Correlations in nonequilibrium diffusive systems

We study the behavior of stationary nonequilibrium two-body correlation functions for diffusive systems with equilibrium reference states (DSe). We describe a DSe at the mesoscopic level by $M$ locally conserved continuum fields that evolve through coupled Langevin equations with white noises. The dynamic is designed such that the system may reach equilibrium states for a set of boundary conditions. In this form, we make the system driven to a nonequilibrium stationary state by changing the equilibrium boundary conditions. We decompose the correlations in a known local equilibrium part and another one that contains the nonequilibrium behavior and that we call correlation's excess $\overline{C}(x,z)$. We formally derive the differential equations for $\overline{C}$. To solve them order by order, we define a perturbative expansion around the equilibrium state. We show that the $\overline{C}$'s first-order expansion, ${\overline{C}}^{\left(1\right)}$, is always zero for the unique field case, $M=1$. Moreover, ${\overline{C}}^{\left(1\right)}$ is always long range or zero when $M>1$. We obtain the surprising result that their associated fluctuations, the space integrals of ${\overline{C}}^{\left(1\right)}$, are always zero. Therefore, fluctuations are dominated by local equilibrium up to second order in the perturbative expansion around the equilibrium. We derive the behaviors of ${\overline{C}}^{\left(1\right)}$ in real space for dimensions $d=1$ and 2 explicitly. Finally, we derive the two first perturbative orders of the correlation's excess for a generic $M=2$ case and a hydrodynamic model.

Figure 1

The basic structure function $\widehat{S}(n,m;k,\theta )$. It is shown only the $(n,m)$ values where $\widehat{S}$ is nonzero. The sublattices with equal parity, i.e., both $n$ and $m$ even or odd, have $\widehat{S}=0$.

Figure 2

The $S(x,z;\theta )$ vs $(\overline{x}=2x/L-1,\overline{z}=2z/L-1)$ for $\theta =0.5$. The red line shows the function $S(x,x;\theta )$. The black line is a reference $(\overline{x},\overline{x},0)$.

Figure 3

Left: $S$ vs $(x,z)$ for $\theta =0.5$ using the expression (90). Right: Figure 2 expanded around $(\overline{x},\overline{z})=(-1,-1)$. The red curve is $[x,x,F(x,x\left)\right]$. The black line is a reference $(\overline{x},\overline{z},0)$.

Figure 4

Several behaviors of $S$ for $\theta =0.5$ from Eq. (93). Top left: $S$ vs $(z_1,x_2-z_2)$ for $x_1=1$. We see the singularity at $z_1=1$. $S$ vs $(z_1,x_1)$ for $x_2-z_2=1$ (top right), $x_2-z_2=0.1$ (bottom left), and $x_2-z_2=0$ (bottom right), respectively.

Figure 5

$D_{11}$ vs $(\overline{x},\overline{z})$ numerically computed from Eq. (104) for $d=1$. Top left and right: $\theta =0.5$ and 1 respectively. The red curve is $[\overline{x},\overline{x},D_{11}(\overline{x},\overline{x})]$. The gray plane is the zero reference $(\overline{x},\overline{z},0)$. The bottom figure is the ratio $R_{11}=D_{11}(\theta =0.5)/D_{11}(\theta =1)$ to show that they are not related by a constant factor.

Figure 6

Numerical computation of $D_{12}=-{\overline{C}}_{12}\pi \lambda {\left(2\right)}^2/a_{12}$ vs $(\overline{x},\overline{z})$ for dimension $d=1$ and $\theta =0.5$. The matrix $g$ is given by Eq. (110). The red curve is $[\overline{x},\overline{x},D_{12}(\overline{x},\overline{x})]$. The gray plane is the reference $(\overline{x},\overline{z},0)$.

Figure 7

$D_{11}(x,z;\theta =0.5)$ from Eq. (114). Top: Three-dimensional contour plot. Each surface correspond to a fix value of $D_{11}$: $-0.5,-0.4,-0.3$, $-0.2$, $-0.1$, $-0.08$, $-0.06$, $-0.04$, and $-0.02$ from inside to outside. Bottom: Behavior for fix value of $x_1=1$ (left) or fix value of $|x_2-z_2|=1$ (right). The dashed lines show the asymptotic behavior along the directions (see text).

Figure 8

Three-dimensional contour plot of $D_{12}(x,z;g)=$ for $\theta =0.5$ from Eq. (116). Each surface correspond to a fix value of $D_{12}$ (see legend).

Figure 9

$D_{12}(x,z;g)$ for $\theta =0.5$ from Eq. (116). Behavior for fixed value of $x_1=1$ (top left) and fixed value of $|x_2-z_2|=1,0.1$ (top right and bottom respectively). The dashed lines show the asymptotic behavior along the directions (see text).

Figure 10

Second-order fluctuations for the hydrodynamic two-field model computed using Eq. (133). Black dots: ${\overline{\mathrm{\Delta }}}_{11}^{\text{neq},\left(2\right)}\left(\omega \right)={\mathrm{\Delta }}_{11}^{\text{neq},\left(2\right)}\left(\omega \right)T_1^2/\overline{n}$. Red dots: ${\overline{\mathrm{\Delta }}}_{12}^{\text{neq},\left(2\right)}\left(\omega \right)={\mathrm{\Delta }}_{12}^{\text{neq},\left(2\right)}\left(\omega \right)T_1/\overline{n}$. The black dashed line is the analytic asymptotic behavior for small values of $\omega$. ${\overline{\mathrm{\Delta }}}_{11}^{\text{neq},\left(2\right)}\left(1\right)=0.061685$, ${\overline{\mathrm{\Delta }}}_{12}^{\text{neq},\left(2\right)}\left(1\right)=0,$ and ${\overline{\mathrm{\Delta }}}_{12}^{\text{neq},\left(2\right)}\left(0\right)=-0.00210306.$

Figure 11

Contour $C=({\bigcup }_n(C_n\bigcup D_n)\bigcup C^{\prime }$ used to evaluate integral (F4). $P_n$ are the simple poles at $w\left(P_n\right)=n\pi$.

Figure 12

Contour $C={\bigcup }_{i=1}^6{C}_i$ used to evaluate integral (F14). $P_{1,2}$ are the poles at $w_{P_{1,2}}=\pm ai$ and $B_{1,2}$ are the branch points at $w_{B_{1,2}}=\pm ia\sqrt{1+1/\theta }$. The branch lines are shown in red.