Mean-Field Convective Phase Separation under Thermal Gradients

Abstract

Nonequilibrium conditions fundamentally change how systems undergo phase separation. In systems with temperature gradients, attractive particles have been shown to form periodic patterns and steady convective currents, but a clear theoretical explanation for this behavior is still missing. Here, we present a dynamical mean-field model that describes the mechanism behind this convective phase separation. Using linear stability analysis, we show that the transition from a uniform state to a periodic pattern is driven by the emergence of a dominant unstable mode. Numerical simulations confirm the predicted phase diagram and demonstrate that these convective currents are a robust feature of the steady state, appearing regardless of the initial conditions. These results provide a direct approach for understanding how temperature gradients drive the formation of steady-state convective patterns.

Figures (7)

• Figure 1: Linear instability: periodic density modulation and convective currents. (a) Temperature profile in Eq. \ref{['eq_temperature_profile_example']} for $\beta_\mathrm{mean}=0.75$ and $\beta_\mathrm{amp} = 0.08$. Blue dots indicate $T_{\bm{j}} < T_c$. (b) Dispersion relation: largest eigenvalue $\sigma_{m_*} (k_y)$ of the linearized dynamics in Eq. \ref{['eq_linearized_dynamics']} near the stability limit for $\beta_\mathrm{mean} = 0.75$ and various $\beta_\mathrm{amp}$. (c) Most unstable eigenvector for $\beta_\mathrm{mean}=0.75$ and $\beta_\mathrm{amp} = 0.08$: density modulation (grayscale) and current pattern (colored lines, brightness indicates magnitude). (d) Phase diagram based on linear stability. White (blue) regions indicate stability (instability) of the uniform density state toward convective patterns. Colored crosses denote parameters for panel (b). The thick gray line and black dot mark the unstable region and critical point for macroscopic phase separation at homogeneous temperature. For all panels, $(L_x, L_y) = (40, 80)$ and $\bar{\rho} = 1 / 2$.
• Figure 2: Dynamics of convective phase separation. Time evolution of the density field $\rho_{\bm{j}}(t)$ (grayscale) starting from a uniform initial state (I) (top panel) and a segregated state (II) (bottom panel). Blue dots indicate $T_{\bm{j}} < T_c$. Simulations are performed at $\beta_{\mathrm{mean}} = 0.83$ and $\beta_{\mathrm{amp}} = 0.09$, where the uniform state is linearly unstable. Colored arrows represent the steady-state current pattern overlaid on the final density field. Parameters: $(L_x, L_y) = (40, 80)$ and $\bar{\rho} = 1/2$.
• Figure 3: Steady-state phase diagram. (a) Heatmap of the order parameter $\Delta \rho^\mathrm{ss}$ for (I) uniform and (II) segregated initial states. The black line denotes the phase boundary predicted by linear stability analysis [cf. Fig. \ref{['fig_linear']}(d)]. (b) Heatmap of the number of high-density clusters $N_{\mathrm{cl}}^\mathrm{ss}$ for each initial state; digits indicate the cluster count within each region. Results for the uniform initial state [(a1) and (b1)] are averaged over 10 independent noise realizations. Parameters: $(L_x, L_y) = (40, 80)$ and $\bar{\rho} = 1/2$.
• Figure 4: Comparison between the mean-field dynamics [Eqs. \ref{['eq_model']} and \ref{['eq_model_current']}] and the stochastic lattice gas in Ref. MS1. (a) Temperature profile $T_{\bm{j}}$ for $\beta_{\mathrm{mean}} = 1.1/T_c$ and $\beta_{\mathrm{amp}} = 0.2/T_c$, used for panels (b) and (c). (b) Steady-state density field and currents for the mean-field model. (c) Time-averaged occupation field and currents for the stochastic model MS1. (d--f) Corresponding results for $\beta_{\mathrm{mean}} = 1.2/T_c$ and $\beta_{\mathrm{amp}} = 0.2/T_c$. Blue dots indicate $T_{\bm{j}} < T_c$. Parameters: $(L_x, L_y) = (40, 80)$ for mean-field, $(L_x, L_y) = (164, 328)$ for lattice gas, $\bar{\rho} = 1/2$ in both cases.
• Figure S1: Growth rate of Fourier modes for the case of homogeneous temperature. (a), (b) Surface plot of the growth rate $r_{\bm{k}}$ as a function of $(k_x, k_y)$ at (a) $c = \bar{\rho} (1 - \bar{\rho}) / T = 0.2$ or (b) $c = 0.205$. Blue and red colors indicate negative and positive $r_{\bm{k}}$, respectively. (c) Growth rate $r_{k_x, k_{y*}}$, which is maximized for $k_y$, as a function of $k_x$. Different colors indicate different values of $c$. (d) Expanded view of panel (c).