  Validity of local equilibrium has been questioned for non-equilibrium systems
which are characterized by delayed response. In particular, for systems with
non-zero thermodynamic inertia, the assumption of local equilibrium leads to
negative values of the entropy production, which is in contradiction with the
second law of thermodynamics. In this paper we address this question by
suggesting a variational formulation of irreversible evolution of a system with
non-zero thermodynamic inertia. We introduce the Lagrangian, which depends on
the properties of the normal and the so-called "mirror-image" systems. We show
that the standard evolution equations, in particular the
Maxwell-Cattaneo-Vernotte equation, can be derived from the variational
procedure without going beyond the assumption of local equilibrium. We also
argue, that the second law of thermodynamics should be understood as a
consequence of the variational procedure and the property of local equilibrium.
For systems with instantaneous response this leads to the standard requirement
of the local instantaneous entropy production being always positive. However,
if a system is characterized by delayed response, the formulation of the second
law of thermodynamics should be altered. In particular, the quantity, which is
always positive, is not the instantaneous entropy production, but the entropy
production averaged over the period of the heat wave.
