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A billiard-theoretic approach to elementary one-dimensional elastic collisions
1.D. Halliday, R. Resnick, and J. Walker, Fundamental of Physics, 5th ed., (Wiley, New York, 1997), Vol. 1.
2.J. Walker, The Flying Circus of Physics (Wiley, New York, 1977), Example 2.18.
3.G. Galperin and A. Zemlyakov, Mathematical Billiards (in Russian) (Nauka, Moscow, 1990).
4.V. V. Kozlov and D. V. Treshshëv, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (American Mathematical Society, Providence, RI, 1991).
5.S. Tabachnikov, Billiards (Société Mathématique de France; American Mathematical Society, Providence, RI, 1995).
6.E. Gutkin, “Billiard in polygons: Survey of recent results,” J. Stat. Phys. 81, 7–26 (1996).
7.This problem was apparently first posed by Sinai. See, for example, Ya. Sinai, Introduction to Ergodic Theory (Princeton University Press, Princeton, NJ, 1978).
8.S. L. Glashow and L. Mittag, “Three rods on a ring and the triangular billiard,” J. Stat. Phys. 87, 937–941 (1996).
9.S. G. Cox and G. J. Ackland, “How efficiently do three pointlike particles sample phase space?” Phys. Rev. Lett. 84, 2362–2365 (2000).
10.J. Rouet, F. Blasco, and M. R. Feix, “The one-dimensional Boltzmann gas: The ergodic hypothesis and the phase portrait of small systems,” J. Stat. Phys. 71, 209–224 (1993).
11.See, for example, A. Dhar, “Heat conduction in a one-dimensional gas of elastically colliding particles of unequal masses,” Phys. Rev. Lett. 86, 3554–3557 (2001);
11.P. L. Garrido, P. I. Hurtado, and B. Nadrowski, “Simple one-dimensional model of heat conduction which obeys Fourier’s law,” Phys. Rev. Lett. 86, 5486–5489 (2001);
11.O. Narayan and S. Ramaswamy, “Anomalous heat conduction in one-dimensional momentum-conserving systems,” Phys. Rev. Lett. 89, 200601–1 (2002).
11.For a recent review, see S. Lepri, R. Livi, and A. Politi, “Thermal conduction in classical low-dimensional lattices,” Phys. Rep. 377, 1–80 (2003).
12.For a general review, see H. M. Jaeger, S. R. Nagel, and R. B. Behringer, “Granular solids, liquids, and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996).
13.S. McNamara and W. R. Young, “Inelastic collapse and clumping in a one-dimensional granular medium,” Phys. Fluids A 4, 496–504 (1992);
13.I. Goldhirsch and G. Zanetti, “Clustering instability in dissipative gases,” Phys. Rev. Lett. 70, 1619–1622 (1993);
13.S. McNamara and W. R. Young, “Dynamics of a freely evolving, two-dimensional granular medium,” Phys. Rev. E 53, 5089–5100 (1996).
14.A similar approach to that in the Appendix is given in P. Constantin, E. Grossman, and M. Mungan, “Inelastic collision of three particles on the line as a two-dimensional billiard,” Physica D 83, 409–420 (1995);
14.see also, T. Zhou and L. P. Kadanoff, “Inelastic collapse of three particles,” Phys. Rev. E 54, 623–628 (1996).
15.M. Hasegawa, “Broken ergodic motion of two hard particles in a one-dimensional box,” Phys. Lett. A 242, 19–24 (1998);
15.B. Cipra, P. Dini, S. Kennedy, and A. Kolan, “Stability of one-dimensional inelastic collision sequences of four balls,” Physica D 125, 183–200 (1999).
16.B. Bernu and R. Mazighi, “One-dimensional bounce of inelastically colliding marbles on a wall,” J. Phys. A 23, 5745–5754 (1990).
17.N. D. Whelan, D. A. Goodings, and J. K. Cannizzo, “Two balls in one dimension with gravity,” Phys. Rev. A 42, 742–754 (1990).
18.D. W. Jepsen, “Dynamics of simple many-body systems of hard rods,” J. Math. Phys. 6, 405–413 (1965).
19.T. J. Murphy, “Dynamics of hard rods in one dimension,” J. Stat. Phys. 74, 889–901 (1994).
20.S. Redner, A Guide to First-Passage Processes (Cambridge University Press, New York, 2001).
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