where the sparks occur depends on the detailed microscopic motion of the molecules in the gas, and is therefore potentially quite random.

In an effort to obtain as much randomness as possible, actual devices that work along these lines have typically used progressively smaller components: first vacuum tubes and later semiconductors. And indeed, in a modern semiconductor diode, for example, a breakdown event can be initiated by the motion of just one electron.

But despite such sensitivity to microscopic effects, what has consistently been found in practice is that the output from such devices has significant deviations from perfect randomness.

At first, this is quite surprising. For one might think that microscopic physical processes would always produce the best possible randomness. But there are two important effects which tend to limit this randomness, or indeed any randomness that is obtained through the mechanism of interaction with the environment.

The first of these concerns the internal details of whatever device is used to sample the randomness in the environment.

Every time the device receives a piece of input, its internal state changes. But in order for successive pieces of input to be treated in an independent and uncorrelated way, the device must be in exactly the same state when it receives each piece of input. And the problem is that while practical devices may eventually relax to what is essentially the same state, they can do this only at a certain rate.

In a device that produces a spark, for example, it inevitably takes some time for the hot gas in the path of the spark to be cleared out. And if another spark is generated before this has happened, the path of the second spark will not be independent of the first.

One might think that such effects could be avoided by allowing a certain "dead time" between successive events. But in fact, as we will also see in connection with quantum mechanics, it is a rather general feature of systems that perform amplification that relaxation to a normal state can effectively occur only gradually, so that one would have to wait an infinite time for such relaxation to be absolutely complete.

But even when the device used to sample the environment does no amplification and has no relevant internal structure, one may still not see