  We examine the classical joint source--channel coding problem from the
viewpoint of statistical physics and demonstrate that in the random coding
regime, the posterior probability distribution of the source given the channel
output is dominated by source sequences, which exhibit a behavior that is
highly parallel to that of thermal equilibrium between two systems of particles
that exchange energy, where one system corresponds to the source and the other
corresponds to the channel. The thermodynamical entopies of the dual physical
problem are analogous to conditional and unconditional Shannon entropies of the
source, and so, their balance in thermal equilibrium yields a simple formula
for the mutual information between the source and the channel output, that is
induced by the typical code in an ensemble of joint source--channel codes under
certain conditions. We also demonstrate how our results can be used in
applications, like the wiretap channel, and how can it be extended to multiuser
scenarios, like that of the multiple access channel.
