Hydrodynamic correlations in multiparticle collision dynamics fluids

The emergent fluctuating hydrodynamics of the multiparticle collision dynamics (MPC) approach, a particle-based mesoscale simulation technique for fluid dynamics, is analyzed theoretically and numerically. We focus on the stochastic rotation dynamics implementation of the MPC method. The fluid is characterized by its longitudinal and transverse velocity correlation functions in Fourier space and velocity autocorrelation functions in real space. Particular attention is paid to the role of sound, which leads to piecewise negative correlation functions. Moreover, finite system-size effects are addressed with an emphasis on the role of sound. Analytical expressions are provided for the transverse and longitudinal velocity correlations, which are derived from the linearized Landau-Lifshitz Navier-Stokes equation adopted for an isothermal MPC fluid. The comparison of the analytical results with simulations shows excellent agreement above a minimal length scale. The simulations indicate a breakdown in hydrodynamics on length scales smaller than this minimal length. This demonstrates that we have an excellent analytical description and understanding of the MPC method and its limitations in terms of time and length scales.

Figure 1

Magnitudes of the velocity autocorrelation functions of infinite and finite size systems for the collision time $h/\sqrt{m{a}^2/k_{B}T}=0.02$. (a) Black, straight line: transverse component of an infinitely large system. The positive longitudinal contribution is displayed as solid curve, red, whereas, the negative one is shown as dashed curve, light blue. The inset shows the same curves with a linear $y$ axis. (b) Black, straight line: Asymptotic correlation function for an infinite system. The other solid lines indicate the positive parts of the correlation functions of finite size systems and the dashed lines their negative parts, with green-blue: $L/a=100$ and red-light blue: 1000. The correlation function for $L/a=100$ deviates early from the asymptotic behavior of an infinite system.

Figure 2

From right to left: transverse velocity autocorrelation functions of a MPC fluid for $k=2\pi n/L$ with $n=1,2,3$ and $L/a=60$. The collision time steps are blue, dark gray: $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.5$; green, light gray: 0.1; and red: 0.01. The individual curves are hardly distinguishable. The corresponding theoretical prediction (26) is represented by a dashed line. Inset: universal dependence on $\nu k^{2}t$.

Figure 3

Time dependence of the integrated transverse velocity correlation function (39). The collision time is $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.01\mathrm{and}L/a=60$. From top to bottom, the dashed lines correspond to the $k$ values: $k=2\pi n/L,n=1,...,10$. The solid lines indicate the theoretical expression (27).

Figure 4

Dependence of $T\left(k\right)={\lim }_{t\to \infty }T(k,t)$ (27) on the wave number for the collision times $▴$: $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.01$; $⧫$: 0.1; $\blacksquare$: 0.5; and $•$: 1.0. The thick solid line indicates the dependence $1/k^2$, corresponding to the Oseen tensor. The horizontal lines are the theoretical predictions for the plateau values of $\nu T\left(k\right)$ [Eq. (A5)]. The inset shows the theoretical prediction for the characteristic length scale ${\lambda }_c$; solid line: Eq. (40) and squares: values extracted from the simulations.

Figure 5

Longitudinal velocity autocorrelation functions for the collision times red: $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.01$; green: 0.05; blue: 0.1; and light blue: 1.0; $k=2\pi /L$, and $L/a=60$. The thin solid lines are theoretical predictions according to Eq. (29). Solid lines: the order of the maxima at $t/\sqrt{m{a}^2/\left(k_{B}T\right)}\approx 60$ corresponds to bottom to top: $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.01$, 0.05, 0.1, and 1.0.

Figure 6

Spectra ${\widehat{C}}_v^L(\mathit{k},\omega )$ of the longitudinal velocity autocorrelation function (42) for the collision times red, lower: $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.01$; green, middle: 0.05; and blue, upper: 0.1, and $k=2\pi /\left(60a\right)$. The lines are obtained from the theoretical expression (20).

Figure 7

Symbols: magnitude of the velocity autocorrelation function (35) of a MPC fluid for the collision times (a) $h/\sqrt{m{a}^2/\left(k_{B}T\right)}=0.1$ and (b) 0.02. Filled symbols indicate positive correlation functions, and open symbols indicate negative correlation functions. Solid lines: the theoretical results are obtained from Eq. (33) with Eqs. (26) and (29) where solid lines indicate positive correlation functions, and dashed lines indicate negative ones.