Skip to main content
SpringerLink
Log in

Maximum entropy principle closure for 14-moment system for a non-polytropic gas

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

In this paper, we consider a rarefied polyatomic gas with a non-polytropic equation of state. We use the variational procedure of maximum entropy principle (MEP) to obtain the closure of the binary hierarchy of 14 moments associated with the Boltzmann equation in which the distribution function depends also on the energy of internal modes. The closed partial differential system is symmetric hyperbolic and the Cauchy problem is well-posed. In the limiting case of polytropic gas in which the internal energy is a linear function of the temperature, we find, as a special case, the previous results of Pavić et al. (Physica A 392:1302–1317, 2013). This paper, therefore, completes the equivalence between the closure obtained in the phenomenological rational extended thermodynamics theory and the one obtained by the MEP for general non-polytropic gas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pavić, M., Ruggeri, T., Simić, S.: Maximum entropy principle for rarefied polyatomic gases. Physica A 392, 1302–1317 (2013)

    Article  MathSciNet  Google Scholar 

  2. Ruggeri, T.: Non-linear maximum entropy principle for a polyatomic gas subject to the dynamic pressure. Bull. Inst. Math. Acad. Sin. 11, 1–22 (2016)

    MathSciNet  MATH  Google Scholar 

  3. Bisi, M., Ruggeri, T., Spiga, G.: Dynamical pressure in a polyatomic gas: interplay between kinetic theory and extended thermodynamic. Kinet. Relat. Mod. 11, 71–95 (2017)

    Article  MathSciNet  Google Scholar 

  4. Arima, T., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of rarefied polyatomic gases: 15-field theory incorporating relaxation processes of molecular rotation and vibration. Entropy 20, 301 (2018)

    Article  Google Scholar 

  5. Borgnakke, C., Larsen, P.S.: Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. J. Comput. Phys. 18, 405–420 (1975)

    Article  Google Scholar 

  6. Bourgat, J.-F., Desvillettes, L., Le Tallec, P., Perthame, B.: Microreversible collisions for polyatomic gases. Eur. J. Mech. B Fluids 13, 237–254 (1994)

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  8. Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Recent results on nonlinear extended thermodynamics of real gases with six fields. Part I: general theory. Ric. Mat. 65, 263–277 (2016)

    Article  MathSciNet  Google Scholar 

  9. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271–292 (2011)

    Article  Google Scholar 

  10. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of the extended thermodynamics. Contin. Mech. Thermodyn. 1, 3–20 (1989)

    Article  MathSciNet  Google Scholar 

  11. Struchtrup, H.: The BGK model for an ideal gas with an internal degree of freedom. Transp. Theory Stat. Phys. 28, 369–385 (1999)

    Article  MathSciNet  Google Scholar 

  12. Rahimi, B., Struchtrup, H.: Capturing non-equilibrium phenomena in rarefied polyatomic gases: a high-order macroscopic model. Phys. Fluids 26, 052001 (2014)

    Article  Google Scholar 

  13. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511 (1954)

    Article  Google Scholar 

  14. Bhatia, A.B.: Ultrasonic Absorption: An Introduction to the Theory of Sound Absorption and Dispersion in Gases, Liquids, and Solids. Dover, New York (1985)

    Google Scholar 

  15. Mason, W.P. (ed.): Physical Acoustics: Principles and Methods, vol. II-A. Academic, New York (1965)

    Google Scholar 

  16. Stupochenko, Y., Losev, S., Osipov, A.: Relaxation in Shock Waves. Springer, Berlin (1967)

    Book  Google Scholar 

  17. Kustova, E.V.: On the simplified state-to-state transport coefficients. Chem. Phys. 270, 177–195 (2001)

    Article  Google Scholar 

Download references

Acknowledgements

This paper is supported by GNFM/INdAM and is dedicated to my friends Masaru Sugiyama and Giuseppe Toscani for their 70-birthday. With Giuseppe I have come a long way together during our career even if I don’t have any jointly paper, he is certainly one of my closest friends and one of the best Italian mathematical physicists. Although I have many co-authors Masaru is the co-author with whom I have the most number of papers. We have known each other for over 20 years and we have always worked very well and in a complementary way on non-linear waves and in the construction of Extended Thermodynamics that also includes polyatomic gases. A relationship of great friendship was established with him over time. He is not only an excellent Mathematical Physicists but also is an exquisite, kind and lovable person. I express my most sincere wishes to both.

Author information

Authors and Affiliations

  1. Department of Mathematics and Alma Mater Research Center on Applied Mathematics, AM2, University of Bologna, Bologna, Italy

    Tommaso Ruggeri

Authors
  1. Tommaso Ruggeri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommaso Ruggeri.

Ethics declarations

Conflict of interest

The author states that there is no conflict of interest.

Additional information

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ruggeri, T. Maximum entropy principle closure for 14-moment system for a non-polytropic gas. Ricerche mat 70, 207–222 (2021). https://doi.org/10.1007/s11587-020-00510-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-020-00510-y

Keywords

Mathematics Subject Classification

Search

Navigation