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- W3204677836 abstract "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 Ω. 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 function of a quantum mechanical particle, and with 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." @default.
- W3204677836 created "2021-10-11" @default.
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- W3204677836 date "2021-10-01" @default.
- W3204677836 modified "2023-10-02" @default.
- W3204677836 title "Chaitin’s Omega and an algorithmic phase transition" @default.
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- W3204677836 doi "https://doi.org/10.1016/j.physa.2021.126458" @default.
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