Fluctuation-Dominated Phase Ordering Driven by Stochastically Evolving Surfaces

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We study a new kind of phase ordering phenomenon in coarse-grained depth (CD) models of the hill-valley profile of fluctuating surfaces with zero overall tilt, and for hard-core particles sliding on such surfaces under gravity. For Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) surfaces, our analytic and numerical results show that CD models exhibit coarsening to an ordered steady state in which the order parameter has a broad distribution even in the thermodynamic limit. Moreover, the distribution of particle cluster sizes decays as a power-law (with an exponent θ), and the scaled 2-point spatial correlation function has a cusp (with an exponent α= 1/2) at small values of the argument. The latter feature indicates a deviation from the Porod law. For linear CD models with dynamical exponent z, we show that α= (z - 1)/2 for z < 3, while α= 1 for z > 3, and there are logarithmic corrections for z = 3. This implies α= 1/2 for the EW surface and 1 for the Golubovic-Bruinsma-Das Sarma-Tamborenea (GBDT) surface. Within the independent interval approximation we show that α+ θ= 2. The scaled density-density correlation function of the sliding particle model shows a cusp with exponent α≃ 0.5, and 0.25 for the EW and KPZ surfaces. The particles on the GBDT surface show conventional coarsening (Porod) behavior with α≃ 1.