  Experimental work has shown that non-equilibrium concentration fluctuations
arise during free diffusion in fluids and theoretical analysis has been carried
on. The results show that, in usual three-dimensional fluids, the phenomenon is
extremely weak, in terms of amplitude of the fluctuations and of corrugation of
the diffusion wave fronts. In this paper, we show that the phenomena strongly
depends on the dimensionality of the system: by extending the theory to two
dimensional systems, we show that the root mean square amplitude of the
fluctuations and the wave front corrugation become much stronger. We also
present an evaluation of the Hausdorf dimension of the expected fluctuations.
Experimentally, two-dimensional liquid systems can be realised as freely
suspended liquid films; experiments and theoretical works on diffusion in such
systems showed that the dynamics is deeply affected by the viscous drag exerted
by the fluids (e.g. air) surrounding the film. We provide an evaluation of the
drag on the fluctuations. In particular, we study the case of a concentration
profile that is initially gaussian: it can be directly compared with the
results from a fluorescence recovery after photobleaching (FRAP) experiment. We
propose that this theory and the related experiments can be relevant for
describing the diffusion along the cellular membranes of living organisms.
