In my script the case of the 2D classic harmonic oscillation is taken into consideration. We are given an example as to how it is related to ergodicity/ an ergotic system. This example might be a bit long but I find it very interesting and extremely important because it takes into consideration several concepts present in statistical mechanics and it relates them to one another. But it is hard to understand what it's meant with the following commentary, so I would appreciate any explanation. It goes like this:

We consider the following classical hamiltonian:

$H= \frac{p^2}{2m} + \frac{m \omega^2 q^2}{2}$.

We are asked to give our opinion on whether the time averaged canonical partition function is similar to the statistical averaged one and to compare it, the partition function of the 2D case with that of the 1D case.

This is the commentary that follows:

The basic assumption when applying ensemble theory is that the systems under consideration behave ergodic. Ergodic means at least that the system comes arbitrarily close to any point in the phase space that is compatible with the conservation laws in a sufficiently long time. The assumption is that the ensemble mean of an observable is O, which is calculated using statistical physics:

$$\langle \hat O \rangle_E=\int \rho(\{\vec q\},\{\vec p\}) O(\{\vec q\},\{\vec p\})d\Pi$$

where $d\Pi=\frac {1}{N!}\Pi_i^N \frac{d \vec p_i d\vec q_i}{(2\pi\hbar)^d}$, N subsystems in d dimensions.

(here ρ is the phase space density and dΠ is a phase space element) agrees with the usual (time) mean of the system:

$$\langle \hat O \rangle_E=\langle \hat O \rangle_T=lim_{t\rightarrow \infty}\frac 1 t\int_0^t O(\{\vec q(t)\},\{\vec p(t)\})dt'$$

Under this assumption, one can determine the expected values of the system using the phase space density without explicitly solving all the equations of motion, which would be necessary to calculate the temporal expected value. Interestingly, this assumption almost always seems to be correct in many-particle systems: With ensemble theory we can calculate correct expectation values. At the same time, it is very difficult to test the assumption in a many-body system.

This part is relatively clear to me. I simply included it because it is part of the whole explanation. Then it goes on:

For the (single-particle) example in the exercise, however, one can see that the assumption is not fulfilled: All positions within the potential circle are energetically allowed.

What is meant with potential circle here?

However, a particle that is pushed in the potential circle will never reach certain positions. This is easiest to see if the particle starts from the center of the circle: it will then move back and forth in a straight line forever.

I don't understand this commentary

So the system is obviously not ergodic and we can not expect the ensemble mean to give us correct (temporal) expectation values.

Why did we decide that the system is not ergodic

Indeed, one finds that the time average of $(\vec x)^2$ (which in this simple case can be computed explicitly) does not agree with the ensemble mean.

The situation is different for the single 1D oscillator that we considered before: given a certain position, two velocities with the same amount are energetically allowed, which are also passed through in each period. All phase space points are thus reached.

I don't understand this commentary

We are lucky that not all systems are ergodic. Another example of a non-ergodic system is the orbit of the earth! Chaos theory studies the conditions under which systems are ergodic. In the circular potential example given above, one can modify the system a little to get such an ergodic system; e.g. a billiard form of the potential. Also in the quantum mechanical equivalent of such systems one sees non-ergodic properties (quantum chaos). For example, the back and forth movement of a quantum particle in such a potential leads to so-called "quantum scars". of quantum dynamics.



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