Nonequilibrium currents in stochastic field theories: A geometric insight

We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Figure 1

Measurements of the stationary probability currents in the planes $[\rho \left({\mathbf{r}}_0\right),{\partial }_x\rho \left({\mathbf{r}}_0\right)]$ at representative points in phase-separated systems, using numerical resolution of Eq. (9). The rows correspond to $2\lambda +{\kappa }^{\prime }<0$ (top), $2\lambda +{\kappa }^{\prime }=0$ (center), and $2\lambda +{\kappa }^{\prime }>0$ (bottom), respectively. The average stationary profiles are shown in the left column. Other columns show the current vector fields measured at the corresponding points ${\mathbf{r}}_0=A,B,C$ in the phase-separated profiles. (Arrow colors encode their angles with respect to ${\mathbf{e}}_x$.) Parameters: $a=-1$, $b=1$, $\kappa =0.1$, average density ${\rho }_0=-0.4$, $D={10}^{-3}$, $L_x=L_y=10$, and $\lambda =-2$ (top), 0 (center), and 2 (bottom). See Appendices pp1 and pp2 for numerical details.

Figure 2

Evolution of a perturbation $\delta \rho$ around the equilibrium profile ${\rho }_s$ under the AMB dynamics (9) using periodic boundary conditions and $\delta \rho (x,y,t=0)=\varepsilon cos(100\pi x/L_x)$. (a)–(c) Kymographs representing the evolution of $\delta \rho (x,L_y/2,t)$. (d) A cut of ${\rho }_s$ at $y=L_y/2$. Parameters: ${\rho }_0=-0.45$, $a=-1$, $b=1$, $\kappa =0.15$, $dt={10}^{-7}$, $dx=dy={10}^{-2}$, $L_x=L_y=10$, $\varepsilon =0.05$, and $\lambda =-4$, 0, and 4 for (a)–(c), respectively. Time axis unit of the kymographs is $\mathrm{\Delta }t={10}^{-5}$.

Figure 3

Analysis of the currents shown in Fig. 1 predicting the mode propagation reported in Fig. 2. Consider the case $2\lambda +{\kappa }^{\prime }<0$ and a point ${\mathbf{r}}_0=(x_0,L_y/2)$ on the left boundary of the droplet (top row, column $A$ of Fig. 1). In the central panel, we show the circulation induced by the current in the $[\delta \rho \left({\mathbf{r}}_0\right),{\partial }_x\delta \rho \left({\mathbf{r}}_0\right)]$ plane. Consider a perturbation such that $\delta \rho \left({\mathbf{r}}_0\right)$ is a local maximum at $t=0$ [panel (1), red star]. As time goes on, the current drives the fluctuation sequentially from (1) to (2) (orange dot), to (3) (yellow triangle), and to (4) (green square). Each panel shows the fluctuation profile $\delta \rho (x,L_y/2)$ around $x_0$ (blue curves). These successive states of $\delta \rho$ show that the probability current corresponds to a leftward propagation of $\delta \rho$ in the real space, from the liquid to the gas phase. Inspection of Fig. 1 allows prediction of all the dynamics reported in Fig. 2.

Figure 4

Kymographs showing the zero-noise relaxation of a perturbation $\delta h\left(x\right)=0.1sin\left(40\pi x\right)$, added at time $t=0$ to a linear profile $h\left(x\right)=10x$, under the KPZ dynamics (43) with Dirichlet boundary conditions. Parameters: $\nu =2$ and $\lambda =-16,0,16$ for (a)–(c), respectively. System size $L=1$. Space and time discretization: $dx={10}^{-3}$ and $dt={10}^{-7}$.