  Energy consumption of computing has found increasing prominence but the area
still suffers from the lack of a consolidated formal theory. In this paper, a
theory for the energy consumption of computing is structured as an axiomatic
system. The work is pure theoretical, involving theorem proving and
mathematical reasoning. It is also interdisciplinary, so that while it targets
computing, it involves theoretical physics (thermodynamics and statistical
mechanics) and information theory. The theory does not contradict existing
theories in theoretical physics and conforms to them as indeed it adopts its
axioms from them. Nevertheless, the theory leads to interesting and important
conclusions that have not been discussed in previous work. Some of them are:
(i) Landauer's principle is shown to be a provable theorem provided that a
precondition, named macroscopic determinism, holds. (ii) It is proved that real
randomness (not pseudo randomness) can be used in computing in conjunction with
or as an alternative to reversibility to achieve more energy saving. (iii) The
theory propounds the concept that computers that use real randomness may
apparently challenge the second law of thermodynamics. These are computational
counterpart to Maxwell's daemon in thermodynamics and hence are named daemon
computers. (iv) It is proved that if we do not accept the existence of daemon
computers (to conform to the second law of thermodynamics), another type of
computers, named clairvoyant computers, must exist that can gain information
about other physical systems through real randomness. This theorem probably
provides a theoretical explanation for strange observations about real
randomness made in the global consciousness project at Princeton University.
