Abstract:
In our former work [K. Tadaki, Local Proceedings of CiE 2008, pp. 425-434,
2008], we developed a statistical mechanical interpretation of algorithmic information
theory by introducing the notion of thermodynamic quantities at temperature T, such as
free energy F(T), energy E(T), and statistical mechanical entropy S(T), into the theory.
These quantities are real functions of real argument T > 0. 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 program-size complexity. Furthermore, we
showed that this situation holds for the temperature itself as a thermodynamic quantity.
Namely, the computability of the value of partition function Z(T) gives a sufficient
condition for T Є (0,1) to be a fixed point on partial randomness. In this paper, we
show that the computability of each of all the thermodynamic quantities above gives
the sufficient condition also. Moreover, we show that the computability of F(T) gives
completely different fixed points from the computability of Z(T).