  Irreversibility is a critical property of non-equilibrium transport
processes. An opinion has long been insisted that the entropy production rate
is a Lyapunov function for all kinds of processes, that is, the principle of
minimum entropy production. However, such principle is based on some strong
assumptions that are rarely valid in practice. Here, the common features of
parabolic-like transport processes are discussed. A theorem is then put forward
that the dot products of fluxes and corresponding forces serve as Lyapunov
function for parabolic-like transport processes. Such fluxes and forces are
defined by their actual constitutive relations (e.g., the Fourier's law, the
Fick's law, etc.). Then, some typical transport processes are analyzed.
Particularly for heat conduction, both the theoretical and numerical analyses
demonstrate that its Lyapunov function is the entransy dissipation rather that
the entropy production, when the Fourier's law is valid. The present work could
be helpful for further understanding on the irreversibility and the
mathematical interpretation of non-equilibrium processes.
