# Probability conservation versus Liouville's theorem

This question arises to my mind while studying notes on Kinetic theory written by Prof. David Tong. There he derived the Liouville’s equation. The outline of his derivation goes as follows:

Our interest in this section will be in the evolution of a probability distribution,$f(\vec{r}_i, \vec{p}_i;t)$ over the $2N$ dimensional phase space. This function tells us the probability that the system will be found in the vicinity of the point $(\vec{r}_i, \vec{p}_i)$. As with all probabilities, the function is normalized as $$\int\prod_{i=1}^{N}dr_{i}dp_{i}f(\vec{r}_i, \vec{p}_i;t)=1$$

Furthermore, because probability is locally conserved, it must obey a continuity equation: any change of probability in one part of phase space must be compensated by a flow into neighbouring regions. The continuity equation of the probability distribution is then $$\frac{\partial f}{\partial t}+\frac{\partial}{\partial\vec{r}_{i}}.(\vec{r}_{i}f)+\frac{\partial}{\partial\vec{p}_{i}}.(\vec{p}_{i}f)=0\tag{1}$$

Then after few turn and twist expression (1) gives the Lioville's equation $$\frac{\partial f}{\partial t}+\left\{f,H\right\}_{P.B} =0\tag{2}$$

So far so good. Now what i know is Lioville's equation is fundamental for the analysis of phase space. And according to the notes this result is a consequence of conservation of probability (in its local form) plus the Hamiltonian evolution of points in phase space.

My question is: which one is more fundamental, the local conservation of probability or Liouville's equation? As there is some other way one can derive the Liouville's equation without saying anything on probability conservation.

• if we express the local conversation of probability as (1) and Liouville's equation as (2), aren't they logically equivalent (i.e., can't we automatically go from (2) to (1) the same as we went from (1) to (2))? Maybe you might want to add what the other way to derive Liouville's equation is? – Sanya Oct 20 '16 at 16:24
• @Sanya, Thanks for mentioning the point. Equation (1) is more general then i initially thought. It arises in theories describing flows of conserving quantities (say, particles of fluids and gases). For the present case it is the Hamiltonian flow. So the the result in fact follows in the logical order. – AMS Oct 20 '16 at 16:38