Hi Stephen P. King
Cool. Go for it.
[Roger Clough], [rclo...@verizon.net]
"Forever is a long time, especially near the end." - Woody Allen
----- Receiving the following content -----
From: Stephen P. King
Time: 2013-01-16, 19:29:33
Subject: Algorithmic Thermodynamics
Dear Bruno and Friends,
The paper that I have been waiting a long time for. ;-)
John C. Baez, Mike Stay
(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
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