Is ergodic hypothesis in contradiction with the notion of equilibrium?

Question

From wikipedia:

In physics and thermodynamics, the ergodic hypothesis1 says that, 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, i.e., that all accessible microstates are equiprobable over a long period of time.

So if I understood it right, given enough time the system will move through all possible states. However, from thermodynamics we know that state of equilibrium is in a sense the "final state" in which system will get once and won't move to other states after that.

Aren't these two things in contradiction? If ergodic hypothesis is true then wouldn't that mean that system which is already in state of equilibrium will spontaneously move out of equilibrium into some other state (after enough time has passed)?

Answer 1

You have to be careful to distinguish between microstates and macrostates. Thermodynamic equilibrium is a macrostate which consists of a mixture of all possible microstates of energy $E$ weighted by a Boltzmann weight $e^{- \beta E} / Z$. A state in macroscopic thermal equilibrium can be thought of as "moving through phase space" ergodically (i.e. the microstate is constantly changing, but the fraction of time spent in each microstate is fixed to the Boltzmann weight).

Answer 2

As far as I understand, the answer is: not only ergodicity but Poincare reccurence theorem "contradicts a bit" with the second law of thermodynamics.

The point is that actually time that every ergodic system (for instance, Boltzmann billiard, as Sinai proved) is in some measurable part of full phase space of system is proportional to the phase volume of this part. But due to some kind of large numbers law almost all of the phase space belongs to the parameters of the maximum entropy state.

So if you remove the boundary between two parts of the half-empty volume, system will periodically return in half-empty state (without boundary now), but the phase volume of this family of states is quite small (in fact, devastatingly tiny), so the proportion of time system belongs to this volume is devastatingly tiny too.

If you want, you can set up computer experiment with 1,2,3,...,10 molecules to see the character of rapid decreasing of the phase volume of small entropy (half-empty rectangle, for instance) while the number of molecules increases.

So in fact entropy is not increasing, it achieves its maximum possible value just when you "open the door" for it, and it persists for very long time interval (transcendentally huge, yes).

Sorry for terrible English.

Answer 3

To be concrete lets picture a box with an ideal mono-atomic gas in equilibrium and containing a constant energy. Let's not consider the deviations from the mean values of the atom momenta, so all the atoms have the same mean value of the momentum (I know this isn't realistic, but it is a good approximation of the situation). There are a lot of distributions of the momenta and positions of the atoms in phase space corresponding to the same, macroscopic equilibrium state of the gas. And the bigger the volume, the more of these distributions are possible (the distributions for which the positions of the atoms are, say, all concentrated in one corner of the box, or the momenta are separated into one high momentum part and a low momentum part are not considered, and I already stated that the momenta of the atoms are to be considered equal).

So you don't have to worry that the atoms in the box suddenly show a non-equilibrium sign (like all the atoms residing in one corner: for this to take place you have to wait for a much very much longer time than the time mentioned in the quote).

There is a nice metaphor for ergodicity: Imagine a lonely man taking a random walk, from the entrance to the exit, in the park each day. You can draw his path for many days after each other. This will give (more or less) the same result if you draw the random paths of a lot of people taking a random walk in the park one one single day.

Answer 4

Ergodic hypothesis is not contradiction to notion of equilibrium. In fact, it is the pillar of equilibrium statistical physics. The measurable quantities(like pressure, temperature,..) which are evaluated at the equilibrium assume that ergodic hypothesis is valid at equilibrium. One starts with finding the partition function(Z) $$ Z = \sum_{\epsilon} e^{-\beta \epsilon} $$ and later on one finds the observable quantities (here we assumed discrete energy levels and we have taken Canonical Ensemble). Here we have summed over all the possible values of energy. In other words, ergodic hypothesis can also be stated as "at equilibrium, time average is equivalent to ensemble average" which we have used while finding the canonical ensemble partition function. We have summed over all the values of microstates as ergodic hypothesis states that " all accessible microstates are equiprobable at equilibrium ".