  This work presents the thermodynamical analysis of the ballistic heat
equation from the viewpoint of two approaches: Classical Irreversible
Thermodynamics (CIT) and Extended Irreversible Thermodynamics (EIT). A formula
for calculation of the entropy within the framework of EIT for the ballistic
heat equation is derived in this work. Entropy is calculated for a sinusoidal
initial temperature perturbation by using both approaches. The results obtained
from CIT show that the entropy is a non-monotonic function and the entropy
production can be negative. The results obtained with EIT show that the entropy
is a monotonic function and the entropy production is nonnegative. An
approximative formula for the asymptotic behavior of the entropy for the
ballistic heat equation is obtained. A comparison with the ordinary
Fourier-based heat equation and hyperbolic heat equation is made. A crucial
difference in asymptotic behaviour of the entropy for ballistic and classical
heat conduction equation is shown. It is shown that mathematical time
reversibility of partial differential ballistic heat equation is not consistent
with its physical irreversibility. The processes described by the ballistic
heat equation are irreversible because of the entropy increase.

Statistical Mechanics