  In this note, we discuss the low and high temperature contribution of
Thermodynamic Bethe Ansatz (TBA) dressed excitations in the thermodynamics and
energy-magnetization relaxation within the Generalized Hydrodynamics approach
in the linear response regime. In particular, we show how the temperature
dependent dispersions of the excitations reproduce well known behavior of the
specific heat, magnetic susceptibility, spin and energy Drude weights. In this
context, we derive a further formulation of the Drude weights from the finite
wavevector relaxation. Furthermore, we contrast the TBA description of
thermodynamics and dynamics in terms of a multitude of string excitations to
that in terms of a single quasi-particle in low energy effective theories.
