  Thermodynamical arguments are known to be useful in the construction of
physically motivated Lyapunov functionals for nonlinear stability analysis of
spatially homogeneous equilibrium steady states in thermodynamically isolated
systems. Unfortunately, the limitation to thermodynamically isolated systems is
essential, and standard arguments are not applicable even for some very simple
thermodynamically open systems.
  On the other hand, the nonlinear stability of thermodynamically open systems
is usually investigated using the so-called energy method. Unfortunately, the
designation "energy method" is clearly a misnomer. The mathematical quantity
that is traditionally referred to as the "energy" is by no means linked to the
energy in the physical sense of the word. Consequently, it would seem that
genuine thermodynamical concepts are of no use in the nonlinear stability
analysis of thermodynamically open systems.
  We show that this is not true. In particular, we propose a construction that
in the case of simple heat conduction problem leads to a physically
well-motivated Lyapunov functional, which effectively replaces the artificial
Lyapunov functional used in the standard energy method. The proposed
construction seems to be general enough to be applied in complex
thermomechanical settings.
