How does $\rho(\dot{q_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f )$ represent the no. of systems that would enter the volume in $\mathrm d t\;?$

Question

I've been following Reif's Fundamentals of Statistical and Thermal Physics; there I came before the derivation of Liouville's theorem:

There I couldn't understood few things.

I could conceive the change in the number of systems in $\mathrm dt$ is given by $$\frac{\partial \rho}{\partial t}\; \mathrm dt\; (\mathrm d q_1,\mathrm dq_2, \ldots,\mathrm d q_f; \mathrm d p_1, \mathrm dp_2,\ldots, \mathrm d p_f)$$ where

• $$\rho(q_1,q_2,\ldots, q_f; p_1,p_2,\ldots, p_f ; t)\;\mathrm d q_1,\mathrm dq_2, \ldots,\mathrm d q_f; \mathrm d p_1, \mathrm dp_2,\ldots, \mathrm d p_f = \textrm{no of systems in the ensemble at $t$ in the phase-space volume}\;(\mathrm d q_1,\mathrm dq_2, \ldots,\mathrm d q_f; \mathrm d p_1, \mathrm dp_2,\ldots, \mathrm d p_f)$$

But then, I couldn't understand why the number of systems 'entering this volume in time $dt$ through the face $q_1$= constant' is given by the quantity $\rho(\dot{q_1}\mathrm dt, \mathrm dq_2, \ldots,\mathrm dp_f )\;.$

My questions are:

$\bullet$ How does $\rho(q_1,q_2,\ldots, q_f; p_1,p_2,\ldots, p_f ; t)((\dot{q_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f ))$ represent the number of systems that would enter the volume in time-interval $\mathrm d t\;?$

$\bullet$ How does the evaluation of $\dot q_i$ at $q_1+\mathrm dq_1$ yield $\dot q_i +\dfrac{\partial \dot q_i}{\partial q_1}\mathrm dq_1 \;?$

Answer 1

In the figure the volume element is moving from right to left. So the shaded region on the left multiplied by the "coarse-grained" density at that region is the number of systems entering the volume element.

The width of the region on the left is equal to the change in $q_1$ in time $dt$ is equal to $\dot{q_1}dt$. So the area is given by $(\dot{q_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f )$. So the number of particles entering the volume element is given by \begin{equation} \rho\times (\dot{q_1}|_{q_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f ) \end{equation}

At the same instant the systems in the shaded region on the right are leaving the volume element. That number is given by $\rho\times(\dot{q_1}|_{q_1+dq_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f )$.

Suppose $\dot{q}=f(q)$ some function of $q$.

$$\dot{q}|_{q+dq}=f(q+dq)=f(q)+\frac{\partial f}{\partial q}dq=\dot{q}+\frac{\partial \dot{q}}{\partial q}dq\;.$$

Putting this in the expression for shaded area (right) and subtracting this from expression for shaded area (left) will give the change in particle number in the volume element.