Abstract
The problem is posed of the prescription of the so-called Boltzmann–Grad limit operator (\(\mathcal {L}_{BG}\)) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator \(\mathcal {L}_{BG}\), the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente (Found Phys 45:404, 2015) that there is “no time-asymmetric ingredient” in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The proof is couched on the “ab initio” axiomatic approach to the classical statistical mechanics recently developed (Tessarotto et al. in Eur Phys J Plus 128:32, 2013). Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
C. Cercignani private communication to M.T. (2002).
Nevertheless, the crucial question concerning how the particle mass m should behave in the continuum limit (9) remains unanswered. In fact, both cases in which respectively \(m\rightarrow 0\) or is kept finite are manifestly unphysical. In fact in the first case no dynamics exists (Newton’s equations become invalid), while in the second a “gravitational catastrophe” occurs since the total mass Nm enclosed in the bounded domain \(\Omega \) necessarily becomes infinite.
References
Boltzmann, L.: Weitere studien über das w ärmegleichgewicht unter gasmolekülen. Sitz. Akad. Wiss. 66, 275 (1872)
Grad, H.: Principles of the Kinetic theory of Gases. Handbook der Physik, vol. XII, p. 205. Springer, Berlin (1958)
Loschmidt, J.: Uber den Zustand des Warmegleichgewichtes eines Systems von Korpern mit Rucksicht auf die Schwerkraft. Akad. Wiss. Wien 73, 128 (1876)
Boltzmann, L.: Weitere Studien iiber das Warmegleichgewicht unter Gasmolekule. Wien. Ber.66, 275 (1877) (in Boltzmann: Uber die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Warmetheorie und der Wahrscheinlichkeitsrechnung, vol. II, pp. 112–148 (1909))
Ehrenfest, P., Ehrenfest-Afanassjewa, T.: The Conceptual Foundations of the Statistical Approach in Mechanics. Cornell University Press, New York (1912)
Cercignani, C.: H-Theorem and trend to equilibrium in the kinetic theory of gases. Arch. Mech. 34, 231 (1982)
Lebowitz, J.L.: Boltzmann’s entropy and time’s arrow. Phys. Today 46, 32 (1994)
Drory, A.: Is there a reversibility paradox? Recentering the debate on the thermodynamic time arrow. Stud. Hist. Phil. Sci. Part B 39, 889–913 (2008)
Tessarotto, M., Cremaschini, C.: Axiomatic foundations of entropic theorems for hard-sphere systems. Eur. Phys. J. Plus 130, 91 (2015)
Tessarotto, M., Cremaschini, C., Tessarotto, M.: On the conditions of validity of the Boltzmann equation and Boltzmann H-theorem. Eur. Phys. J. Plus 128, 32 (2013)
Uffink, J., Valente, G.: Lanford’s theorem and the emergence of irreversibility. Found Phys. 45, 404 (2015)
Lanford, O.E.: Time Evolution of Large Classical Systems. Lecture Notes in Physics, vol. 38. Springer, Berlin (1975)
Lanford, O.E.: On the derivation of the Boltzmann equation. Soc. Math. France Asterisque 40, 117 (1976)
Lanford, O.E.: Hard-sphere gas in the Boltzmann-Grad limit. Physica 106A, 70 (1981)
Cercignani, C.: Theory and Applications of the Boltzmann Equation. Scottish Academic Press, Edinburgh (1975)
Tessarotto, M., Asci, C., Cremaschini, C., Soranzo, A., Tironi, G., Tessarotto, M.: The Lagrangian dynamics of thermal tracer particles in Navier-Stokes fluids. Eur. Phys. J. Plus 127, 36 (2012)
Ardourel, V.: Irreversibility in the derivation of the Boltzmann equation. Found Phys. 47, 471 (2017)
Enskog, D.: Kinetiche Theorie dr Wärmeleiting, Reibung und Selbs-diffusion in gewissenverdichteten Gasen und Flü ssigkeiten. Svensk Vetenskps Akad. 63, 4 (1921) (English trans: Brush, S.G., Kinetic Theory, vol. 3, Pergamon, New York, 1972)
Tessarotto, M., Cremaschini, C.: Theory of collisional invariants for the Master kinetic equation. Phys. Lett. A 378, 1760 (2014)
Tessarotto, M., Cremaschini, C.: The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system. Eur. Phys. J. Plus 129, 157 (2014)
Boltzmann, L.: Vorlesungen über Gasstheorie, vol. 2. J.A. Barth, Leipzig (1896–1898) (English trans: Brush, H., Lectures on Gas Theory, vol. 1, Sect. 12; vol. 2, Sect. 22. University of California Press, California)
Boltzmann, L.: Vorlesungen über Gasstheorie, vol. 2. J.A. Barth, Leipzig (1896–1898) (English trans: Brush, H., Lectures on Gas Theory, vol. 1, Sect. 8. University of California Press, California)
Boltzmann, L.: Vorlesungen über Gasstheorie, vol. 2. J.A. Barth, Leipzig (1896–1898) (English trans: Brush, H., Lectures on Gas Theory, vol. 2, Sect. 38. University of California Press, California)
Cercignani, C.: 134 years of Boltzmann equation. In: Gallavotti, G., Reiter, W., Yngvason, J. (eds.) Boltzmann’s Legacy. ESI Lecture in Mathematics and Physics. European Mathematical Society, Prague (2008)
Munster, A.: Statistical Thormodynamics. Springer, New York (1969)
Villani, C.: Entropy Production and Convergence to Equilibrium for the Boltzmann Equation. In: Zambrini, J.C. (ed.) 14th Int. Congress on Math. Physics, 28 July–2 August 2003, University of Lisbon, Portugal. World Scientific Publishing Co., Singapore (2006)
Tessarotto, M., Cremaschini, C.: Modified BBGKY hierarchy for the hard-sphere system. Eur. Phys. J. Plus 129, 243 (2014)
Tessarotto, M., Cremaschini, C.: Theory of collisional invariants for the Master kinetic equation. Phys. Lett. A 379, 1206 (2015)
Tessarotto, M., Asci, C., Cremaschini, C., Soranzo, A., Tironi, G.: Global validity of the Master kinetic equation for hard-sphere systems. Eur. Phys. J. Plus 130, 160 (2015)
Tessarotto, M., Asci, C.: Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems. Phys. Lett. A 381, 1484 (2017)
Tessarotto, M., Mond, M., Asci, C.: Microscopic statistical description of incompressible Navier-Stokes granular fluids. Eur. Phys. J. Plus 132, 213 (2017)
Cercignani, C.: Mathematical Methods in Kinetic Theory. Plenum Press, New York (1969)
Asci, C.: Integration over an infinite-dimensional banach space and probabilistic applications. Int. J. Anal. (2014). https://doi.org/10.1155/2014/404186
Asci, C.: Differentiation theory over infinite-dimensional banach spaces. J. Math. (2016). https://doi.org/10.1155/2016/2619087
Acknowledgements
The authors are grateful to the International Center for Theoretical Physics (Miramare, Trieste, Italy) for the hospitality during the preparation of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tessarotto, M., Cremaschini, C., Mond, M. et al. On the Boltzmann–Grad Limit for Smooth Hard-Sphere Systems. Found Phys 48, 271–294 (2018). https://doi.org/10.1007/s10701-018-0144-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-018-0144-5