  By Girsanov's thoerem and using the existing log-Harnack inequality for
distribution independent SDEs, the log-Harnack inequality is derived for
path-distribution dependent stochastic Hamiltonian systems. As an application,
the exponential ergodicity in relative entropy is obtained by combining with
transportation cost inequality. In addition, the quantitative propagation of
chaos in the sense of Wasserstein distance, which together with the coupling by
change of measure implies the quantitative propagation of chaos in total
variation norm as well as relative entropy are obtained.
