Dear Bruno and Friends,

    The paper that I have been waiting a long time for.  ;-)

 Algorithmic Thermodynamics

John C. Baez <,math/1/au:+Baez_J/0/1/0/all/0/1>,Mike Stay <,math/1/au:+Stay_M/0/1/0/all/0/1>
(Submitted on 11 Oct 2010)

   Algorithmic entropy can be seen as a special case of entropy as
   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 maximizes entropy subject to constraints on 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 which we call the 'algorithmic temperature' T,
   'algorithmic pressure' P and algorithmic potential' mu, since they
   are analogous to the temperature, pressure and chemical potential.
   We derive an analogue of the fundamental thermodynamic relation dE =
   T dS - P d V + mu dN, and use it to study thermodynamic cycles
   analogous to those for heat engines. We also investigate the values
   of T, P and mu 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 nontrivial algorithmic entropy.

Now to discuss how this is useful to define a local notion of a measure for COMP.



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