  Counterion-mediated attractions between like-charged fluid membranes are
long-ranged and non-pairwise additive at high temperatures. At zero
temperature, however, they are pairwise additive and decay exponentially with
the membrane separation. Here we show that the nature of these attractions is
determined by the dominant modes of fluctuations in the density of counterions.
While the non-pairwise additive interactions arise from long-wavelength
fluctuations and vanish at zero temperature, the short-ranged pairwise additive
interactions arise from short-wavelength fluctuations and are stronger at low
temperatures.
