The fundamental postulate states, roughly, that each microstate of a system consistent with certain known constraints (e.g. on the total energy of the system), is equally likely to occur.

The ergodic hypothesis w.r.t. a system states roughly that "all microstates are equally likely to occur over a longer period of time"

I don't fully understand the ergodic hypothesis, but it seems to me that, at least the way I have formulated them here, the ergodic hypothesis is less strict than the fundamental postulate, even though it is the other way around.

So what exactly is the relation between these two? and where am I misunderstanding them?


Ergodic hypothesis (EH): Over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region.

Let's say that you partition your phase space in cells of volume $\Delta p \Delta q$. If the system is ergodic, the time spent by the system in one of these regions is

$$\tau \propto \Delta p \Delta q$$

If you make an observation of the system, the probability $P$ that you find its microstate to lay in a certain cell in phase space is proportional to $\tau$:

$$P \propto \tau \propto\Delta p \Delta q$$

In the limit in which the volume of the cell becomes vanishingly small, you have

$$dP \equiv \rho(p,q)dp dq \propto dpdq$$

where we have introduced the probability density $\rho(p,q)$. We have therefore found that

$$\rho(p,q) = constant $$

i.e., the probability density is uniform: the system is equally likely to be found in any of the microstates corresponding to the macrostate.

Postulate of equal a-priori probability (PEAPP): At equilibrium, all microstates compatible with a given macrostate are equally probable.

The PEAPP is less strict than the EH because it tells you nothing about time evolution. As far as the PEAPP is concerned, there could be no (microscopic) time evolution at all.

Let's put it this way: the PEAPP tells you that if you have an infinite number of copies of a system in a certain macrostate $\mathcal M$, the probability that you find one of these system to be in one of the microstates compatible with $\mathcal M$ is a constant. However, it is also possible that if you were able to follow the time evolution of a single one of these systems, you would observe no time evolution at all: this is compatible with the PEAPP.

The PEAPP allows you to do statistical mechanics, in the sense that it gives you a "prior" (the microcanonical distribution) from which you can start to build statistical mechanics. The EH does much more: it provides you a link between phase-space averages and time evolution. A consequence of the EH is indeed that time averages performed over an long (infinite) time interval are equivalent to phase space averages:

$$\bar A = \lim_{T \to \infty} \int_0^T A(p(t),q(t)) dt = \langle A \rangle = \int A(p,q) \rho(p,q) dp dq \tag{1}$$

This result gives a justification of the validity of statistical mechanics (even if it's not the only possible way to do so), because in real life we don't have an infinite number of copies of the system (the so-called "ensemble") available, but we can just do measurements on a single copy of the system.

Since microscopic time scales are of the order of $10^{-15}$ s, and macroscopic observation times are of the order of $1$ s, we can expect $(1)$ to work even if we cannot actually take the limit $T\to \infty$.

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