We study the mobility of a particle coupled to a one dimensional interacting
fermionic system, a Luttinger liquid. We bosonize the Luttinger liquid and find
the effective interaction between the particle and the bosonic system. We show
that the dynamics of this system is completely equivalent to the acoustic
polaron problem where the interaction has purely electronic origin. This
problem has a zero mode excitation, or soliton, in the strong coupling limit
which corresponds to the formation of a polarization cloud due to the
fermion-fermion interaction around the particle. We obtain that, due to the
scattering of the residual bosonic modes, the soliton has a finite mobility and
diffusion coefficient at finite temperatures which depend on the
fermion-fermion interaction. We show that at low temperatures the mobility and
the diffusion coefficient are proportional to T−4 and T5 respectively
and at high temperatures the mobility vanishes as T−1 while the diffusion
increases as T.Comment: 9 pages, Revtex, UIUC preprin