Time dependence in Gibbs ensemble density
I am trying to understand why there is time dependence in a Gibbs ensemble density. I am loosely following <PERSON>'s Statistical Mechanics of Particles, but here is the argument as I understand it.
Let $\Gamma$ be the phase space of our system, let M be a macrostate of our system with $\mathscr{N}$ microstates corresponding to it. We define $d\mathscr{N}(g)$ to be the number of such microstates in some small neighborhood of $g \in \Gamma$ so that $\int d\mathscr{N} = \mathscr{N}$. We then define $\rho (g) d \Gamma = d\mathscr{N}/\mathscr{N} $ so that $\rho (g) \geq 0 $ everywhere and $\int \rho d \Gamma = 1$. This tells us that $\rho$ is a probability density function.
Now I am going to work out what I think is a generic and representative case, and it leads me to a conflict with what I am reading everywhere.
Let's assume I have a dilute, non-interacting gas consisting of N particles. I will consider a macrostate corresponding to a fixed energy E. The microstates corresponding to this are all {$ p_i $} such that
$$E = \sum_i \frac{p_i ^2}{2m_i}$$
is fixed at some fixed value E. 749
So if I am trying to imagine what my $\rho$ looks like in this case, I would imagine that it has some non-zero value for any point in phase space where the momenta satisfy the above equation and zero everywhere else. Crucially, this has no time dependence whatsoever, this is just a function of the position and momenta.
Now what every book does at this point is the following. They say that yes, $\rho$ only depends on p and q, but they themselves depend on time. To which I say that's only because you are sloppy with your notations.
$\Gamma$ is some space with a density function on it which we call $\rho$, it is only a function of the points in $\Gamma$. Period. Now, we also look at curves in $\Gamma$ i.e $\gamma : \mathbb{R} \to \Gamma$, these trajectories have to satisfy <PERSON>'s equations, etc.
Now, sometimes we are sloppy and instead of writing $\gamma(t)$ we write its image in $\Gamma$ as $p_i(t),q_i(t)$. Thats all fine, but how does putting a curve in your space satisfying <PERSON>'s equations suddenly make $\rho$ depend on time?
Where am I going wrong? 474