Box 1. The role of thermodynamics in computation

The fact that entropy and information are intimately linked has been known since Maxwell introduced his famous 'demon' well over a century ago¹. Maxwell's demon is a hypothetical being that uses its information-processing ability to reduce the entropy of a gas. The first results in the physics of information processing were derived in attempts to understand how Maxwell's demon could function^{1, 2, 3, 4}. The role of thermodynamics in computation has been examined repeatedly over the past half century. In the 1950s, von Neumann¹⁰ speculated that each logical operation performed in a computer at temperature T must dissipate energy k_BTln2, thereby increasing entropy by k _Bln2. This speculation proved to be false. The precise, correct statement of the role of entropy in computation was attributed to Landauer⁵, who showed that reversible, that is, one-to-one, logical operations such as NOT can be performed, in principle, without dissipation, but that irreversible, many-to-one operations such as AND or ERASE require dissipation of at least k_Bln2 for each bit of information lost. (ERASE is a one-bit logical operation that takes a bit, 0 or 1, and restores it to 0.) The argument behind Landauer's principle can be readily understood³⁷. Essentially, the one-to-one dynamics of hamiltonian systems implies that when a bit is erased the information that it contains has to go somewhere. If the information goes into observable degrees of freedom of the computer, such as another bit, then it has not been erased but merely moved; but if it goes into unobservable degrees of freedom such as the microscopic motion of molecules it results in an increase of entropy of at least k_Bln2.

In 1973, Bennett^{28, 29, 30} showed that all computations could be performed using only reversible logical operations. Consequently, by Landauer's principle, computation does not require dissipation. (Earlier work by Lecerf²⁷ had anticipated the possibility of reversible computation, but not its physical implications. Reversible computation was discovered independently by Fredkin and Toffoli³¹.) The energy used to perform a logical operation can be 'borrowed' from a store of free energy such as a battery, 'invested' in the logic gate that performs the operation, and returned to storage after the operation has been performed, with a net 'profit' in the form of processed information. Electronic circuits based on reversible logic have been built and exhibit considerable reductions in dissipation over conventional reversible circuits^{33, 34, 35}.

Under many circumstances it may prove useful to perform irreversible operations such as erasure. If our ultimate laptop is subject to an error rate of bits per second, for example, then error-correcting codes can be used to detect those errors and reject them to the environment at a dissipative cost of k_BT_Eln2 J s^-1, where T_E is the temperature of the environment. (k_B Tln2 is the minimal amount of energy required to send a bit down an information channel with noise temperature T (ref. 14).) Such error-correcting routines in our ultimate computer function as working analogues of Maxwell's demon, getting information and using it to reduce entropy at an exchange rate of k_BTln2 joules per bit. In principle, computation does not require dissipation. In practice, however, any computer — even our ultimate laptop — will dissipate energy.

The ultimate laptop must reject errors to the environment at a high rate to maintain reliable operation. To estimate the rate at which it can reject errors to the environment, assume that the computer encodes erroneous bits in the form of black-body radiation at the characteristic temperature 5.87 10⁸ K of the computer's memory²¹. The Stefan–Boltzmann law for black-body radiation then implies that the number of bits per unit area than can be sent out to the environment is B = ²k_B³T ³/60ln(2)³c² = 7.195 10⁴² bits per square meter per second. As the ultimate laptop has a surface area of 10^-2 m² and is performing 10⁵⁰ operations per second, it must have an error rate of less than 10^-10 per operation in order to avoid over-heating. Even if it achieves such an error rate, it must have an energy throughput (free energy in and thermal energy out) of 4.04 10²⁶ W — turning over its own resting mass energy of mc² 10¹⁷ J in a nanosecond! The thermal load of correcting large numbers of errors clearly indicates the necessity of operating at a slower speed than the maximum allowed by the laws of physics.