# Why do we only use energy in the micro canonical ensemble?

The microcanonical ensemble is given by $$\rho = \frac{\delta(H-E)}{\int dxdp\;\delta(H-E)}$$. This corresponds to a uniform distribution on the energy shell. The interpretation of this is that we don't know the microstate of the system, we only know the total energy so we average over all possibilities.

However we also know other conserved quantities of the system. For example the momentum, the angular momentum or the center of gravity.

For example if I have a (perfectly isolating) balloon filled with an ideal gas that is floating in the air, I can clearly say that the gas inside has no total momentum, no total angular momentum and I also know its center of gravity. However when I use the micro canonial ensemble to describe it I only fix the energy to a specific value. This means I also allow for states where all particles move in the same direction (which is clearly not the case).

Shouldn't we change the micro canonical ensemble to something like $$\rho = \frac{\delta(H-E)\delta(\sum_i p_i - P)\delta(\sum_i x_i \times p_i - L)\delta(\sum_i x_i - R)}{\int dxdp\; \delta(H-E)\delta(\sum_i p_i - P)\delta(\sum_i x_i \times p_i - L)\delta(\sum_i x_i - R)}?$$

• You are confusing internal energy of the system given by its components and macroscopic energy of the whole system. The momentum, angular momentum and center of gravity are dependent on an external referential which is outside of the system. The micro canonical ensemble defines the Hamiltonian in the reference of the system in which P, L and R can be set at zero. Else they would be constants which would make no difference as we are only interested in the fact that the quantity is conserved. – G.Clavier May 1 at 16:32

• If I choose a reference frame in which the total momentum and angular momentum vanish I should at least have a $\delta(\sum_i p_i)\delta(\sum_i x_i \times p_i)$. Otherwise I also consider microstates with all particles moving in one direction (which I don't want to consider, do I?). PS: The center of gravity is an integral of motion (In the sense that $R(t)-Pt = const$). I know its trivial but formally it is an independent conserved quantity. – toaster May 1 at 21:11
• Ok I understand your point. So for example if I try to describe the gas in my room, then it is clear. But if I try to describe gas for example in a balloon then I should formally add $\delta(\sum_i p_i)$ (do you agree with that?). However I guess it will not change the behavior of the macroscopic state significantly, so I can probably omit it. – toaster May 2 at 15:12