Abstract
The specific heat of Fermi–Pasta–Ulam systems has until now been estimated through the energy fluctuations of a suitable subsystem, and opposite answers were apparently provided concerning its possible vanishing for vanishing temperatures. In the present paper a more “realistic” numerical implementation of the specific heat measurement is discussed, which mimics the interaction of the FPU system with a calorimeter. It is found that there exists a “freezing” critical temperature below which the relaxation times to equilibrium between FPU system and calorimeter become relevant, so that the system presents aging and hysteresis features very similar to those familiar in glasses and spin glasses. In particular, in the framework of such a point of view involving finite long times, the specific heat appears to vanish for vanishing temperatures.
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REFERENCES
P. Bocchieri, A. Scotti, B. Bearzi, and A. Loinger, Phys. Rev. A 2:2013 (1970).
G. Benettin, Ordered and chaotic motions in dynamical systems with many degrees of freedom, in Molecular-Dynamics Simulation of Statistical Mechanical Systems, G. Ciccotti and W. G. Hoover, eds. (North-Holland, Amsterdam, 1986).
R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani, Phys. Rev. A 31:1039 (1985).
R. Livi, M. Pettini, S. Ruffo, and A. Vulpiani, Phys. Rev. A 31:2740 (1985).
L. Galgani, The quest for Planck's constant in classical physics, in Probabilistic Methods in Mathematical Physics, F. Guerra, M. Loffredo, and C. Marchioro, eds. (World Scien-tific, Singapore, 1992).
M. Pettini and M. Landolfi, Phys. Rev. A 41:768–783 (1990).
L. Casetti, C. Clementi, and M. Pettini, Phys. Rev. E 54:5969–5984 (1996).
R. Livi, M. Pettini, S. Ruffo, and A. Vulpiani, J. Stat. Phys. 48:539 (1987).
A. Perronace and A. Tenenbaum, Classical specific heat of an atomic lattice at low temperature, revisited, Phys. Rev. E 57, No. 1 (1998).
M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik, Z. Phys. 35:557–615 (1926); translated in B. L. van der Waerden, Sources of Quantum Mechanics (Dover, New York, 1968).
H. Poincaré, Revue Générale des Sciences Pures et Appliquées 5:513–521 (1894), in Oeuvres X, pp. 246–263.
H. Poincaré, J. Phys. Théor. Appl. 5:5–34 (1912), in Oeuvres IX, pp. 626–653.
W. Thomson (Lord Kelvin), Works, pp. 646–648.
G. Parisi, in Workshop on Statistical Mechanics (Parma University, 1996), unpublished.
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, L. Galgani, and A. Giorgilli, Comm. Math. Phys. 121:557 (1989).
W. Götze, Aspects of structural glass transitions, in Les Houches School, session LI, J. P. Hansen, D. Levesque, and J. Zinn-Justin, eds. (North-Holland, Amsterdam, 1991).
R. G. Palmer, Adv. Phys. 31:669 (1982).
J. H. Jeans, The Dynamical Theory of Gases, 2nd Ed. (Cambridge University Press, Cambridge, 1916), see p. 374. Notice that the relevant chapter was eliminated in the subsequent editions.
L. D. Landau and E. Teller, Physik. Z. Sowjetunion 10:34 (1936), in Collected Papers of L. D. Landau, D. ter Haar, ed. (Pergamon Press, Oxford, 1965), p. 147.
A. Carati, G. Benettin, and L. Galgani, Towards a rigorous treatment of the Jeans–Landau–Teller method for the energy exchanges of harmonic oscillators, Comm. Math. Phys. 150:331–336 (1992).
T. M. O'Neil and P. G. Hjorth, Collisional relaxation of strongly magnetized pure electron plasma, Phys. Fluids 28:3241–3252 (1985).
T. M. O'Neil, P. G. Hjorth, B. Beck, J. Fajans, and J. H. Malmberg, Collisional relaxation of strongly magnetized pure electron plasma (theory and experiment), in Strongly Coupled Plasma Physics (North-Holland, Amsterdam, 1990), p. 313.
G. Benettin and P. Hjorth, Collisional approach to statistical equilibrium in a 1-d model, J. Stat. Phys., in print.
K. A. Fisher and J. A. Hertz, Spin Glasses (Cambridge University Press, Cambridge, 1991).
L. Boltzmann, Nature 51:413 (1895).
L. Boltzmann, Lectures on Gas Theory, translated by S. G. Brush, University of Cal. Press (1966); see especially Section 45, Comparison with Experiments.
J. H. Jeans, Phil. Mag. 6:279 (1903).
J. H. Jeans, Phil. Mag. 10:91 (1905).
G. Benettin, L. Galgani, and A. Giorgilli, Nature 311:444 (1984).
G. Benettin, L. Galgani, and A. Giorgilli, Phys. Lett. A 120:23 (1987).
O. Baldan and G. Benettin, Classical “freezing” of fast rotations: Numerical test of the Boltzmann–Jeans conjecture, J. Stat. Phys. 62:201–219 (1991).
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Carati, A., Galgani, L. On the Specific Heat of Fermi–Pasta–Ulam Systems and Their Glassy Behavior. Journal of Statistical Physics 94, 859–869 (1999). https://doi.org/10.1023/A:1004531032623
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DOI: https://doi.org/10.1023/A:1004531032623