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Algorithmic thermodynamics

Published online by Cambridge University Press:  06 September 2012

JOHN BAEZ  and
MIKE STAY
JOHN BAEZ
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521, U.S.A.
MIKE STAY
Affiliation:
Computer Science Department, University of Auckland, and Google, 1600 Amphitheatre Pkwy Mountain View, California 94043, U.S.A.
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Abstract

Algorithmic entropy can be viewed as a special case of the entropy studied in statistical mechanics. This viewpoint allows us to apply many techniques developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing machine and let X be the set of programs that halt for this machine. Then we can regard X as a set of ‘microstates’, and treat any function on X as an ‘observable’. For any collection of observables, we can study the Gibbs ensemble that maximises entropy subject to constraints on the expected values of these observables. We illustrate this by taking the log runtime, length and output of a program as observables analogous to the energy E, volume V and number of molecules N in a container of gas. The conjugate variables of these observables allow us to define quantities we call the ‘algorithmic temperature’ T, ‘algorithmic pressure’ P and ‘algorithmic potential’ μ, since they are analogous to the temperature, pressure and chemical potential. We derive an analogue of the fundamental thermodynamic relation dE = TdS − PdV + μdN, and use it to study thermodynamic cycles analogous to those for heat engines. We also investigate the values of T, P and μ for which the partition function converges. At some points on the boundary of this domain of convergence, the partition function becomes uncomputable – indeed, at these points the partition function itself has non-trivial algorithmic entropy.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Special Issue 5: Computability of the Physical , October 2012 , pp. 771 - 787
Copyright
Copyright © Cambridge University Press 2012

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