Quantum Kinetic Theory

The formal appearance of the transport coefficients may be simple, but, in general, their actual calculation is not. The collision integral, even the linearised version, is a formidable operator, and already the proof that $L^{-1}$ exists in the Hilbert space orthogonal to the collision invariants is a non-trivial mathematical problem. It is therefore of some importance to know that there exists a variational method whereby explicit inversion can be avoided.

We need the result that the collision operator is non-negative. This is an immediate consequence of the fact that the - linearised - entropy production (7.3) is non-negative (see exercise 7):

One can check this explicitly for the linear operator (6.8), but one should be aware that this positivity property is an essential ingredient of any kinetic theory which ensures, in particular, that the (direct) transport coefficients are always positive.

Let now $|\varphi )$ be a solution of the transport equation

with $|j)$ a given dissipative current. And suppose that we wish to calculate the transport coefficient

Let us consider states $|\chi )$ such that

The variational principle now states that the solution of (11.2) gives a maximum

within the class of functions that satisfy (11.4). The validity of this principle is shown by considering

Hence, any trial function $|\chi )$, subject to the constraint (11.4), gives a value for the transport coefficient that is certainly less than would follow from the exact solution:

• This may be used in a systematic approximation scheme, sometimes called the Ritz method, wherein the trial function is expanded with respect to some convenient basis set, and the first few terms are retained. If the set is chosen with care, one term of ten suffices to get an accuracy of a few percent.