Ergodicity

Here we will introduce an abstract concept– ergodicity. The idea provides a connection between time-evolution and the statistical description of the system. However, prove a system is ergodic or not is in general difficult. (That’s why I skip the discussion during the lecture. For those who are interested in this concept, please check [Set21] and the related references.) Therefore, the approach is usually we assume our system is ergodic. Then, we argue the time average equals micro canonical averages. We will discuss the concept of ergodicity here and the concept of micro canonical ensemble later.

In the study of statistical mechanics, we are trying to describe the observables of many-body system under time-evolution using statistical description. The bottleneck that we are trying to pass is how to go from time-evolution to statistics? The Liouville’s theorem suggest that understanding the time-evolution in phase space is a good approach as it has the nice properties 3 that we discussed in the last section, i.e. it is possible to have a probability density function \(\rho\) does not depends on \(t\) explicitly. But it does not tell us how to derive the probability density function.

The time average starting at initial condition \(\xi(t=0)\) is

\[ \overline{O(t=0)}\equiv \lim_{T\to\infty} \frac{1}{T}\int_{0}^{T} dt O(\xi(t))\text{.} \]

If \(O\) is well behaved, we expect \(\overline{O(t=0)}=\overline{O(t=t')}\). That is, the long time average should not depends on the value of \(O\) during the time interval \((0,t')\).

\[\begin{split} \overline{O(t=0)} &\equiv \lim_{T\to\infty} \frac{1}{T}\int_{0}^{T} dt O(\xi(t))\\ \overline{O(t=t')} &\equiv \lim_{T\to\infty} \frac{1}{T}\int_{t'}^{T} dt O(\xi(t))\text{.} \end{split}\]

Therefore, we found the long time averages are constant on trajectories.

Since \(\overline{O}\) is constant on trajectories. The microcanonical ensemble average of a constant is still a constant. i.e.

\[ \underbrace{\overline{O}=a^*=\langle \overline{O} \rangle}_{\text{long time average is constant}}=\langle \lim_{T\to\infty} \frac{1}{T}\int_{0}^{T} dt O(\xi(t))\rangle=\lim_{T\to\infty} \frac{1}{T}\int_{0}^{T} dt \langle O(\xi(t))\rangle=\langle O \rangle \]

Because of the nice properties 3 from Liouville’s theorem, the micro canonical ensemble was time independent and the ensemble average equals to the time average.

Note

The microcanonical ensemble that we are going to discuss later means we average over all apossible phase space configuration with equal weight.

\[ \langle O\rangle\equiv \frac{1}{|\Gamma|}\sum_{\xi\in\Gamma} O(\xi)\text{.} \]

The ergodicity support that

\[ \overline{O}=\langle O\rangle\text{.} \]

We will discuss the idea of micro canonical ensemble in the next section such that we will establish

\[ \langle O\rangle=O_{eq}\text{.} \]