  We introduce a variational structure for the Fourier-Cattaneo (FC) system
which is a second-order hyperbolic system. This variational structure is
inspired by the large-deviation rate functional for the Kac process which is
closely linked to the FC system. Using this variational formulation we
introduce appropriate solution concepts for the FC equation and prove an a
priori estimate which connects this variational structure to an appropriate
Lyapunov function and Fisher information, the so-called FIR inequality.
Finally, we use this formulation and estimate to study the diffusive and
hyperbolic limits for the FC system.
