  We analyse phase averaging in one-dimensional interacting mesoscopic systems
with several barriers and show that for incommensurate positions an independent
average over several phases can be induced by finite temperature. For three
strong barriers with conductances G_i and mutual distances larger than the
thermal length, we obtain G ~ sqrt{G_1 G_2 G_3} for the total conductance G.
For an interacting wire, this implies power laws in G(T) with novel exponents,
which we propose as an experimental fingerprint to distinguish temperature
induced phase averaging from dephasing.
