Fronts and fluctuations at a critical surface

TL;DR

This paper analyzes the behavior of phase fronts at a critical surface under inhomogeneous fields, distinguishing between bistable and bifurcational scenarios through shape, response, and fluctuation statistics.

Contribution

It provides a detailed comparison of two generic scenarios for phase fronts at critical surfaces, highlighting fluctuation statistics as a key distinguishing factor.

Findings

Fluctuation statistics differ: Gaussian for bifurcation, double-peak for bistability.

Static and dynamic properties are similar in both scenarios, making them hard to distinguish.

Reanalysis of Drosophila morphogenesis suggests a bistable underlying process.

Abstract

The properties of a front between two different phases in the presence of a smoothly inhomogeneous external field that takes its critical value at the crossing point is analyzed. Two generic scenarios are studied. In the first, the system admits a bistable solution and the external field governs the rate in which one phase invades the other. The second mechanism corresponds to a second order transition that, in the case of reactive systems, takes the form of a transcritical bifurcation at the crossing point. We solve for the front shape and its response to external white noise, showing that static properties and also some of the dynamics features cannot distinguish between the two scenarios. The only reliable indicator turns out to be the fluctuation statistics. These take a Gaussian form in the bifurcation case and a double-peak shape in a bistable system. The results of a recent analysis of the morphogenesis process in Drosophila embryos are reanalyzed and we show, in contrast with the interpretation suggested by Krotov et. al., that the plausible underlying dynamics is bistable and not bifurcational.