The computer makes such transitions only as the accidental result of the random thermal motion of its parts biased by the weak external force. It is nearly as likely to proceed backward along the computational path, undoing the most recent transition, as it is to proceed forward. A small force, provided externally, drives the computation forward. This force can again be as small as we wish, and so there is no minimum amount of energy that must be expended in order to run a Brownian clockwork Turing machine.
According to classical thermodynamics, then, there is no minimum amount of energy required in order to perform a computation. Is the classical thermodynamical analysis in conflict with quantum theory? After all, the quantum-mechanical uncertainty principle states there must be an inverse relation between our uncertainty about how long a process takes and our uncertainty about how much energy the process involves. Some investigators have suggested that any switching process occurring in a short period of time must involve a minimum expenditure of energy.
In fact the uncertainty principle does not require any minimum energy expenditure for a fast switching event. The uncertainty principle would be applicable only if we attempted to measure the precise time at which the event took place. Even in quantum mechanics extremely fast events can take place without any loss of energy. Our confidence that quantum mechanics allows computing without any minimum expenditure is bolstered when we remember that Benioff and others have developed models of reversible quantum-mechanical computers, which dissipate no energy and obey the laws of quantum mechanics.
Thus the uncertainty principle does not seem to place a fundamental limit on the process of computation; neither does classical thermodynamics. Does this mean there are no physical limitations to computing? Far from it. The real limitations are associated with questions that are much harder to answer than those we have asked in this article.
For example, do elementary logic operations require some minimum amount of time? What is the smallest possible gadgetry that could accomplish such operations? Because scales of size and time are connected by the velocity of light, it is likely that these two questions have related answers. We may not be able to find these answers, however, until it is determined whether or not there is, some ultimate graininess in the universal scales of time and length.
At the other extreme, how large can we make a computer memory? How many particles in the universe can we bring and keep together for that purpose? The maximum possible size of a computer memory limits the precision with which we can calculate. It will limit, for example, the number of decimal places to which we can calculate pi. The inevitable deterioration processes that occur in real computers pose another, perhaps related, question: Can deterioration, at least in principle, be reduced to any desired degree, or does it impose a limit on the maximum length of time we shall be able to devote to any one calculation? That is, are there certain calculations that cannot be completed before the computer's hardware decays into uselessness?
Such questions really concern limitations on the physical execution of mathematical operations. Physical laws, on which the answers must ultimately be based, are themselves expressed in terms of such mathematical operations. Thus we are asking about the ultimate form in which the laws of physics can be applied, given the constraints imposed by the universe that the laws themselves describe.