Recently when I deal with 1D electron system, it occurred to my mind that since these electrons are not able to bypass each other during the scattering processes, we can actually label them as the 1st, 2nd,..., Nth electron. As a result, it seems that these electrons now become distinguishable.
so my questions is: does this kind of distinguishability have any deep physical consequences? For instance, for 3D identical particles the wave function has to be either symmetric or anti-symmetric, whereas in 2D case we have the interesting anyons that obey a different statistics. Then what about the 1D case? Further more, what kind of distribution functions should we use (i.e. Fermi-Dirac or Bose-Einstein)? I do remember that in undergraduate condensed matter modules people deal with 1D electron gas using Fermi-Dirac distribution, but now it seems not so natural to me.