[Paper Review] Mean-Field Convective Phase Separation under Thermal Gradients

Meander Van den Brande, François Huveneers+1 more|arXiv (Cornell University)|Mar 6, 2026

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TL;DR

A deterministic mean-field model explains convective phase separation under temperature gradients, identifying the dominant unstable mode via linear stability and validating with simulations.

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.

Motivation & Objective

• Motivate understanding of how temperature gradients alter phase separation in attractive particle systems.
• Develop a deterministic mean-field framework that captures convection-driven pattern formation under inhomogeneous temperatures.
• Derive and analyze a linear stability condition to predict the onset of periodic density modulations.
• Validate the mean-field predictions with nonlinear simulations and compare to stochastic lattice-gas behavior.

Proposed method

• Formulate a density field on a 2D lattice with spatially varying temperature and a conserved mean density.
• Define a local current with linear (diffusive) and nonlinear (attractive) terms through a hyperbolic tangent interaction.
• Linearize the dynamics around the uniform density to obtain a linear evolution operator R.
• Diagonalize a k_y-dependent block matrix S(k_y) to extract the largest eigenvalue σ_m*(k_y) and the most unstable mode.
• Construct a phase diagram from the linear stability analysis and identify the relation to the macroscopic critical temperature T_c.
• Perform nonlinear time integration of the mean-field equations to study pattern formation and steady states.

Figure 1: Linear instability: periodic density modulation and convective currents. (a) Temperature profile in Eq. ( 8 ) 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 th

Experimental results

Research questions

• RQ1 What mechanisms drive convective patterns in an attractive lattice gas under a temperature gradient?
• RQ2 How does spatially varying temperature alter the linear stability and select the most unstable mode of density fluctuations?
• RQ3 Do mean-field predictions of convection-driven phase separation persist in nonlinear simulations and resemble stochastic lattice-gas behavior?

Key findings

• The uniform density state becomes unstable when the dominant eigenvalue Re(λ_n*) crosses zero, predicting a transition to a convective, periodic pattern.
• The most unstable mode has a nonzero wavenumber along the y-axis, indicating spontaneous periodic density modulation.
• Convective currents form robust circulating patterns in the steady state, localized primarily in low-temperature regions (T_j < T_c).
• Phase diagrams show convection can occur only when local temperatures drop below T_c in some region, with a boundary aligning to a computed stability line.
• Nonlinear simulations corroborate the linear prediction: uniform and segregated initial conditions both evolve toward periodic convective states within the predicted region, though final pattern selection can depend on initial conditions.
• Comparison with stochastic Kawasaki-type dynamics shows qualitative agreement in density modulations and current patterns, validating the mean-field approach.

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