  We use the concept of typicality to study the real-time dynamics of spin and
energy currents in spin-1/2 models in one dimension and at nonzero
temperatures. These chains are the integrable XXZ chain and a nonintegrable
modification due to the presence of a staggered magnetic field oriented in z
direction. In the framework of linear response theory, we numerically calculate
autocorrelation functions by propagating a single pure state, drawn at random
as a typical representative of the full statistical ensemble. By comparing to
small-system data from exact diagonalization (ED) and existing short-time data
from time-dependent density matrix renormalization group (tDMRG), we show that
typicality is satisfied in finite systems over a wide range of temperature and
is fulfilled in both, integrable and nonintegrable systems. For the integrable
case, we calculate the long-time dynamics of the spin current and extract the
spin Drude weight for large systems outside the range of ED. We particularly
provide strong evidence that the high-temperature Drude weight vanishes at the
isotropic point. For the nonintegrable case, we obtain the full relaxation
curve of the energy current and determine the heat conductivity as a function
of magnetic field, exchange anisotropy, and temperature.
