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The phase behavior of liquids confined in a slit geometry does not reveal acrossover from a three to a two-dimensional behavior as the gap size decreases.Indeed, the prototypical two-dimensional hexatic phase only occurs in liquidsconfined to a monolayer. Here, we demonstrate that the dimensionality crossoveris apparent in the lateral size dependence of the relaxation dynamics ofconfined liquids, developing a Debye model for the density of vibrationalstates of confined systems and performing extensive numerical simulations. Inconfined systems, Mermin-Wagner fluctuations enhance the amplitude ofvibrational motion or Debye-Waller factor by a quantity scaling as the inversegap width and proportional to the logarithm of the aspect ratio, as a clearsignature of a two-dimensional behaviour. As the temperature or lateral systemsize increases, the crossover to a size-independent relaxation dynamics occurswhen structural relaxation takes place before the vibrational modes with thelongest wavelength develop.

Long-wavelength fluctuations and dimensionality crossover in confined liquids