# Algorithmic Thermodynamics

### Venue

Mathematical Structures in Computer Science, vol. 22 (2012), pp. 771-787

### Publication Year

2012

### Authors

John Baez, Michael Stay

### BibTeX

## Abstract

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.