**
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} $. |

(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.