Abstract (17.0: Equilibrium statistical mechanics and Ergodic Hypothesis)

In this chapter, we study and answer the following fundamental problems concerning classical equilibrium statistical mechanics:

$(A):$ Is the principle of equal a priori probabilities indispensable for equilibrium statistical mechanics?
$(B):$ Is the ergodic hypothesis related to equilibrium statistical mechanics?
$(C):$ Why and where does the concept of "probability" appear in equilibrium statistical mechanics?
Note that there are several opinions for the formulation of equilibrium statistical mechanics. In this sense, the above problems are not yet answered. Thus we propose the measurement theoretical foundation of equilibrium statistical mechanics, and clarify the confusion between two aspects (i.e., probabilistic and kinetic aspects in equilibrium statistical mechanics), that is, we discuss

$\bullet$ $ \begin{cases} \mbox{the kinetic aspect (i.e., causality)} & \cdots \mbox{ in Section 17.1} \\ \mbox{the probabilistic aspect (i.e., measurement) } & \cdots \mbox{ in Section 17.2} \end{cases} $

And we answer the above (A) and (B), that is, we conclude that
$(\sharp )$ (A) is "No", but, (B) is "Yes"
and further, we can understand the problem (C). This chapter is extracted from the following:

$\bullet$ S.Ishikawa, Equilibrium Statistical Mechanics in the Quantum Mechanical World View World Journal of Mechanics, Vol. 2, No. 2, 2012, pp. 125-130