  We present the mathematical ingredients for an extension of the Third Law of
Thermodynamics (Nernst heat postulate) to nonequilibrium processes. The central
quantity is the excess heat which measures the quasistatic addition to the
steady dissipative power when a parameter in the dynamics is changed slowly. We
prove for a class of driven Markov jump processes that it vanishes at zero
environment temperature. Furthermore, the nonequilibrium heat capacity goes to
zero with temperature as well. Main ingredients in the proof are the
matrix-forest theorem for the relaxation behavior of the heat flux, and the
matrix-tree theorem giving the low-temperature asymptotics of the stationary
probability. The main new condition for the extended Third Law requires the
absence of major (low-temperature induced) delays in the relaxation to the
steady dissipative structure.
