The impact of fluctuations on particle systems described by Dean-Kawasaki-type equations

TL;DR

This work probes the role of conservative multiplicative fluctuations in Dean-Kawasaki-type density dynamics for Brownian particles by comparing four models across three descriptions: microscopic particle simulations, DKTE, and deterministic (noise-free) DKTE. It shows that fluctuations can roughen density profiles without changing mean-field behavior (Model I), accelerate front propagation in density-dependent diffusion (Model II), advance the onset of nonlocal-density-driven patterns (Model III), and reduce hysteresis in repulsively interacting systems (Model IV), with some cases even producing structures absent in deterministic descriptions. The results are encapsulated in quantitative findings such as front scaling $x_f(t)\propto t^{1/3}$ with enhanced prefactors, pattern formation thresholds $p_c$ shifted by noise, and structure-function peaks indicating periodic ordering. Overall, the study demonstrates a constructive and nontrivial influence of DK-type fluctuations, underscoring the necessity of stochastic modeling to capture collective particle dynamics accurately.

Abstract

We study the role of fluctuations in particle systems modeled by Dean-Kawasaki-type equations, which describe the evolution of particle densities in systems with Brownian motion. By comparing microscopic simulations, stochastic partial differential equations, and their deterministic counterparts, we analyze four models of increasing complexity. Our results identify macroscopic quantities that can be altered by the conserved multiplicative noise that typically appears in the Dean-Kawasaki-type description. We find that this noise enhances front propagation in systems with density-dependent diffusivity, accelerates the onset of pattern formation in particle systems with nonlocal interactions, and reduces hysteresis in systems interacting via repulsive forces. In some cases, it accelerates transitions or induces structures absent in deterministic models. These findings illustrate that (conservative) fluctuations can have constructive and nontrivial effects, emphasizing the importance of stochastic modeling in understanding collective particle dynamics.

Figures (5)

• Figure 1: Density distributions measured at long times, when statistically steady states are reached, for model I in the $[0,1]$ interval (only a slightly smaller region is shown). In all panels red circles are from the particle simulations and blue triangles are from the DKTE. For the particles, densities are obtained by binning the particle positions in histograms of 300 bins. Instantaneous configurations are displayed in panels (a) and (b), whereas their averages over long times are in panels (c) and (d). There, the error bars indicate the standard deviation of the density fluctuations at each point. (a) and (c) are for $N=1000$ and (b) and (d) for $N=2000$. The black line superimposed on all panels is the exact solution of the deterministic DKTE, corresponding to $N \to \infty$. We note the very good agreement between the variances of the particle and DKTE simulations and of the average densities between them and with the deterministic solution. Simulations of deterministic DTKE and DKTE were performed with $dt = 10^{-6}; dx =3.33\times 10^{-3}$, up to $5\times 10^{6}$ simulation steps, while for the particles $dt = 10^{-6}$ with $10^6$ steps and we binned the particles into a density histogram with the same $dx$ of the DKTE.
• Figure 2: a) $N=1000$. Red line: numerical solution of the DKTE at large times ($t=0.01$); black line: exact solution of the deterministic DKTE at the same time; green line: density obtained from particle simulations of Eq.(\ref{['model2b']}) (using $401$ bins for a simulation with system size $L=5$). b) the same for $N=2000$ particles. c) $N=1000$. Red line is the average over $300$ realizations, all for the same initial condition, of the solution of the DKTE at $t=0.1$; black line is again the analytic solution of the deterministic DKTE at the same time. d) The same as c) for $N=2000$. The numerical simulations for DKTE and deterministic (see Appendix) here are performed with $dt=10^{-8}$ and $dx=5/3000$; particle simulations with $dt=10^{-4}$.
• Figure 3: Front positions vs time for the deterministic equation (black line), the average over $300$ realizations of the DKTE (red line), and the average over $300$ realizations of the density obtained from position histograms of particle simulations (green). The front is located as the rightmost spatial point for which the density is equal or smaller than $0.01$. a) is for $N=1000$, b) is for $N=2000$. The insets show the relation between $\left< \rho^2 \right>$ and $\left< \rho \right>^2$ at time $t=0.1$ at the different spatial points.
• Figure 4: Height of the peak of the structure function, normalized with $N$ as a function of the parameter $p$ for the three frameworks: particle (red), DKTE (green) and deterministic DKTE (black). a) is for $N=10000$ and b) for $N=20000$. The vertical dotted line indicates the value of $p_c$ identified by linear stability analysis of the homogeneous solution in the deterministic case. Simulations of deterministic DTKE and DKTE were performed with $dt = 10^{-4}; dx =7.8\times 10^{-3}$, up to $\{1.5\times 10^{6}; 2.5\times 10^{6} \}$ simulation steps, respectively, while for the particles $dt = 10^{-1}$ with $10^8$ steps.
• Figure 5: Maximum height of the structure function normalized by the number of particles. a) Upper plot is for $N=10000$ and b) bottom one for $N=20000$. Dashed vertical gray line indicates the transition from homogeneous to periodic pattern as predicted by linear stability analysis of the homogeneous solution. Red line with circles is obtained from particle simulations, Green line with triangles is from the DKTE, and the deterministic DKTE gives the black lines. Simulations of deterministic DTKE and DKTE were performed with $dt = 10^{-3}; dx =10^{-2}$, up to $\{3\times 10^{5}; 5\times 10^{5} \}$ simulations steps, respectively, while for the particles $dt = 10^{-5}$ with $10^6$ steps.