  The Onsager linear relations between macroscopic flows and thermodynamics
forces are derived from the point of view of large deviation theory. For a
given set of macroscopic variables, we consider the short-time evolution of
near-equilibrium fluctuations, represented as the limit of finite-size
conditional expectations. The resulting asymptotic conditional expectation is
taken to represent the typical macrostate of the system and is used in place of
the usual time-averaged macrostate of traditional approaches. By expanding in
the short-time, near-equilibrium limit and equating the large deviation rate
function with the thermodynamic entropy, a linear relation is obtained between
the time rate of change of the macrostate and the conjugate initial macrostate.
A Green-Kubo formula for the Onsager matrix is derived and shown to be positive
semi-definite, while the Onsager reciprocity relations readily follow from time
reversal invariance. Although the initial tendency of a macroscopic variable is
to evolve towards equilibrium, we find that this evolution need not be
monotonic. The example of an ideal Knundsen gas is considered as an
illustration.
