If the entire assembly is immersed in an ideal viscous fluid, then the frictional forces that act on the balls will be proportional to their velocity; there will be no static friction. The frictional force will therefore be very weak if we are content to move the balls slowly. In any mechanical system the energy that must be expended to work against friction is equal to the product of the frictional force and the distance through which the system travels. (Hence the faster a swimmer travels between two points, the more energy he or she will expend, although the distance traveled is the same whether the swimmer is fast or slow.) If we move the balls through the Fredkin gates at a low speed, then the energy expended (the product of force and distance) will be very small, because the frictional force depends directly on the balls' speed. Infact, we can expend as little energy as we wish, simply by taking a long time to carry out the operation. There is thus no minimum amount of energy that must be expended in order to perform any given computation.
The energy lost to friction in this model will be very small if the machine is operated very slowly. Is it possible to design a more idealized machine that could compute without any friction? Or is friction essential to the computing process? Fredkin, together with Tommaso Toffoli and others at M.I.T., has shown that it is not.
They demonstrated that it is possible to do computation by firing ideal, frictionless billiard balls at one another. In the billiard-ball model perfect reflecting "mirrors," surfaces that redirect the balls' motion, are arranged in such a way that the movement of the balls across a table emulates the movement of bits of information through logic gates. As before, the presence of a ball in a particular part of the computer signifies a 1, whereas the absence of a ball signifies a 0. If two balls arrive simultaneously at a logic gate, they will collide and their paths will change; their new paths represent the output of the gate. Fredkin, Toffoli and others have described arrangements of mirrors that correspond to different types of logic gate, and they have shown that billiard-ball models can be built to simulate any logic element that is necessary for computing.
To start the computation we fire a billiard ball into the computer wherever we wish to input a 1. The balls must enter the machine simultaneously. Since they are perfectly elastic, they do not lose energy when they collide; they will emerge from the computer with the same amount of kinetic energy we gave them at the beginning.
In operation a billiard-ball computer produces "garbage bits," just as a computer built of Fredkin gates does. After the computer has reached an answer we reflect the billiard balls back into it, undoing the computation. They will come out of the machine exactly where we sent them in, and at the same speed. The mechanism that launched them into the computer can then be used to absorb their kinetic energy. Once again we will have performed a computation and returned the computer to its initial state without dissipating energy.
The billiard-ball computer has one major flaw: it is extremely sensitive to slight errors. If a ball is aimed slightly incorrectly or if a mirror is tilted at a slightly wrong angle, the balls' trajectories will go astray. One or more balls will deviate from their intended paths, and in due course errors will combine to invalidate the entire computation. Even if perfectly elastic and frictionless billiard balls could be manufactured, the small amount of random thermal motion in the molecules they are made of would be enough to cause errors after a few dozen collisions.
Of course we could install some kind of corrective device that would return any errant billiard ball to its desired path, but then we would be obliterating information about the ball's earlier history. For example, we might be discarding information about the extent to which a mirror is tilted incorrectly. Discarding information, even to correct an error, can be done only in a system in which there is friction and loss of energy. Any correctional device must therefore dissipate some energy.