rarefied Gas Dynamics. Plenum Press, New York (1969)

    Google Scholar 

  10. 10.

    Kapur, J.N.: Maximum Entropy Models in Science and Engineering. Wiley, New York (1989)

    Google Scholar 

  11. 11.

    Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Contin. Mech. Thermodyn. 9, 205–212 (1997)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Junk, M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Boillat, G., Ruggeri, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 5, 285–292 (1998)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: beyond the Bethe–Teller theory. Phys. Rev. E 89, 013025 (2014)

    Article  Google Scholar 

  16. 16.

    Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Shock wave structure in rarefied polyatomic gases with large relaxation time for the dynamic pressure. J. Phys. Conf. Ser. 1035, 012009 (2018)

    Article  Google Scholar 

  17. 17.

    Ruggeri, T., Taniguchi, S.: Shock waves in hyperbolic systems of non-equilibrium thermodynamics. In: Berezovski, A., Soomere, T. (eds.) Applied Wave Mathematics II: Selected Topics in Solids, Fluids, and Mathematical Methods and Complexity. Mathematics of Planet Earth, vol. 6, pp. 167–186. Springer, Cham (2019)

    Google Scholar 

  18. 18.

    Brini, F., Ruggeri, T.: Second-order approximation of extended thermodynamics of a monatomic gas and hyperbolicity region. Contin. Mech. Thermodyn. 32, 23–39 (2020)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Mentrelli, A., Ruggeri, T.: Shock structure in extended thermodynamics with second order maximum entropy principle closure (submitted)

  20. 20.

    Pennisi, S., Ruggeri, T.: Classical limit of relativistic moments associated with Boltzmann Chernikov equation: optimal choice of moments in classical theory (submitted)

  21. 21.

    Kremer, G.M.: Extended thermodynamics of ideal gases with 14 fields. Ann. I. H. P. Phys. Theor. 45, 419–440 (1986)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Brini, F., Ruggeri, T.: Entropy principle for the moment systems of degree associated to the Boltzmann equation. Critical derivatives and non controllable boundary data. Contin. Mech. Thermodyn. 14, 165–189 (2002)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760–R5763 (1995)

    Article  Google Scholar 

  24. 24.

    Ruggeri, T.: Shock waves in hyperbolic dissipative systems: non equilibrium gases. In: Proceedings of the Euromech Colloquium 270 (Reggio Calabria, Italy; September 1990) (1991)

  25. 25.

    Kosuge, S., Aoki, K.: Shock-wave structure for a polyatomic gas with large bulk viscosity. Phys. Rev. Fluids 3, 023401-1/42 (2018)

    Article  Google Scholar 

  26. 26.

    Kosuge, S., Kuo, H.-W., Aoki, K.: A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure. J. Stat. Phys. 177, 209–251 (2019)

    MathSciNet  Article  Google Scholar 

Download references


This work has been partially supported by GNFM/INdAM and by the Italian MIUR through the PRIN2017 project “Multiscale phenomena in Continuum Mechanics: singular limits, off-equilibrium and transitions”(Project Number: 2017YBKNCE).

Author information



Corresponding author

Correspondence to Andrea Mentrelli.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

It is my pleasure to dedicate this paper to Masaru Sugiyama and Giuseppe Toscani, who recently celebrated their 70th birthday. I have not been lucky enough to collaborate with Giuseppe Toscani yet, but having enjoyed several of his talks I have developed a true admiration for him. With Masaru Sensei Sugiyama I was more lucky, since I have had (and still have) the chance of collaborating with him in Nagoya and Bologna over the years. Not only have I learned a lot from him, but I also have the most sincere esteem of him. I sincerely wish to both of them all the best.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mentrelli, A. Shock structure in the 14 moment system of extended thermodynamics with high order closure based on the maximum entropy principle. Ricerche mat (2020).

Download citation


  • Extended thermodynamics
  • Shock structure
  • Maximum entropy principle

Mathematics Subject Classification

  • 82C35
  • 76L05
  • 82C40
  • 35L60