Billiard balls

Hamiltonian quantum mechanical operator for energy, hard sphere assumption that atoms are like hard billiard balls, which is implemented by having an infinite potential inside the sphere radius and zero potential outside the radius Hartree atomic unit of energy... 

Kletz, T. A. (1991a). Billiard Balls and Polo Mints. The Chemical Engineer, 495 (25 April), 21-22. 

Fig. 6.10 (a) Billiard balls of radius r = l/ /2 traveling on a unit-spaced lattice balls B and... 

Fig. 8.2 An example of a Partitioning CA reversible rule, /, mapping (2x 2)-blocks of two-valued states to (2 X 2-blocks / (2 X 2) —> (2 X 2). Note that this rule conserves the total number of I s (indicated by a solid circle) and O s (indicated by an empty square). The system that evolves under this rule is in fact a universal CA (see Billiard Ball Model, later in this section).

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

Not every collision, not every punctilious trajectory by which billiard-ball complexes arrive at their calculable meeting places leads to reaction. 

Consider the motion of a billiard ball. After it is struck by the cue, it moves until it strikes a cushion, front which it bounces, apparently with undiminished velocity. It rolls along in a new direction until it strikes another cushion, changing its direction again. It may continue to roll until it has hit the cushions six or seven times. The billiard ball seems almost tireless as it rebounds time and again from the walls of the billiard table. Could there be a connection between the untiring motion of a billiard ball and the untiring pressure of a gas in a balloon ... 

Billiard balls have long fascinated both idle and curious men. The latter group has found that the motion of a billiard ball can be described... 

Fig. 1-4. A rebounding billiard ball suggests a possible explanation of gas pressure.

Possible Answer Perhaps the gas put in the balloon consists of a collection of small particles that rebound from the wall of the balloon just as billiard balls rebound from the cushions of a billiard table. As the gas particles rebound from the balloon wail, they push on it. When more gas particles are added, the number of such wall collisions per second increases, hence the outward push on the balloon wall increases. The balloon expands. 

This is the characteristic pattern of an explanation. It begins with a Why question that asks about a process that is not well understood. An answer is framed in terms of a process that is well understood. In our example, the origin, of gas pressure in the balloon is the process we wish to clarify. It is difficult even to sense the presence of a gas. The air around us usually cannot be seen, tasted, nor smelled (take away smog) it cannot be heard or felt if there is no wind. So we attempt to explain the properties of a gas in terms of the behavior of billiard balls. These objects are readily seen and felt their behavior has been thoroughly studied and is well understood. 

Now, perhaps, you can see that answering the question Why is merely a highly sophisticated form of seeking regularities. It is indeed a regularity of nature that gases and billiard balls have properties in common. The special creativ-... 

This model is useful, first, because we can calculate in mathematical detail just how much push a billiard ball exerts on a cushion at each rebound, and, second, because exactly the same mathematics describes the pressure behavior of gas in a balloon. The success of the model leads to new directions of thought. For example, we might now wonder whether the pressure-volume behavior of oxygen, as shown in Table l-II (p. 14), can be explained in terms of the particle model of a gas. 

The amount of work performed fixed W. Measurements of mass and velocity of the rubber band tell us, experimentally, the magnitude of (KE),. How do we know (PE)%1 How are we sure that (PE)2 is equal to W and to (KE),1 The evidence we have is that we put an amount of energy into the system and can recover all of it later at will. It is natural to say the energy is stored in the meantime. Then we can say that the rubber band is just like the billiard ball collision energy is conserved at all times. 

It is easier to explain why W, = Q3 if we say that the energy fV, was stored in the chemical substances H2(g) and O (g). We assign to these (and all other) substances the capacity to store energy and we call it heat content. This permits us to say that energy is conserved at all times during a chemical reaction as it is in billiard ball collisions and in stretched rubber bands. 

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