Fluctuation dissipation theorems and irreversible thermodynamics.

We investigate the statistics of fluctuations in macroscopic systems described by thermodynamics. We begin by reviewing fluctuations in the context of linear irreversible thermodynamics and show that a more direct characterization of the fluctuations is possible, if velocity fluctuations are explicitly included in the second variation of the entropy, delta2S, about the equilibrium state. A similar procedure is then applied to what is the main goal of this paper: elucidating the nature of fluctuations in hyperbolic macroscopic systems, where signals have a finite transmission velocity. We find that, once again, velocity fluctuations have to be explicitly included, which takes us outside of extended irreversible thermodynamics as it is often defined. We find the explicit form of the fluctuation-dissipation theorem in this case, and determine the statistics of the stochastic variables in terms of the quantities appearing in the deterministic dynamics. The fluctuating theory is then reformulated in order to elucidate the relationship between the extended theory and linear irreversible thermodynamics. This has the effect of bringing out the general structure more clearly: the real, frequency-independent transport coefficients of linear irreversible thermodynamics are replaced by their complex, frequency-dependent counterparts in the extended theory.