Using transmission electron microscopy, we investigate the thermally activated motion of domain walls (DWs) between two positions in Permalloy (Ni80Fe20) nanowires at room temperature. We show that this purely thermal motion is well described by an Arrhenius law, allowing for a description of the DW as a quasiparticle in a one-dimensional potential landscape. By injecting small currents, the potential is modified, allowing for the determination of the nonadiabatic spin torque: beta(t) = 0.010 +/- 0.004 for a transverse DW and beta(v) = 0.073 +/- 0.026 for a vortex DW. The larger value is attributed to the higher magnetization gradients present.