Microcanonical Ensemble: Unequal Weighting of Points

[nonequilibrium]

Hello,

I was wondering if it is a well known fact that the microcanonical ensemble (i.e. [itex]W = \int \delta(\mathcal H(\vec x, \vec p) - E) \mathrm d \vec x \mathrm d \vec p[/itex]) does not weight every point equally, in the sense that in the integral (which was just quoted) some points on the energy hypersurface are weighted more heavily than others.

I'm just wondering cause I had been wrong myself for quite some time and I haven't found a reference to this fact in standard books (but maybe I haven't looked well enough!).
Concrete example: take a 2D phase space and a system where the energy "hypersurfaces" are ellipses (e.g. harmonic oscillator), then the W (as defined above) is not simply the circumference of the ellipse.

 

[LightHero]

Thanks!</code>Yes, it is well known that the microcanonical ensemble does not weight every point equally. This is because the phase space can be partitioned into regions of different energies, and the integrand in the definition of the microcanonical ensemble will be weighted differently depending on which region it is in. For example, in the case of a two-dimensional phase space and a system where the energy "hypersurfaces" are ellipses (such as the harmonic oscillator), the integral will be weighted more heavily towards the regions with higher energies, and thus the circumference of the ellipse will not be the same as the value for the microcanonical ensemble. This phenomenon is known as "Gibbs paradox" and has been studied extensively in statistical mechanics.