17.1: Equilibrium statistical mechanical phenomena concerning Axiom 2 (causality)

Equilibrium statistical mechanical phenomena

Hypothesis 17.1 [Equilibrium statistical mechanical hypothesis]. Assume that about $N ({{\approx}} 10^{24} \approx 6.02 \times 10^{23}$ $\approx$ "the Avogadro constant") particles (for example, hydrogen molecules) move in a box with about $20$ liters. It is natural to assume the following phenomena @ $\text{--}$ C
 @ Every particle obeys Newtonian mechanics.
 A Every particle moves uniformly in the box. For example, a particle does not halt in a corner.
 B Every particle moves with the same statistical behavior concerning time.
 C The motions of particles are $($approximately$)$ independent of each other.

In what follows we shall devote ourselves to the problem:

 (D) how to describe the above equilibrium statistical mechanical phenomena @-C in terms of quantum language ( =measurement theory).

In Newtonian mechanics, any state of a system composed of $N ({}\approx 10^{24}{})$ particles is represented by a point $({}q , p{})$ $\bigl(\equiv$ (position, momentum) $=$ $({} q_{1n}, q_{2n}, q_{3n} ,$ $p_{1n} , p_{2n} ,$ $p_{3n}{})_{n=1}^N$ $\bigl)$ in a phase (or state) space ${\mathbb R}^{6N}$. Let ${\cal H}: {\mathbb R}^{6N} \to {\mathbb R}$ be a Hamiltonian such that

\begin{align} & {\cal H} \big( ({} q_{1n}, q_{2n}, q_{3n} , p_{1n} , p_{2n} , p_{3n}{})_{n=1}^N \big) = \text{momentum energy}+\text{potential energy} \nonumber \\ = & [\sum\limits_{n=1}^N \sum\limits_{k=1,2,3} \frac{(p_{kn})^2}{2 \times \text{particle's mass}} ] \! + \! U( ({} q_{1n}, q_{2n}, q_{3n} )_{n=1}^N ). \tag{17.2} \end{align}

Fix a positive $E>0$. And define the measure $\nu_{{}_E}$ on the energy surface ${{\Omega}}_{{}_E}$ ($\equiv$ $\{ ({}q, p{}) \in {\mathbb R}^{6N}{} \; | \; {\cal H}({}q,p{}) = E \}$) such that

\begin{align} {{ \nu}_{{}_E} }({}B) = \int_B | \nabla {\cal H}({}q,p{}) |^{-1} d m_{6N-1} \qquad ({}\forall B \in {\cal B}_{{{{\Omega}}_{{}_E}}}, \text{ the Borel field of }{{\Omega}}_{{}_E} ) \nonumber \end{align} where $$| \nabla {\cal H}({}q,p{}) |= [\sum\limits_{n=1}^N \sum\limits_{k=1,2,3} \{ (\frac{\partial {\cal H}}{\partial p_{kn}})^2 + (\frac{\partial {\cal H}}{\partial q_{kn}})^2 \}]^{1/2}$$

and $d m_{6N-1}$ is the usual surface Lebesgue measure on ${{\Omega}}_{{}_E}$. Let $\{ {{{}} \psi}^{{}_E}_t \}_{ - \infty < t < \infty }$ be the flow on the energy surface ${{\Omega}}_{{}_E}$ induced by the Newton equation with the Hamiltonian ${\cal H}$, or equivalently, Hamilton's canonical equation:

\begin{align} & \frac{dq_{kn}}{dt}= \frac{\partial {\cal H}}{\partial p_{kn}}, \quad \frac{dp_{kn}}{dt}=- \frac{\partial {\cal H}}{\partial q_{kn}}, \tag{17.3} \\ & \;\; (k=1,2,3, \;\; n=1,2,\ldots,N). \nonumber \end{align}

Liouville's theorem says that the measure ${ \nu}_{{}_E}$ is invariant concerning the flow $\{ {{{}} \psi}^{{}_E}_t \}_{ - \infty < t < \infty }$. Defining the normalized measure ${\overline \nu}_{{}_E}$ such that ${\overline \nu}_{{}_E}$ $=$ $\frac{ { \nu}_{{}_E} }{ { \nu}_{{}_E} ({}{{\Omega}}_{{}_E}{}) }$, we have the normalized measure space $({}{{\Omega}}_{{}_E} , {\cal B }_{{{\Omega}}_{{}_E} } , {\overline \nu}_{{}_E} {})$.

Putting ${\cal A}=C_0(\Omega_{{}_E})$ $=C(\Omega_{{}_E})$ (from the compactness of $\Omega_{{}_E}$), we have the classical basic structure:

\begin{align} [C(\Omega_{{}_E}) \subseteq L^\infty( \Omega_{{}_E}, \nu_{{}_E} ) \subseteq B( L^2 ( \Omega_{{}_E}, \nu_{{}_E} ) )] \nonumber \end{align}

Thus, putting $T={\mathbb R}$, and solving the (17.3), we get $\omega_t = (q(t),p(t))$, $\phi_{t_1.t_2} = \psi_{t_2 -t_1}^E$, $\Phi^*_{t_1. t_2 } \delta_{\omega_{t_1}} = \delta_{\phi_{t_1. t_2} (\omega_{t_1} )}$ $(\forall \omega_{t_1} \in \Omega_{{}_E})$, and further we define the sequential deterministic causal operator $\{\Phi_{t_1, t_2 }: L^\infty (\Omega_{{}_E}) \to L^\infty (\Omega_{{}_E}) \}_{(t_1.t_2) \in T_{\le}^2}$.

Now let us begin with the well-known ergodic theorem. For example, consider one particle $P_1$. Put

\begin{align} S_{P_1}=\{\omega \in \Omega_{{}_E} \;|\; \mbox{ a state $\omega$ such that the particle $P_1$ stays around a corner of the box $\}$ } \nonumber \end{align}

Clearly, it holds that $S_{P_1} \subsetneq \Omega_{{}_E}$. Also, if $\psi^{{}_E}_t ( S_{P_1} ) \subseteq S_{P_1}$ $(0 {{\; \leqq \;}}\forall t < \infty )$, then the particle $P_1$ must always stay a corner. This contradicts A Therefore, A means the following:

 A' [Ergodic property]: If a compact set $S(\subseteq \Omega_{{}_E}, S \not= \emptyset )$ satisfies $\psi^{{}_E}_t (S) \subseteq S$ $(0 {{\; \leqq \;}}\forall t < \infty )$, then it holds that $S=\Omega_{{}_E}$.

The ergodic theorem says that the above A$'$ is equivalent to the following equality:

\begin{align} & \displaystyle{ { \mathop {\int_{\Omega_{{}_E}} f( \omega ) {\overline \nu}_{{}_E} ({}d \omega )}_{ {\text{((state) space average)}} } } } = \displaystyle{ { \mathop { \lim_{ T \to \infty} \frac{1}{T} \int_\alpha^{\alpha+T} f( \psi^{{}_E}_{t} (\omega_0) ) dt }_{ {\text{(time average)}} } } } \tag{17.4} \\ & \qquad (\forall \alpha \in {\mathbb R}, \forall f \in C( {{\Omega}}_{{}_E} ), \quad \forall \omega_0 \in {{\Omega}}_{{}_E} ) \nonumber \end{align}

After all, the ergodic property A$'$ ($\Leftrightarrow$ (17.4) ) says that if $T$ is sufficiently large, it holds that

\begin{align} \displaystyle{ { \mathop {\int_{\Omega_{{}_E}} f( \omega ) {\overline \nu}_{{}_E} ({}d \omega )}_{ } } } {{\approx \;}} \displaystyle{ { \mathop { \frac{1}{T} \int_\alpha^{\alpha +T} f( \psi^{{}_E}_{t} (\omega_0) ) dt }_{ } } }. \tag{17.5} \end{align}

Put ${\overline m}_{{}_T}(dt) = \frac{dt}{T}$. The probability space $([\alpha, \alpha+ T], {\cal B}_{[\alpha, \alpha + T]}, {\overline m}_{{}_T})$ (or equivalently, $([0,T], {\cal B}_{[0,T]},$ ${\overline m}_{{}_T})$ ) is called a (normalized) first staying time space, also, the probability space $({}{{\Omega}}_{{}_E} , {\cal B}_{ {{\Omega}}_{{}_E} } , {\overline \nu}_{{}_E} {})$ is called a (normalized) second staying time space. Note that these mathematical probability spaces are not related to "probability".

Put $K_N = \{ 1,2,\ldots, N ({}{{\approx}} 10^{24}{}) \}$. For each $k$ $({}\in K_N )$, define the coordinate map ${\pi}_k{}: {{\Omega}}_{{}_E} ({}\subset {\mathbb{R}}^{6N}{}) \to {\mathbb{R}}^6$ such that

\begin{align} {\pi}_k(\omega) = {\pi}_k (q,p) = & {\pi}_k (({} q_{1n}, q_{2n}, q_{3n} , p_{1n} , p_{2n} , p_{3n}{})_{n=1}^N{}) \nonumber \\ = & ({} q_{1k}, q_{2k}, q_{3k} ,p_{1k} , p_{2k} , p_{3k}{}) \label{eq'176} \end{align}

for all $\omega= (q,p)$ $=$ $({} q_{1n},$ $q_{2n},$ $q_{3n} ,$ $p_{1n} ,$ $p_{2n} , p_{3n}{})_{n=1}^N$ $\in$ ${{\Omega}}_{{}_E} ({}\subset {\mathbb{R}}^{6N}{})$.

Also, for any subset $K$ $({}\subseteq K_N {{=}}$ $\{1,2,$ $\ldots,N$ $({}{{\approx}} 10^{24}{}) \}{})$, define the distribution map $D_{K}^{({}\cdot{})}$ $: {{\Omega}}_{{}_E}$ $({}\subset {\mathbb{R}}^{6 N}{})$ $\to {\cal M}_{+1}^m ({}{\mathbb{R}}^6{})$ such that

\begin{align*} D_K^{({}q, p{}) } = \frac{1}{\sharp [{}K] } \sum\limits_{ k \in K } \delta_{ {\pi}_k ({}q,p{}) } \quad ( \forall (q,p{}) \in {{\Omega}}_{{}_E} ({}\subset {\mathbb{R}}^{6 N}{}) ) \end{align*}

where ${\sharp [{}K] }$ is the number of the elements of the set $K$.

Let $\omega_0 ( \in \Omega_{{}_E})$ be a state. For each $n$ $(\in K_N )$, we define the map $X_n^{\omega_0}: [0, T] \to {\mathbb R}^6$ such that

\begin{align} X_n^{\omega_0} (t) = {\pi}_n ( \psi^{{}_E}_t (\omega_0) ) \qquad (\forall t \in [0,T]). \tag{17.7} \end{align}

And, we regard $\{X_n^{\omega_0}\}_{n=1}^N$ as random variables (i.e., measurable functions ) on the probability space $([0,T], {\cal B}_{[0,T]}, {\overline m}_{{}_T})$. Then, B and C respectively means

 B' $\{X_n^{\omega_0}\}_{n=1}^N$ is a sequence with the approximately identical distribution concerning time. In other words, there exists a normalized measure $\rho_{{}_E}$ on ${\mathbb R}^6$ $(${}i.e., $\rho_{{}_E} \in {\cal M}^m_{+1} ({}{\mathbb R}^6{})${}$)$ such that: \begin{align} & {\overline m}_{{}_T}( \{ t \in [0,T] \;: \; X_n^{\omega_0} ( t) \in \Xi \} ) {{\approx \;}} \rho_{{}_E}(\Xi ) \label{eq'178} \\ & \quad (\forall \Xi \in {\cal B}_{{\mathbb R}^6}, n=1,2,\ldots, N) \nonumber \end{align}
 C$'$ $\{X_n^{\omega_0}\}_{n=1}^N$ is approximately independent, in the sense that, for any $K_0 \subset \{ 1,2,$ $\ldots,$ $N ({}{{\approx}} 10^{24}{})\}$ such that $1 {{\; \leqq \;}} \sharp[{}K_0{}] \ll N{}$ ( that is, $\frac{\sharp[{}K_0{}] }{N} {\approx} 0$ ), it holds that \begin{align*} & {\overline m}_{{}_T}( \{ t \in [0,T] : X_{k}^{\omega_0} ( t) \in \Xi_{k} (\in {\cal B}_{{\mathbb R}^{6}} ), {k} \in K_0 \}) \\ {{\approx}} & {\Large \times}_{{k} \in K_0 } {\overline m}_{{}_T}( \{ t \in [0,T] : X_{k}^{\omega_0} ( t) \in \Xi_{k} (\in {\cal B}_{{\mathbb R}^{6}} ) \}). \end{align*}
Here, we can assert the advantage of our method in comparison with Ruelle's method as follows.

Remark 17.2 [About the time interval $[0,T]$].

For example, as one of typical cases, consider the motion of $10^{24}$ particles in a cubic box (whose long side is 0.3m). It is usual to consider that "averaging velocity"=$5\times10^{2} {\rm m}/{\rm s}$, "mean free path"=$10^{-7}{\rm m}$. And therefore, the collisions rarely happen among $\sharp[{}K_0{}]$ particles in the time interval $[0, T]$, and therefore, the motion is {\lq\lq}almost independent".

For example, putting $\sharp[{}K_0{}]=10^{10}$, we can calculate the number of times a certain particle collides with $K_0$-particles in [0,T] as $(10^{-7} \times \frac{10^{24}}{10^{10}})^{-1} \times {(5 \times 10^2)} \times T$ $\approx 5 \times 10^{-5} \times T$. Hence, in order to expect that \textcircled{\scriptsize 3}$'$ and \textcircled{\scriptsize 4}$'$ hold, it suffices to consider that $T \approx 5$ seconds.

Also, we see, by (17.7) and (17.5), that, for $K_0 (\subseteq K_N)$ such that $1 \le \sharp[{}K_0{}] \ll N{}$,

\begin{align} & {\overline m}_{{}_T}( \{ t \in [0,T] \;:\; X_{k}^{\omega_0} ( t) \in \Xi_{k} (\in {\cal B}_{{\mathbb R}^{6}} ), {k} \in K_0 \}) \nonumber \\ = & {\overline m}_{{}_T}( \{ t \in [0,T] : {\pi}_{k}(\psi^{{}_E}_t ( \omega_0) \in \Xi_{k} (\in {\cal B}_{{\mathbb R}^{6}} ), {k} \in K_0 \}) \nonumber \\ = & {\overline m}_{{}_T}( \{ t \in [0,T] : \psi^{{}_E}_t ( \omega_0) \in (({\pi}_k)_{k\in K_0})^{-1} ( \! {\Large \times}_{k\in K_0} \! \Xi_{k} ) \}) \nonumber \\ {{\approx}} & \; {\overline \nu}_{{}_E} \big( (({\pi}_k)_{k\in K_0})^{-1} ({\Large \times}_{k\in K_0 }\Xi_{k} ) \big) \nonumber \\ \equiv & \big( {\overline \nu}_{{}_E} \circ (({\pi}_k)_{k\in K_0})^{-1} \big) ({\Large \times}_{k\in K_0 }\Xi_{k} ). \label{eq'179} \end{align} Particularly, putting $K_0=\{k\}$, we see: \begin{align} & {\overline m}_{{}_T}( \{ t \in [0,T] \;:\; X_k^{\omega_0} ( t) \in \Xi \}) {{\approx \;}} ({\overline \nu }_{{}_E} \circ {\pi}_k^{-1} )(\Xi) \nonumber \\ & \qquad \qquad (\forall \Xi \in {\cal B}_{{\mathbb R}^{6}} ). \label{eq'1710} \end{align} Hence, we can describe the B and C in terms of $\{{\pi}_k\}$ in what follows.

Hypothesis 17.3 [B and C ].

Put $K_N$ $=$ $\{ 1,2,$ $\ldots,$ $N ( {{\approx}} 10^{24}) \}$. Let ${\cal H}$, $E$, ${\nu_{{}_E}}$, ${\overline \nu}_{{}_E}$, ${\pi}_k:{{\Omega}}_{{}_E} \to {\mathbb{R}}^6$ be as in the above. Then, summing up B and C, by (17.7) we have:

 (E) $\{ {\pi}_k:\Omega_{{}_E} \to {\mathbb{R}}^6 \}_{k=1}^N$ is approximately independent random variables with the identical distribution in the sense that there exists $\rho_{{}_E}$ $(\in {\cal M}_{+1}^m ({\mathbb{R}}^6))$ such that \begin{align} \bigotimes_{ k \in K_0 } {\rho_{{}_E}} (=\text{"product measure"}) {{\approx}} & \; {\overline \nu}_{{}_E} \circ ({} ({}{\pi}_k{})_{ k \in K_0 }{})^{-1}. \label{eq'1711} \end{align} for all $K_0 \subset K_N$ and $1 {{\; \leqq \;}}$ $\sharp[{}K_0{}]$ $\ll N{}$.

Also, a state $(q,p) (\in \Omega_{{}_E})$ is called an equilibrium state , if it satisfies $D_{K_N}^{ ({}q , p {})} {{\approx}} \rho_{{}_E}$.

17.1.5. Ergodic Hypothesis

Now, we have the following theorem:

Theorem 17.4 [Ergodic hypothesis].

Assume Hypothesis 17.3 $($ or equivalently, B and C $)$. Then, for any $\omega_0 = (q(0), p(0)) \in \Omega_{{}_E}$, it holds that

\begin{align} & [D_{K_N}^{ ({}q({}t) , p ({}t) {})} ](\Xi) {{\approx \;}} {\overline m}_{{}_T}( \{ t \in [0,T] \;:\; X_k^{\omega_0} ( t) \in \Xi \}) \nonumber \\ & \qquad ( \forall \Xi \in {\cal B}_{{\mathbb R}^6}, k=1,2,\ldots, N ({}{{\approx}} 10^{24}{}){}) \label{eq'1712} \end{align}

for almost all $t$. That is, $0 {{\; \leqq \;}}$ ${\overline m}_{{}_T}(\{t \in [0,T] : \mbox{(17.12) does not hold}\}$ $\ll 1$.

Proof. Let $K_0 \subset K_N$ such that $1 \ll \sharp[{}K_0{}] \equiv N_0 \ll N{}$ (that is, $\frac{1}{\sharp[{}K_0{}] } {\approx} 0 {\approx} \frac{\sharp[{}K_0{}] }{N}$ ). Then, from Hypothesis 17.3, the law of large numbers says that

\begin{align} D_{K_0}^{ ({}q({}t) , p ({}t) {})} {{\approx \;}} {\overline \nu }_{{}_E} \circ {\pi}_k^{-1} \; (\; {{\approx \;}} {\rho_{{}_E}} \;{}) \qquad \label{eq'1713} \end{align}

for almost all time $t$. Consider the decomposition $K_N$ $=$ $\{ K_{ ({}1{})} , K_{ (2 {})} ,\ldots,$ $K_{ (L {})} \}$. (i.e., $K_N=\bigcup_{l=1}^L K_{(l)}$, $K_{(l)} \cap K_{(l')}=\emptyset \;\; (l \not= l')$ ), where $\sharp [{}K_{ ({}l{})} {}] {{\approx}} N_0$ $({}l=1,2,\ldots, L{})$. From (7.13), it holds that, for each $k$ $({}= 1,2,\ldots,N$ $({}{{\approx}} 10^{24}{}){})$,

\begin{align} & D_{K_N}^{ ({}q({}t) , p ({}t) {})} = \frac{1}{N} \sum\limits_{l=1}^L [ \sharp [{}K_{(l)}{}] \times D_{K_{(l)}}^{ ({}q({}t) , p ({}t) {})} ] \nonumber \\ {{\approx \;}} & \frac{1}{N} \sum\limits_{l=1}^L [ \sharp [{}K_{(l)}{}] \times {\rho_{{}_E}} ] {{\approx \;}} {\overline \nu }_{{}_E} \circ {\pi}_k^{-1} \; ({}\; {{\approx \;}} {\rho_{{}_E}} \;), \label{eq'1714} \end{align}

for almost all time $t$. Thus, by (17.10), we get (17.12). Hence, the proof is completed.

We believe that Theorem 17.4 is just what should be represented by the "ergodic hypothesis" such that

\begin{align*} & \text{"population average of $N$ particles at each $t$"} \\ = & \text{"time average of one particle"}. \end{align*}

Thus, we can assert that the ergodic hypothesis is related to equilibrium statistical mechanics. Here, the ergodic property A$'$ (or equivalently, equality (17.5) and the above ergodic hypothesis should not be confused. Also, it should be noted that the ergodic hypothesis does not hold if the box ( containing particles ) is too large.

Remark 17.5 [The law of increasing entropy].

The entropy $H(q,p)$ of a state $(q,p) (\in \Omega_{{}_E})$ is defined by

\begin{align*} H(q,p) = k \log [ \nu_{_E} ( \{(q',p') \in \Omega_{{}_E}:D_{K_N}^{ ({}q , p )} {{\approx \;}} D_{K_N}^{ ({}q' , p' )} ) \} ) ] \end{align*} where $$k= [\text{Boltzmann constant}] / ( {[\text{Plank constant}]^{3N}N!} )$$

Since almost every state in $\Omega_{{}_E}$ is equilibrium, the entropy of almost every state is equal $k \log \nu_{{}_E}(\Omega_{{}_E})$. Therefore, it is natural to assume that the law of increasing entropy holds.