  We consider the statistical mechanical ensemble of bit string histories that
are computed by a universal Turing machine. The role of the energy is played by
the program size. We show that this ensemble has a first-order phase transition
at a critical temperature, at which the partition function equals Chaitin's
halting probability $\Omega$. This phase transition has curious properties: the
free energy is continuous near the critical temperature, but almost jumps: it
converges more slowly to its finite critical value than any computable
function. At the critical temperature, the average size of the bit strings
diverges. We define a non-universal Turing machine that approximates this
behavior of the partition function in a computable way by a super-logarithmic
singularity, and discuss its thermodynamic properties. We also discuss
analogies and differences between Chaitin's Omega and the partition functions
of a quantum mechanical particle, a spin model, random surfaces, and quantum
Turing machines. For universal Turing machines, we conjecture that the ensemble
of bit string histories at the critical temperature has a continuum formulation
in terms of a string theory.
