Hydrodynamic fluctuations at the convective instability

TLDR

Thermal fluctuations in convective instability are studied, revealing that the system’s free energy lacks cubic terms and thus undergoes a mean‑field second‑order transition, though Brazovskii’s analysis predicts a first‑order transition in three dimensions, and connections to Graham and Pleiner’s work are examined. Near the instability, the Langevin equations for hydrodynamic fluctuations are equivalent to a crystallization model, leading to a discontinuous (though negligibly small) transition in the convective system, which effectively behaves two‑dimensionally for large lateral dimensions.

Abstract

The effects of thermal fluctuations on the convective instability are considered. It is shown that the Langevin equations for hydrodynamic fluctuations are equivalent, near the instability, to a model for the crystallization of a fluid in equilibrium. Unlike the usual models, however, the free energy of the present system does not possess terms cubic in the order parameter, and therefore the system undergoes a second-order transition in mean-field theory. The effects of fluctuations on such a model were recently discussed by Brazovskii, who found a first-order transition in three dimensions. A similar argument also leads to a discontinuous transition for the convective model, which behaves two dimensionally for sufficiently large lateral dimensions. The magnitude of the jump is unobservably small, however, because of the weakness of the thermal fluctuations being considered. The relation of the present analysis to the work of Graham and Pleiner is discussed.