# The Liouville operator and the Poisson bracket

From the last lecture, we saw that Liouville's equation could be cast in the form

$\frac {\partial f}{\partial t} + \nabla _x \cdot \dot {x} f = 0$

The Liouville equation is the foundation on which statistical mechanics rests. It will now be cast in a form that will be suggestive of a more general structure that has a definite quantum analog (to be revisited when we treat the quantum Liouville equation).

Define an operator

$iL = \dot {x} \cdot \nabla _x$

known as the Liouville operator ( $$i = \sqrt {-1}$$ - the i is there as a matter of convention and has the effect of making $$L$$ a Hermitian operator). Then Liouville's equation can be written

$\frac {\partial f}{\partial t} + iLf = 0$

The Liouville operator also be expressed as

$iL = \sum _{i=1}^N \left [ \frac {\partial H}{\partial p_i} \cdot \frac {\partial}{\partial r_i} - \frac {\partial H}{\partial r_i} \cdot \frac {\partial}{\partial p_i} \right ] \equiv \left \{ \cdots , H \right \}$

where $$\{ A, B \}$$ is known as the Poisson bracket between $$A(x)$$ and $$B (x)$$:

$\left \{ A, B \right \} = \sum _{i=1}^N \left [ \frac {\partial A}{\partial r_i} \cdot \frac {\partial B}{\partial p_i} - \frac {\partial A}{\partial p_i} \cdot \frac {\partial B}{\partial r_i} \right ]$

Thus, the Liouville equation can be written as

$\frac {\partial f}{\partial t} + \left \{ f, H \right \} = 0$

The Liouville equation is a partial differential equation for the phase space probability distribution function. Thus, it specifies a general class of functions $$f (x, t)$$ that satisfy it. In order to obtain a specific solution requires more input information, such as an initial condition on f, a boundary condition on f, and other control variables that characterize the ensemble.