Abstract
The ergodic properties of many-body systems with repulsive-core interactions are the basis of classical statistical mechanics and are well established. This is not the case for systems of purely-attractive or gravitational particles. Here we consider two examples, (i) a family of one-dimensional systems with attractive power-law interactions, \(|x_i -x_j|^{\nu} \;, \; \nu > 0\), and (ii) a system of N gravitating particles confined to a finite compact domain. For (i) we deduce from the numerically-computed Lyapunov spectra that chaos, measured by the maximum Lyapunov exponent or by the Kolmogorov–Sinai entropy, increases linearly for positive and negative deviations of ν from the case of a non-chaotic harmonic chain (ν = 2). For \(2 < \nu \le 3\) there is numerical evidence for two additional hitherto unknown phase-space constraints. For the theoretical interpretation of model (ii) we assume ergodicity and show that for a small-enough system the reduction of the allowed phase space due to any other conserved quantity, in addition to the total energy, renders the system asymptotically stable. Without this additional dynamical constraint the particle collapse would continue forever. These predictions are supported by computer simulations.
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References
P. Ehrenfest and T. Ehrenfest, The Conceptual Foundation of the Statistical Approach in Mechanics (Cornell University Press, Ithaca, 1959).
D. Szász, in Hard Ball Systems and the Lorentz Gas (Springer Verlag, Berlin, 2000).
Y. G. Sinai, Russ. Math. Surv. 25(2):137 (1970).
See the review of N. Simányi in Ref. [2].
V. Donnay, Ergodic Theory of Dynamical Systems 16:975 (1996).
H. Narnhofer and W. Thirring, in Ergodic Properties of Quantum Field Theories in Mathematical Physics Towards the 21st Century, R. N. Sen and A. Gersten, eds. (Negev Press, 1994).
M. K.-H. Kiessling, J. Stat. Phys. 55:203 (1989).
E. V. Votyakov, A. De Martino and D. H. E. Gross, Eur. Phys. J. B29:593 (2002).
L. Milanović, H. A. Posch and W. Thirring, Phys. Rev. E 57:2763 (1998).
J. H. Oort, Bull. Astron. Inst. Netherlands 6:289 (1932).
G. L. Camm, Mon. Notices R. Astron. Soc. 110:305 (1950).
G. B. Rybicki, Astrophys. Space Sci. 14:56 (1971).
C. J. Reidl, Jr. and B. N. Miller, Phys. Rev. A 46:837 (1992).
C. J. Reidl, Jr. and B. N. Miller, Phys. Rev. E 48:4250 (1993).
C. Dellago and H. A. Posch, Phys. Rev. E 55:R9 (1997).
H. A. Posch and W. Thirring, J. Math. Phys. 41:3430 (2000).
F. Calogero, J. Math. Phys. 12:419 (1971).
B. Sutherland, Phys. Rev. A 4:2019 (1971).
D. Gómez-Ullate, A. González-López and M. A. Rodríguez, Phys. Lett. B 511:112 (2001).
G. Severne and M. Luwel, Astrophys. Space Sci. 122:299 (1986).
F. Hohl and M. R. Feix, Astrophys. J. 167:1166 (1967).
L. Milanović, Simulation eindimensionaler Vielteilchensysteme mit rein anziehender Wechselwirkung, diploma thesis (University of Vienna, 1995).
A. Noullez, D. Fanelli and E. Aurell, J. Computational Phys. 186:697 (2003).
J. B. Pesin, Usp. Mat. Nauk. [Russ. Math. Surv.] 32(4):55 (1977).
T. Tsuchiya, T. Konishi and N. Gouda, Phys. Rev. E 50:2607 (1994).
T. Tsuchiya and N. Gouda, Phys. Rev. E 61:948 (2000).
L. Milanović, H. A. Posch and W. G. Hoover, Molec. Phys. 95:281 (1998).
C. Forster and H. A. Posch, Lyapunov modes in soft-disk fluids. New J. Phys. 7:32 (2005). arXive nlin.CD/0409019
G. Benettin, C. Froeschle and J. P. Scheidecker, Phys. Rev. A 19:2454 (1979).
K. R. Yawn and B. N. Miller, Phys. Rev. E 68:056120 (2003).
C. J. Reidl, Jr. and B. N. Miller, Phys. Rev. E 51:884 (1995).
H. A. Posch and R. Hirschl, Simulation of Billiards and of Hard-Body Fluids, in Hard Ball Systems and the Lorenz Gas, Vol. 101, pp. 269–310. D. Szasz, ed. Encyclopedia of the mathematical sciences (Springer Verlag, Berlin 2000).
L. Milanović and H. A. Posch, J. Molec. Liquids 96-97:221–244 (2002).
C. Forster, R. Hirschl, H. A. Posch and W. G. Hoover, Phys. D 187:294 (2004).
J.-P. Eckmann, C. Forster, H. A. Posch and E. Zabey, Lyapunov Modes in Hard-Disk Systems. J. Stat. Phys. 118:813 (2005).
C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Clifs, 1971).
C. P. Dettmann and G. P. Morriss, Phys. Rev. E 53:R5541 (1996).
D. Ruelle, J. Stat. Phys. 95:393 (1999).
D. J. Evans, E. G. D. Cohen and G. P. Morriss, Phys. Rev. A 42:5990 (1990).
F. Kuypers, Klassische Mechanik (Wiley-VCH Verlag, Weinheim, 2003).
P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, 1998).
P. P. Ewald, Ann. Phys. 64:253 (1921).
M. D. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987).
P. Kustaanheimo and E. Stiefel, J. Reine Angew. Math. 218:204 (1965).
E. L. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics (Springer-Verlag, Berlin, 1971).
S. Mikkola and S. J. Aarseth, Celestial Mech 57:439 (1993).
J. Stoer and R. Bulirsch, Numerische Mathematik, vol. 2, (Springer-Verlag, Berlin, 1990).
V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies (Academic Press, New York, 1967).
P. Klinko and B. N. Miller, Phys. Rev. Lett. 92:021102 (2004).
W. Thirring, A Course in Mathematical Physics Vol. 1: Classical Dynamical Systems (Springer, New York, 1978).
M. Bottaccio, A. Amici, P. Miocchi, R. Capuzzo Dolcetta, M. Montuori and L. Pietronero, Europhys. Lett. 57:315 (2002).
I. Ispolatov and E. G. D. Cohen, Phys. Rev. Lett. 87:210601 (2004).
P. Klinko and B. N. Miller, Phys. Lett. A 333:187 (2004).
H. A. Posch, H. Narnhofer and W. Thirring, Phys. Rev. A 42:1880 (1990).
W. Thirring, Classical Mathematical Physics: Dynamical Systems and Field Theories (Springer, New York, 1997).
P. Hertel and W. Thirring, Annals of Phys. 63:520 (1971).
E. H. Lieb and W. Thirring, Annals of Phys. 155:494 (1984).
E. H. Lieb and H. T. Yau, Commun. Math. Phys. 112:147 (1987).
M. K.-H. Kiessling, How to Implement Boltzmann's Probabilistic Ideas in a Relativistic World?, in Chance in Physics: Foundations and Perspectives, Vol. 574, p. 83. J. Bricmont, D. D¨rr, M. C. Galavotti, G. Ghirardi, F. Petruccione and N. Zanghi, eds. Lecture Notes in Physics (Springer, Berlin, 2001).
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PACS numbers: 05.45.Pq, Numerical simulation of chaotic systems, 05.20.−y, Classical statistical mechanics, 36.40.Qv, Stability and fragmentation of clusters, 95.10.Fh, Chaotic dynamics.
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Milanović, L., Posch, H.A. & Thirring, W. Gravitational Collapse and Ergodicity in Confined Gravitational Systems. J Stat Phys 124, 843–858 (2006). https://doi.org/10.1007/s10955-006-9095-x
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DOI: https://doi.org/10.1007/s10955-006-9095-x