A statistical mechanical interpretation of algorithmic information theory: Total statistical mechanical interpretation based on physical argument

Abstract

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 samelevel of mathematical strictness as normal statistical mechanics in physics, we develop a totalstatistical mechanical interpretation of AIT which actualizes a perfect correspondence to normalstatistical mechanics. We do this by identifying a microcanonical ensemble in the framework ofAIT. As a result, we clarify the statistical mechanical meaning of the thermodynamic quantitiesof AIT.