We present and discuss the classical statistical mechanics of the dynamic properties associated with a discretized sine-Gordon and 4 system by using the molecular-dynamics technique. To calibrate the interpretation, various approximate expressions for the dynamic form factors of interest are derived on the basis of an interacting phonon gas and an ideal kink gas. On this basis, kink, phonon, and second-sound excitation branches are identified. The occurrence of the kink excitation branch turns out to be restricted to small wave numbers and frequencies and to low temperatures. Evidence for breather features, distinct from anharmonic contributions, is also given. © 1981 The American Physical Society.