# Well known? Microcanonical ensemble does NOT weight every phase space point equally!

1. Nov 8, 2011

### nonequilibrium

Hello,

I was wondering if it is a well known fact that the microcanonical ensemble (i.e. $W = \int \delta(\mathcal H(\vec x, \vec p) - E) \mathrm d \vec x \mathrm d \vec p$) 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.