17.2: Equilibrium statistical mechanical phenomena concerning Axiom 1 ( Measurement)

In this section we shall study the probabilistic aspects of equilibrium statistical mechanics. For completeness, note that

 (F) the argument (e.g., Ergodic theorem) in the previous section is not related to "probability"

since Axiom 1 (measurement; $\S$2.7) does not appear in Section 17.1. Also, Recall the linguistic interpretation ($\S$3.1) : there is no probability without measurement .

Note that the (17.12) implies that the equilibrium statistical mechanical system at almost all time $t$ can be regarded as:

 (G) a box including about $10^{24}$ particles such as the number of the particles whose states belong to $\Xi$ $({}\in {\cal B}_{{\mathbb R}^6 }{})$ is given by $\rho_{{}_E} ({}\Xi{}) \times 10^{24}$.
Thus, it is natural to assume as follows.
 (H) if we, at random, choose a particle from $10^{24}$ particles in the box at time $t$, then the probability that the state $(q_1, q_2, q_3,$ $p_1, p_2, p_3)$ $(\in {\mathbb R}^6)$ of the particle belongs to $\Xi$ $({}\in {\cal B}_{{\mathbb R}^6 }{})$ is given by $\rho_{{}_E} ({}\Xi{})$.

In what follows, we shall represent this (H) in terms of measurements. Define the observable ${\mathsf O}_0=({\mathbb R}^6 , {\cal B}_{{\mathbb R}^6}, F_0)$ in $L^\infty( {\Omega}_{{}_E} )$ such that

\begin{align} & [F_0( \Xi ) ]( q,p) = [ D_{K_N}^{({}q, p{}) }](\Xi) \Big(\equiv \frac{\sharp[\{ k \;|\;{\pi}_k ({}q,p{})\in \Xi \} ]}{\sharp [{}K_N] } \Big) \nonumber \\ & \quad \qquad ( \forall \Xi \in {\cal B}_{{\mathbb R}^6}, \forall (q,p{}) \in {{\Omega}}_{{}_E} ({}\subset {\mathbb R}^{6 N}{}) ). \tag{17.15} \end{align}

Thus, we have the measurement ${\mathsf M}_{L^\infty ( {\Omega}_{{}E} )}( {\mathsf O}_0:= ({\mathbb R}^6 , {\cal B}_{{\mathbb R}^6}, F_0), S_{[ \delta_{\psi_t ( q_{{}_0} , p_{{}_0})}]} )$. Then we say, by Axiom 1 (measurement; $\S$2.7), that

 (I) the probability that the measured value obtained by the measurement ${\mathsf M}_{L^\infty ( {\Omega}_{{}E} )}( {\mathsf O}_0:= ({\mathbb R}^6 , {\cal B}_{{\mathbb R}^6}, F_0), S_{[ \delta_{\psi_t ( q_{{}_0} , p_{{}_0} )}]} )$ belongs to $\Xi ( \in {\cal B}_{{\mathbb R}^6})$ is given by $\rho_{{}_E} ( \Xi )$. That is because Theorem 17.4 says that $[ F_0( \Xi ) ]( \psi_t ( q_{{}_0} , p_{{}_0}) )$ $\approx \rho_{{}_E} ( \Xi )$ $\text{(almost every time$t$)}$.

Also, let $\Psi^{{}_E}_t: L^\infty (\Omega_{_E}) \to L^\infty (\Omega_{_E})$ be a deterministic Markov operator determined by the continuous map $\psi^{{}_E}_t: \Omega_{{}_E} \to \Omega_{{}_E}$ (cf. Section 17.1.2). Then, it clearly holds $\Psi^{{}_E}_t{\mathsf{O}_0}={\mathsf{O}_0}$. And, we must take a ${\mathsf{M}}_{L^\infty ( \Omega_{{}_E} )}( {\mathsf{O}_0}, S_{[{(q(t_k),p(t_k))}]} )$ for each time $t_1,t_2,\ldots,t_k, \ldots, t_n$. However, the linguistic interpretation ($\S$3.1) : ( there is no probability without measurement) says that it suffices to take the simultaneous measurement ${\mathsf{M}}_{C( \Omega_{{}_E} )}({\Large \times}_{k=1}^n {\mathsf{O}_0} ,$ $S_{[\delta_{(q(0),p(0))}]} )$.

Remark 17.6 [The principle of equal a priori probabilities ]. The (H) (or equivalently, (I)) says "choose a particle from $N$ particles in box"$\!$, and not "choose a state from the state space $\Omega_{{}_E}$". Thus, the principle of equal (a priori) probability is not related to our method. If we try to describe Ruele's method in terms of measurement theory, we must use mixed measurement theory (cf. Chapter 9). However, this trial will end in failure.

17.3 Conclusions

Our concern in this chapter may be regarded as the problem: {\lq\lq}What is the classical mechanical world view?" Concretely speaking, we are concerned with the problem: $$\mbox{ "our method" vs. "Ruele's method ( which has been authorized for a long time )" }$$ And, we assert the superiority of our method to Ruele's method in Remarks 17.2, 17.5 17.6.