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Time asymmetry in a dynamical model of the one-dimensional ideal gas
1.Thermodynamic time-asymmetry can also be demonstrated using simple toy models. See P.-M. Binder, J. M. Pedraza, and S. Garzón, “An invertibility paradox,” Am. J. Phys. 67(12), 1091–1093 (1999), which uses a chaotic mapping;
1.V. Ambegaokar and A. A. Clerk, “Entropy and time,” Am. J. Phys. 67(12), 1068–1073 (1999), which uses Ehrenfest’s double-urn model; and
1.G. R. Fowles, “Time’s arrow: A numerical experiment,” Am. J. Phys. 62(4), 321–328 (1994), which uses a toy model involving plane waves.
2.The computer program used to perform these simulations will be provided upon request.
3.Here we are only considering entropy in the context of the ideal gas. An overview of entropy in other contexts is given in K. Andrew, “Entropy,” Am. J. Phys. 52(6), 492–496 (1984).
3.The concept of entropy can also be introduced by explicitly counting microstates in discrete toy models; this approach is discussed in T. A. Moore and D. V. Schroeder, “A different approach to introducing statistical mechanics,” Am. J. Phys. 65(1), 26–36 (1997);
4.By “extremely long” we mean long compared to the Poincaré recurrence time; that is, long enough that the system has a chance to sample the entire state space. By looking over such time scales, we ensure that the fraction of time that the system spends in a given region is independent of the initial conditions.
5.See L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Pergamon, Tarrytown, 1980), Sec. 40.
6.For a given distribution , we can define a quantity that gives the probability that an atom is within of ; for the Maxwell distribution . We can show that if we average over the entire state space, the root-mean-square deviation of from is proportional to in the limit of large . See K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987), Sec. 4.3.
7.Time-reversal invariance is discussed in J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994), Sec. 4.4
8.There are also reversible thermodynamic processes such as the quasi-static process that we consider in Sec. VI. See M. Samiullah, “What is a reversible process?” Am. J. Phys. 75(7), 608–609 (2007).
9.For a more extensive discussion of irreversibility see R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, 1963), Vol. 1, Chap. 46.
10.This apparent inconsistency was pointed out by Loschmidt in a 1876 paper that criticized Boltzmann’s derivation of the -theorem, and is known as Loschmidt’s paradox. The historical development of the paradox is discussed in S. G. Brush, The Kind of Motion We Call Heat (North-Holland, Amsterdam, 1976).
11.In the limit of large , randomly choosing from the entire state space is equivalent to randomly choosing from the subspace , because in this limit almost all the states in state space have distributions close to .
12.The atoms are uniformly distributed in space for the initial states that we will consider, and will tend to remain uniformly distributed as the system evolves in time. Hence, this assumption is justified.
13.To evolve the system backward in time, we motion-reverse the initial state, evolve it forward in time, and then motion-reverse the time-evolved state.
14.To obtain state we choose position and momentum values for each of the atoms by randomly sampling the distribution . From Eq. (13), it follows that randomly sampling amounts to taking and , where and are randomly chosen from the interval [0,1] using a uniform probability distribution.
15.This statement is only probabilistically true: it is possible, though unlikely, that the entropy will decrease when the system is evolved away from initial state . The probability of such an entropy decrease is quantified by the fluctuation theorem, which was first proposed in D. J. Evans, E. G. D. Cohen, and G. P. Morriss, “Probability of second law violations in shearing steady states,” Phys. Rev. Lett. 71, 2401–2404 (1993).
16.Note that the second law does not say that the entropy increases when the system is evolved away from any low entropy state. For example, we can define a low entropy state by evolving state to ; for state the entropy increases when evolved forward, but decreases when evolved backward.
17.A simulation of the evolution of the velocity distribution for the two-dimensional ideal gas is discussed in J. Novak and A. B. Bortz, “The evolution of the two-dimensional Maxwell-Boltzmann distribution,” Am. J. Phys. 38(12), 1402–1406 (1970).
18.The derivation of this result is left as an exercise for students; note that the equation of motion for the piston is , and that for quasi-static expansion and contraction is a constant, so .
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