  In the theory of extended irreversible thermodynamics (EIT), the
flux-dependent entropy function plays a key role and has a fundamental
distinction from the usual flux-independent entropy function adopted by
classical irreversible thermodynamics (CIT). However, its existence, as a
prerequisite for EIT, and its statistical origin have never been justified. In
this work, by studying the macroscopic limit of an \epsilon-dependent Langevin
dynamics, which admits a large deviations (LD) principle, we show that the
stationary LD rate functions of probability density p_{\epsilon}(x; t) and
joint probability density p_{\epsilon}(x; \dot{x}; t) actually turn out to be
the desired fluxindependent entropy function in CIT and flux-dependent entropy
function in EIT respectively. The difference of the two entropy functions is
determined by the time resolution for Brownian motions times a Lagrangian, the
latter arises from the LD Hamilton-Jacobi equation and can be used for
constructing conserved Lagrangian/Hamiltonian dynamics.
