  Chemical waves constitute a known class of dissipative structures emerging in
reaction-diffusion systems. They play a crucial role in biology, spreading
information rapidly to synchronize and coordinate biological events. We develop
a rigorous thermodynamic theory of reaction-diffusion systems to characterize
chemical waves. Our main result is the definition of the proper thermodynamic
potential of the local dynamics as a nonequilibrium free energy density and
establishing its balance equation. This enables us to identify the dynamics of
the free energy, of the dissipation, and of the work spent to sustain the wave
propagation. Two prototypical classes of chemical waves are examined. From a
thermodynamic perspective, the first is sustained by relaxation towards
equilibrium and the second by nonconservative forces generated by chemostats.
We analytically study step-like waves, called wavefronts, using the
Fisher-Kolmogorov equation as representative of the first class and oscillating
waves in the Brusselator model as representative of the second. Given the
fundamental role of chemical waves as message carriers in biosystems, our
thermodynamic theory constitutes an important step toward an understanding of
information transfers and processing in biology.
