On linearized hydrodynamic modes in statistical physics

We formulate the linearized generalized Boltzmann equation as an (asymmetric) eigenvalue problem. This problem has five eigenvalues which tend to zero when the uniformity parameter tends to zero: to second order in this parameter, they correspond to damped sound (two modes), diffusing shear flow (two modes), and diffusing entropy flow (one mode). The microscopic expressions deduced from these results for the transport coefficients agree with the correlation-function formulas. Moreover, the corresponding eigenfunctions are explicitly displayed to lowest order in the uniformity parameter: they are microscopic analogs, in terms ofone-particle distribution functions, of the well-known linearized hydrodynamic modes of macroscopic physics. All results are established to all orders in the interactions.