Variational principles for relativistic transport coefficients

Variational expressions for the transport coefficients of a one-component, relativistic gas are derived from the linearized relativistic Boltzmann equation for both quantum and classical gases. These expressions depend on functions χ of the energy of the particles comprising the gas in such a way that: a) if χ differs from a solution of the linearized Boltzmann equation by ε, then the value of the variational expression calculated with this χ differs from the true value of the corresponding transport coefficient by ε2; and b) the value of the variational expression is always less than this true value. It is shown that values of the transport coefficients obtained by expanding χ in a particular set of orthogonal polynomials and keeping only the first nontrivial term in the expansion are equivalent to those obtained using the Grad method of moments. It follows therefore that values obtained using this later method represent lower bounds on the true values. We also show that one can obtain simple, closed-form expressions for the various transport coefficients corresponding to an arbitrary number of terms in an expansion of the trial function χ in the above-mentioned set of orthogonal polynomials. Finally we point out that all of our results can be carried over to the nonrelativistic case by taking the limit c → ∞.