Abstract
Around 1900, J. H. Jeans suggested that the “abnormal” specific heats observed in diatomic gases, specifically the lack of contribution to the heat capacity from the internal vibrational degrees of freedom, in apparent violation of the equipartition theorem, might be caused by the large separation between the time scale for the vibration and the time scale associated with a typical binary collision in the gas. We consider here a simple 1D model and show how, when these time scales are well separated, the collisional dynamics is constrained by a many- particle adiabatic invariant. The effect is that the collisional energy exchanges betgween the translational and the vibrational degrees of freedom are slowed down by an exponential factor (as Jeans conjectured). A metastable situation thus occurs, in which the fast vibrational degrees of freedom effectively do not contribute to the specific heat. Hence, the observed “freezing out” of the vibrational degrees of freedom could in principle be explained in terms of classical mechanics. We discuss the phenomenon analytically, on the basis of an approximation introduced by Landau and Teller (1936) for a related phenomenon, and estimate the time scale for the evolution to statistical equilibrium. The theoretical analysis is supported by numerical examples.
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REFERENCES
J. H. Jeans, On the vibrations set up in molecules by collisions, Phil. Mag. 6:279–286 (1903).
J. H. Jeans, The Dynamical Theory of Gases, 2nd Ed., Chapter XIV (Cambridge University Press, Cambridge, 1916).
L. Boltzmann, On certain questions of the theory of gases, Nature 51:413–415 (1895).
L. Boltzmann, Vorlesungen über Gastheorie (Barth, Leipzig, 1896), Vol. I; (1898), Vol. II [English translation: Lectures on Gas Theory (University of Cal. Press, 1966)].
L. Landau and E. Teller, Zur Theorie der Schalldispersion, Physik. Z. Sowjetunion 10:34–43 (1936). Engl. translation: On the theory of sound dispersion, in Collected Papers of L. D. Landau, D. Ter Haar, ed. (Pergamon Press, Oxford 1965), p. 147.
D. Rapp, Complete classical theory of vibrational energy exchange, J. Chem. Phys. 32:735 (1960).
G. Benettin, A. Carati, and P. Sempio, On the Landau–Teller approximation for the energy exchanges with fast degrees of freedom, J. Stat. Phys. 73:175 (1993).
G. Benettin, A. Carati, and G. Gallavotti, A rigorous implementation of the Jeans–Landau–Teller approximation for adiabatic invariants, Nonlinearity 10:479–505 (1997).
G. Benettin, A. Carati, and F. Fassò, On the conservation of adiabatic invariants for a system of coupled rotators, Physica D 104:253–268 (1997).
G. Benettin and F. Fassò, From Hamiltonian perturbation theory to symplectic integrators, and back, Appl. Num. Math. 29:73–87 (1999).
G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near to the identity symplectic mappings, with application to symplectic integration algorithms, J. Stat. Phys. 73:1117–1144 (1994).
T. M. O'Neil and P. G. Hjorth, Phys. Fluids 28:3241 (1985); M. E. Glinsky, T. M. O'Neil, M. N. Rosenbluth, K. Tsuruta, and S. Ichimaru, Phys. Fluids B 4:1156 (1992).
T. M. O'Neil, P. G. Hjorth, B. Beck, J. Fajans, and J. Malmberg, in Strongly Coupled Plasma Physics (North-Holland, Amsterdam, 1990), p. 313.
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Benettin, G., Hjorth, P. & Sempio, P. Exponentially Long Equilibrium Times in a One-Dimensional Collisional Model of Classical Gas. Journal of Statistical Physics 94, 871–891 (1999). https://doi.org/10.1023/A:1004583016693
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DOI: https://doi.org/10.1023/A:1004583016693