Hydrodynamics of a hard-core active lattice gas

TL;DR

This work addresses motility-induced phase separation (MIPS) in a hard-core active lattice gas by constructing a fluctuating hydrodynamics derived from a quasi-1D two-lane lattice model. The authors employ a Martin-Siggia-Rose path-integral approach to obtain coupled stochastic equations for density ρ and polarization m, derive linear-spinodal stability bounds in terms of Pe and ρ_0, and compute two-point correlations that exhibit exponential decay with analytically determined lengths. The theory quantitatively agrees with Monte Carlo simulations and extends to higher dimensions, while revealing a breakdown of the hydrodynamic description in strictly one-dimensional single-file geometries due to micro-phase separation. Overall, the paper provides a bottom-up, minimal macroscopic description of MIPS in active matter that connects microscopic dynamics to collective phase behavior and fluctuations.

Abstract

We present a fluctuating hydrodynamic description of an active lattice gas model with excluded volume interactions that exhibits motility-induced phase separation under appropriate conditions. For quasi-one dimension and higher, stability analysis of the noiseless hydrodynamics gives quantitative bounds on the phase boundary of the motility-induced phase separation in terms of spinodal and binodal. Inclusion of the multiplicative noise in the fluctuating hydrodynamics describes the exponentially decaying two-point correlations in the stationary-state homogeneous phase. Our hydrodynamic description and theoretical predictions based on it are in excellent agreement with our Monte Carlo simulations and pseudospectral iteration of the hydrodynamics equations. Our construction of hydrodynamics for this model is not suitable in strictly one-dimension with single-file constraints, and we argue that this breakdown is associated with micro-phase separation.

Figures (15)

• Figure 1: Quasi-1D model: A schematic of the model defined in items (\ref{['diff_drift']})--(\ref{['lane_cross']}) on a two-lane ladder lattice with periodic boundary condition.
• Figure 2: Test of hydrodynamics: Time evolution of the hydrodynamic density $\rho(x,t)$ and polarization $m(x,t)$, starting from a Gaussian profile $\rho(x,0)$ centered around the middle of the system, with $(+)$ particles placed at the left half and $(-)$ particles at the right half. The solid lines represent noiseless hydrodynamic evolution \ref{['fluctuate_hydro_quasi_1d']}, while the points represent Monte Carlo simulations of the microscopic dynamics in (\ref{['diff_drift']})--(\ref{['lane_cross']}). The dashed lines indicate the steady-state profile from hydrodynamics. The plots are for bulk density $\rho_0=0.66$ and microscopic parameter values $\ell_\mathrm{d}=1024$, $\tau_\mathrm{p}= 2.5 \ell_\mathrm{d}^2$, $\ell_\mathrm{p}=10 \ell_\mathrm{d}$, and $\tau_\times=0.1$, and system size $L=4\ell_\mathrm{d}$, which corresponds to $\mathrm{Pe}=10$.
• Figure 3: Phase diagram: Spinodal and binodal, given by \ref{['rho_lh']} and \ref{['binodal-condition']} respectively, plotted with the solid and dashed lines. Along the dash-dotted line, the prefactor of the exponential in \ref{['eq:correlation']} diverges. The points represent the "liquid" and "gas" densities obtained from Monte Carlo simulations and numerical iteration of hydrodynamics. The error bars indicate the uncertainty due to the width of the density distribution around the peaks shown in the insets. The bimodal density distribution inside the spinodal region is an indication of MIPS, whereas the unimodal density distribution outside the binodal region is an indication of the homogeneous phase. In the inset, the points are from Monte Carlo simulations and the solid lines are from an analytical calculation in Appendix \ref{['numerical_evidence_loc_eq']}. We use $L=4096$ in Monte Carlo simulations and hydrodynamics equation solved numerically using Fourier spectral method with $1024$ modes.
• Figure 4: Correlations: Solid lines plot the analytical results of the correlation functions whereas the points are from numerical simulations. The insets show the asymptotic decay of the correlation functions (with a constant shift in order to show up in the log-linear plot), with the dashed guiding lines plot \ref{['eq:correlation']} with a shift for better visibility. The parameters used in the simulations are $\ell_\mathrm{d}=10.24$, $\tau_\mathrm{p}=2\ell_\mathrm{d}^2$, $\ell_\mathrm{p}=2\ell_\mathrm{d}$, $\tau_\times = 0.1$, $L=100\ell_\mathrm{d}$ and $\rho_0=0.25$, and we average over $10^5$ statistically independent time steps to generate a clean curve for comparing with hydrodynamic predictions.
• Figure 5: MIPS: Monte Carlo simulation for the two-dimensional models on a $512\times 512$ square lattice with $\rho_0=0.62$. Left figure corresponds to the two-species ($\rightarrow$ and $\leftarrow$) generalization and right to the four-species ($\rightarrow$, $\uparrow$, $\leftarrow$ and $\downarrow$) generalization. The colors indicate species (magenta$\equiv\rightarrow$, cyan$\equiv\uparrow$, green$\equiv\leftarrow$, red$\equiv\downarrow$). For the two-species model ${\ell_\mathrm{d}=128}$ and for the four-species model ${\ell_\mathrm{d}=32}$, with $\mathrm{Pe}=12$ and $\tau_\mathrm{p}=\ell_\mathrm{d}^2$ in both cases. The holes in the high-density phase were seen earlier in active systems 2018_Tjhung_Cluster2023_Nejad_Spontaneous. For the full time evolution see Appendix \ref{['supplement_vid']} and S_M.