Convection Patterns in Nonequilibrium Kawasaki Dynamics at Low Temperature

TL;DR

The study shows that a low-temperature Ising-like lattice gas driven by a macroscopic temperature gradient forms robust convection-driven stripe patterns in nonequilibrium steady states, in stark contrast to equilibrium predictions. Through large-scale simulations and a macroscopic framework grounded in local equilibrium and diffusive transport, the authors demonstrate stripe formation, current balance, and scaling laws (e.g., stripe count $\mathcal{N} \sim L_x^{1/2}$ in subcritical regimes) that cannot be explained by free-energy minimization alone. The work highlights how weak nonequilibrium driving and conserved dynamics can reorganize phase-separated states into regular, stripe-like structures with currents confined to convection cells, offering insights into the organization of NESS and prompting questions about entropy production and macroscopic fluctuation theories in driven systems.

Abstract

We study a conservative stochastic lattice dynamics (Kawasaki dynamics) in contact everywhere in the bulk with a heat bath. Particles interact via an Ising Hamiltonian and phase separation occurs at low temperature. We drive the system out of equilibrium by imposing a temperature field that varies spatially on macroscopic scales while preserving local equilibrium. Under these conditions, the usual low-temperature long-range order is replaced by robust convection patterns, featuring regularly spaced stripe structures for suitable geometries. These nonequilibrium states differ markedly from those obtained in an equilibrium dynamics with the same local temperature profile. We develop a macroscopic description that captures these behaviors and provides a unified framework for understanding the observed patterns.

Figures (13)

• Figure 1: Time-averaged densities (black and white) and particle currents (colored lines). Time averaging is performed over the last $25\%$ of the total simulation time $t = 5 \times 10^{7}$, for a single realization. Left panel: $x$-dependent temperature profile as in Eq. \ref{['eq: x temperature profile']} with $T_{\mathrm{mean}} = 0.4$, $T_{\mathrm{amp}}=0.2$, $L_y=2L_x = 400$. Right panel: Mexican-hat temperature profile as in Eqs. (\ref{['eq: radial temperature profile']}) and (\ref{['eq: first radial profile']}) with $T_{\mathrm{mean}} = 0.4$, $T_{\mathrm{amp}}=0.2$, $L_x=L_y=200$.
• Figure 2: Equilibrium (left) and nonequilibrium (right) steady states. Time-averaged particle density at filling factor $\overline{\rho}=0.8$, obtained over the final $25\%$ of a simulation of duration $t = 5 \times 10^{7}$ for a single realization. The temperature varies along $x$ as in Eq. \ref{['eq: x temperature profile']}, with $T_{\mathrm{mean}}=0.4$ and $T_{\mathrm{amp}}=0.2$. The system size is $L_y = 2L_x = 400$. Equilibrium and nonequilibrium transition rates are given in Eqs. \ref{['eq: equilibrium rates']} and \ref{['eq: nonequilibrium rates']}, respectively.
• Figure 3: Time evolution of the density field. Snapshots of the particle density at $0.1\%,\,0.4\%,\,1.6\%,\,6.3\%,\,25\%,$ and $100\%$ of the total simulation time $t=5\times10^{7}$, for a single realization at filling factor $\overline{\rho}=1/2$. Upper row: $x$-dependent temperature profile as in Eq. \ref{['eq: x temperature profile']} with $T_{\mathrm{mean}}=0.4$, $T_{\mathrm{amp}}=0.2$, and system size $L_y=2L_x=400$. Lower row: dip temperature profile as in Eqs. \ref{['eq: radial temperature profile']} and \ref{['eq: second radial profile']} with $T_{\mathrm{mean}}=0.4$, $T_{\mathrm{amp}}=0.2$, and system size $L_x=L_y=200$. The system is initialized from an infinite-temperature (white-noise) configuration.
• Figure 4: Time and $y$-averaged density within high density lines as a function of $x$. Temperature as in Eq. \ref{['eq: x temperature profile']} with $T_{\mathrm{mean}}=0.4$, $T_{\mathrm{amp}}=0.2$, system size $L_y=2L_x=400$, $\overline\rho=1/2$. Blue line: time average taken over the final $75\%$ of a simulation of duration $t=5\times10^{7}$ and spatial average in the $y$ direction taken over a fraction of the sites with $y$-coordinate corresponding to maximal density at $x=L_x/2$ (see main text). Single realization, starting from an initial configuration with $13$ equally spaced stripes. Orange line: equilibrium spontaneous density at the local temperature.
• Figure 5: Number of high-density stripes $\mathcal{N}$ as a function of the system size $L_x$ for fixed aspect ratio $L_y/L_x$ and filling factor $\overline\rho = 1/2$. The temperature varies along $x$ as in Eq. \ref{['eq: x temperature profile']} for $T_{\mathrm{mean}}=0.42$, and $T_{\mathrm{amp}}=0.12$ (orange) or $T_{\mathrm{amp}}=0.087$ (green). Average over 40 samples and over the last $25\%$ of the total time $t=1.33 \times 10^7$. See main text.

Theorems & Definitions (1)

• proof