A statistical mechanical interpretation of algorithmic information theory: Total statistical mechanical interpretation based on physical argument
Kohtaro Tadaki · Journal of Physics Conference Series · 2010
The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed by our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's LNCS, vol. 5407, pp. 422–440, 2009], where we introduced the notion of thermodynamic quantities, such as partition function Z ( T ), free energy F ( T ), energy E ( T ), statistical mechanical entropy S ( T ), and specific heat C ( T ), into AIT. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature T itself, which is one of the most typical thermodynamic quantities. Namely, we showed that, for each of the thermodynamic quantities Z ( T ), F ( T ), E ( T ), and S ( T ) above, the computability of its value at temperature T gives a sufficient condition for T (0,1) to satisfy the condition that the partial randomness of T equals to T . In this paper, based on a physical argument on the same level of mathematical strictness as normal statistical mechanics in physics, we develop a total statistical mechanical interpretation of AIT which actualizes a perfect correspondence to normal statistical mechanics. We do this by identifying a microcanonical ensemble in the framework of AIT. As a result, we clarify the statistical mechanical meaning of the thermodynamic quantities of AIT.