# Pattern formation in a coupled driven diffusive system

## TL;DR

This paper introduces a hybrid lattice model to study pattern formation in driven binary mixtures, revealing new phases and behaviors, and providing a continuum description that captures complex phenomena like irregular and perpendicular stripes.

## Contribution

The study develops a Field-based Lattice Model combining lattice gas and field theory, uncovering new intermediate phases and deriving coupled PDEs to describe pattern formation in driven systems.

## Key findings

- Identification of an intermediate 'irregular stripes' phase with long-range order
- Derivation of coupled PDEs capturing microemulsion and stripe phases
- Discovery of novel behaviors like parallel stripes and chaos

## Abstract

We investigate pattern formation in a driven mixture of two repulsive particles by introducing a Field-based Lattice Model (FLM), a hybrid model that combines aspects of the driven Widom-Rowlison lattice gas (DWRLG) and its statistical field theory. We find that the FLM effectively captures the bulk behavior of the DWRLG in both low- and high-density phases, suggesting that phase transitions in these models may share a universality class. Under the effect of the drive, the FLM additionally reveals an intermediate regime, not reported in the previous DWRLG studies, characterized by "irregular stripes" with widely fluctuating widths, contrasting with the "regular", well-ordered stripes found at higher densities. In this intermediate phase, the system exhibits long-range order, predominantly perpendicular to the drive direction. To construct a continuum description, we derive two coupled partial differential equations via a gradient expansion of the FLM mean mass-transfer equations, supplemented with additive noise. Designing a numerical solver using the pseudospectral method with dealiasing and stochastic time differencing, we reproduce the low-density microemulsion phase (characterized by a non-zero characteristic wavenumber q*) and perpendicular stripes at high density. We identify the non-zero difference in the characteristic velocities of the fields as a necessary condition for perpendicular stripe formation in the high-density phase. The continuum model also uncovers novel behaviors not previously observed in the FLM, such as stripes aligned parallel to the drive and chaotic patterns. This work highlights how the interplay of external drive, particle interactions, and noise can lead to a rich phenomenology in strongly driven binary mixtures.