Classical Mechanical Incarnation

The BBM gas consists of an arbitrary number of hard spheres (or balls) of finite diameter that collide elastically both among themselves and with any solid walls (or mirrors) that they may encounter during their motion. Starting out on some site of a two-dimensional Euclidean lattice, each ball is allowed to move only in one of four directions (see figure 6.10). The lattice spacing, d = l/ /2 (in arbitrary units), is chosen so that balls collide while occupying adjacent sites. Unit time is 

B2 are shown undergoing a right-angle elastic collision (b) An example of an interaction gate (see text for discussion). 

The first observation consists merely of an alternative but natural interpretation of the presence or absence of balls the movement of balls is equated with the communication of binary signals. Once this interpretation is made, the second observation also becomes a natural one wherever balls collide, either among themselves or with some collection of rigid mirrors, the effect of the collision may be viewed as a Boolean logic gate. 

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This chapter is organized as follows. In Section II, we briefly summarize the findings of the geometric TST for autonomous Hamiltonian systems to the extent that it is needed for the present discussion. Readers interested in a more detailed exposition are referred to Ref. 35, where the field has recently been reviewed in depth. We restrict our discussion to classical mechanics. Semiclassical extensions of geometric TST have been developed in Refs. 70-75. Section III discusses the notion of the TS trajectory in general and its incarnation in different specific settings. Section IV demonstrates how the TS trajectory allows one to carry over the central concepts of geometric TST into the time-dependent realm. 

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