Langevin dynamics of fluctuation-induced first-order phase transitions: Self-consistent Hartree approximation

The Langevin dynamics of a system with a scalar-order parameter exhibiting a fluctuation-induced first-order phase transition is solved within the self-consistent Hartree approximation. Competition between interactions on short and long length scales gives rise to spatial modulations in the order parameter, such as stripes in $2d$ and lamellae in $3d$. We show that when the time scale of observation is small compared with the time needed for the formation of modulated structures, the dynamics is dominated by a standard ferromagnetic contribution plus a correction term. However, once these structures are formed, the long-time dynamics is no longer purely ferromagnetic. After a quench from a disordered state to low temperatures, the system develops growing domains of stripes (lamellae). Due to the character of the transition, the paramagnetic phase is metastable at all finite temperatures, and the correlation length diverges only at $T=0$. Consequently, the temperature is a relevant variable: for $T>0$ the system ends up forming domains of stripes with a finite correlation length while for $T=0$ a scaling behavior in space and time, characteristic of smectic order, is obtained.

Figure 1

Path of integration for the inverse Laplace transform.