Pattern formation in a coupled driven diffusive system

TL;DR

This work introduces a Field-based Lattice Model (FLM) as a minimal, stochastic lattice-field framework bridging the driven Widom-Rowlinson lattice gas (DWRLG) and its continuum description. Through a gradient expansion, two coupled stochastic PDEs for density- and charge-fluctuation fields are derived, with drive entering via $\Delta v$ and nonlinear couplings; the linear coefficients depend on density and exhibit a sign change signaling a disordered-to-ordered transition. Simulations show three regimes: a low-density microemulsion with a nonzero $q^*$, an intermediate irregular-stripe phase with long-range order perpendicular to the drive, and a high-density regular-stripe phase; the continuum model reproduces these features and reveals additional patterns like stripes parallel to the drive and chaotic states. A key finding is that a nonzero difference in characteristic velocities $\Delta v$ is necessary for perpendicular stripe formation, highlighting how drive, interactions, and noise jointly generate rich nonequilibrium phenomenology in driven binary mixtures. This work suggests a unified framework for understanding stripe formation and microemulsions in driven systems and points to future refinements of field theories for nonequilibrium phase behavior.

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.

Figures (13)

• Figure 1: Summary of steady-states in the FLM. See text for details. Upper panels (a)-(b): Typical charge fluctuation field configurations in a $L = 192$ system, also observed in the other sizes studied. (a): $\delta = 1$. For $\rho < \rho_\ell$ (orange background), the system is well mixed and homogeneous, with the structure factors exhibiting a discontinuity at the origin and a peak at $(q_x,q_y)=(q^*,0)$. The intermediate regime (blue background), $\rho_\ell < \rho < \rho_u$, is marked by irregular stripes which suffer long-wavelength instabilities and display a range of widths. For $\rho>\rho_u$ (green background), the stripes are fully developed and long-wavelength instabilities are not sufficient to destroy long-range order in the $x$(drive)-direction. This regime is referred to as the regular stripe regime. (b): $\delta =0$. For $\rho< \rho_c^{(0)}$ (orange background) the system is well mixed and homogeneous, with the structure factors displaying the Ornstein-Zernike form. For $\rho> \rho_c^{(0)}$ (blue background) the system is heterogeneous, exhibiting coarsening dynamics that follows the Lifshitz-Slyozov law. For both $\delta=0$ and $1$, the system undergoes a glassy transition when $\rho>\rho_g$ (yellow background). Lower panels (c)-(d): order parameter plots [Eq. \ref{['eq:op']}] for $\delta = 0$ (c) and $\delta = 1$ (d); using $L=64$ and averaged over $100$ realizations. The blue Roman numerals (i-v) near $\rho=0.2, 0.4, 0.5, 0.7, 0.9$ correspond to the typical configurations shown in (a) and (b). Red stars are our crude estimates of the critical density; $\rho_c^{(0)} \simeq 0.39$ and $\rho_\ell \simeq 0.40$. Red squares are estimates of the glassy transition density, $\rho_g \simeq 0.90$. For $\delta =1$, the red triangle marks the transition to regular stripes at the density $\rho_u \simeq 0.65$. A movie related to this figure is available in movies.
• Figure 2: Charge fluctuation field configurations of the driven FLM for $\rho=0.60,\ L=256$, at times $5.40 \times 10^{5}$ (a), $2.26 \times 10^{6}$ (b), and $1.02 \times 10^{7}$ (c) MCS, after starting from a uniform initial state. Panel (a) shows quite regular, thin stripes. Breakup and merging take place in panel (b). Panel (c) shows a typical configuration at late times, with few stripes of widely differing widths.
• Figure 3: (a): Peak position of the charge fluctuation structure factor (CF SF) along the drive, $q^*$, versus the density $\rho$, for $\delta =1$ and $L=128$, averaged over $100$ realizations starting from a uniform initial condition. Color plots: typical field configurations of the CF (b) and the density fluctuation (DF) (c), for: (i) $\rho= 0.475$, (ii) $\rho = 0.625$, (iii) $\rho=0.7$ and (iv) $\rho = 0.8$. Observe that the DF has minima at the interfaces of the CF. The color map scale for the DF is shifted by $\rho$. Since the DF signal for $\rho = 0.475$ is very weak, we mark the interfaces (regions where $\phi_+ \sim-\rho$) in white.
• Figure 4: Data collapse for the order parameter in the different density regimes, when scaled by different powers of $L$. (a): $\delta = 0$. (b): $\delta = 1$. Dashed lines mark the transition densities. In each case, the different regimes, where data best collapse, are highlighted in red (green) for $\delta =0(1)$. Error bars are mainly smaller than the size of the symbols. Observe the absence of a third regime in the undriven $\delta=0$ case (middle-(a) panel); see text for details.
• Figure 5: $\delta = 1$: The charge fluctuation structure factor, as a function of $q_{x}$ for $q_{y}\,L/2\pi=0,...,10$ (using $L=128$), in the three regimes: microemulsion ($\rho=0.39$) (a), irregular stripes ($\rho=0.60$) (b), and regular stripes ($\rho=0.80$) (c). Note that the $S_{-}$ axis is logarithmic, so that, for the ordered case, there is an extremely sharp “ ridge” at $q_{y}=0$. For the irregular stripe regime in (b), this ridge is quite sharp also, but not as extreme.