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
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:
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
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
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))}]} )$.
(F)
the argument (e.g., Ergodic theorem) in the previous section
is not related to
"probability"
Thus, it is natural to
assume as follows.
(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{})
$.
(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 $t$)}
$.
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.
17.2: Equilibrium statistical mechanical phenomena concerning Axiom 1 ( Measurement)
This web-site is the html version of "Linguistic Copehagen interpretation of quantum mechanics; Quantum language [Ver. 4]" (by Shiro Ishikawa; [home page] )
PDF download : KSTS/RR-18/002 (Research Report in Dept. Math, Keio Univ. 2018, 464 pages)