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By Christopher Jarzynski

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Thus, Eq. 17 should be valid. Both terms on the right hand side of Eq. 17 scale like e2. Comparing 22 this equation with Eq. 7, we find the leading terms of gl and g2 to scale like e and e2, respectively. 18) . 19) It now remains to obtain the O(e 2) term of gl. To accomplish this, we will make use of Liouville's theorem. We begin by rewriting Eq. 6 in terms of the distribution of "enclosed phase space volumes", ((f_, t), rather than the distribution T/(E, t). That is, define ( so that ((ft, t)dr/gives, of energies at time t, the number of systems found on energy shells enclosing a volume of phase space between ft and _/-t- d_.

2 PRELIMINARIES • We take the time-dependent rather than a dynamical, independently shape of the container to be an externally quantity: imposed, the shape evolves in a pre-determined of the gas of particles. Each bounce of a particle walls of the container is taken to be specular the angle of incidence) in the instantaneous way, off the moving (the angle of reflection is equal to rest frame of the local piece of wall at which the collision occurs. Effectively, these bounces constitute elastic collisions in which the inertia of the wall is infinitely greater than that of the particle.

6 RELATION In this section TO PREVIOUS RESULTS we show that one can consistently treat a chaotic adiabatic billiard gas as an example of a chaotic adiabatic ensemble, by treating the container as the limiting case of a smooth following expressions potential well. 4 for the billiard gas. I We begin by recalling the definitions of fi and C(s). f-/(z, c-t) at time t. 37) t where the curly brackets indicate an average over all points z on the energy shell E of Ha, and Oa(s) is a time evolution operator which acts to the right, evolving a point z for a time s under the frozen Hamiltonian Ha.

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