In the Wikipedia article about the Ergodic hypothesis (https://en.wikipedia.org/wiki/Ergodic_hypothesis) this is what the hypothesis say:

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.

If the system is in equilibrium, which means that the system spends infinite amout of time in some region (equivalently $\rho(\vec r, \vec p,t)$ is non-zero constant at some region and zero everywhere else), according to the first part of the hypothesis (about it's proportionality to the volume) doesn't this imply that the volume in phase space is infinite, which obviously is wrong.

I consider the case of MCE,where the region represents the surface of the hypersphere (6N-1 Dimensional, for a system of N particles in the 3D case). In equilibrium, in accordance with what the hypothesis say about the equiprobable, that is true in this case. Each microstate in this region has probability $P=\frac 1\Omega$, where $\Omega $ is the multiplicity of the macrostate for our system in equilibrium.

But the first part, would imply infinite volume, which is not true.

So what is it that I am getting wrong ?


1 Answer 1


In order to define probability you need a normalized measure (of the phase space volume): you have to set the total volume to 1, for example. Then, IF the system is ergodic, a region with a volume 0.25, or 1/4 of the total volume, contains microstates such that, over a long period of time, the system will access one of them with probability 1/4.


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