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The Liouville operator and the Poisson bracket

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  • 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.

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