# Concept of Volume of a System in Statistical Mechanics

I was reading my professor's notes on the microcanonical ensemble and I've been having some trouble understanding some of the concepts introduced. First of all, I'll start with some necessary definitions and notations I use. I know that the following expression $$\int dq^{3N}dp^{3N}\delta(\mathcal{H}(p,q)-E) = \Gamma(E)$$ describes a hypersurface in the $$6N$$-dimensional space, which I'll call energy surface. Then, I define the volume occupied by the microcanonical ensemble in the phase space $$\Gamma$$ as: $$\Gamma_{\Delta}(E) = \int_{E < \mathcal{H}(p,q) < E+\Delta } dq^{3N}dp^{3N}\tag{1}$$ Finally, I define entropy as the following function: $$S(E,V) = k_B\log\Gamma_{\Delta}(E)$$ Here come my doubts: it is written in my notes that if we combine two sufficiently large systems (whose reciprocal interactions can be ignored) with entropies $$S_1=k_B\log\Gamma^{1}_{\Delta}(E_1), S_2=k_B\log\Gamma^{2}_{\Delta}(E_2)$$, then the resulting system will have energy $$E \text{ s.t. }$$ $$E < E_1+E_2< E + 2\Delta \tag{2}$$ and for every possible choice of the couple $$E_1,E_2$$, the total volume will be given by $$\Gamma^{tot}_{\Delta} =\Gamma^{2}_{\Delta}(E_2) \cdot \Gamma^{1}_{\Delta}(E_1) \tag{3}$$ Now, I'd like it if someone could explain to me how to obtain the last inequality in $$(1)$$ and its physical meaning. More importantly, I'd like to understand why volumes in the phase space wouldn't simply add up ($$\Gamma^{2}_{\Delta}(E_2) + \Gamma^{1}_{\Delta}(E_1)$$) but needs to be multiplied. It honestly does not make any sense, especially if you see it from a purely visual standpoint. Maybe there's some non trivial physical meaning behind $$(3)$$ that I'm missing, or maybe I'm just not grasping the whole concept of volume of an ensemble as defined in (1). Any help is much appreciated.

## 1 Answer

OP asked: More importantly, I'd like to understand why volumes in the phase space wouldn't simply add up (Γ2Δ(𝐸2)+Γ1Δ(𝐸1) ) but needs to be multiplied

The volume of phase space can be seen as the number of microstates a system can occupy (given some constraints, like energy in your case).

Say I have two systems; one can take 2 microstates and the other 3. Then, a combination of these two systems will allow for $$2 \times 3$$ microstates and not $$2 + 3$$.