Statistical mechanics of active Ornstein-Uhlenbeck particles

We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.

Figure 1

Steady-state distribution of an AOUP in a confining potential $\mathrm{\Phi }\left(x\right)=x^4/4$. Top: For $\tau =0.01$, the series (20) converges rapidly and coincides with the numerics. The latter are obtained by integrating Eqs. (1) and (2) using the Heun algorithm (see [70] for detail). (a) The truncated Borel also describes correctly the data (b). Bottom: For $\tau =0.2$, the series (20) is rapidly diverging and very far from the numerics (c). On the contrary, the truncated Borel sum describes very well the data (d).

Figure 2

Steady-state distributions of AOUPs evolving in 1D with $D=1$ and $\tau \left(x\right)=1+\varepsilon cos(4\pi x/L)$ (blue crosses) and with $\tau =1$ and $D={[1+\varepsilon cos(4\pi x/L)]}^{-1}$ (red circles), compared with the theoretical prediction (36) (black line). Parameters: $L=40, \varepsilon =0.1$

Figure 3

Simulations of $N$ AOUPs evolving with dynamics (40) and (41) and interacting via potential (39) in a $400\times 400$ domain with periodic boundary conditions. Parameters: $D=10, \varepsilon =r_0=\mu =1$. In (a) and (b), snapshots taken after a time $t=10000$ show the occurrence of motility-induced phase separation for $\tau =90$. The average densities are ${\rho }_0=0.5$ and ${\rho }_0=0.9$, respectively. Varying the overall density alters the size of the dense and dilute phases, but leaves their respective density unchanged. This can be seen from (c), which presents histograms of the local density measured in boxes of size $10\times 10$. The three curves correspond to ${\rho }_0=0.50,0.70,0.90$. Finally, the phase diagram shown in (d) is obtained by measuring the densities of the dilute and dense phases in simulations with an average density 0.9 and different values of $\tau$. The densities are estimated from the maxima of histograms obtained as in (c).

Figure 4

Snapshots of $N$ AOUPs interacting via quorum sensing through Eq. (55) with $\tau =1$ and $D(\mathbf{r},\left[\widehat{\rho }\right])=D_0{e}^{-\lambda \varphi arctan(\tilde{\rho }\left(\mathbf{r}\right)/\varphi )}$. The field $\tilde{\rho }\left(\mathbf{r}\right)$ measures the density field $\widehat{\rho }$ averaged over a disk of radius 1 centered in $\mathbf{r}$. A linear instability will lead to MIPS whenever $\lambda \varphi >2$, according to Eq. (63). Simulation parameters: $D_0=1, {\rho }_0=N/L^2=25, L=50, \lambda =0.05, \varphi =44.72$, so that $\lambda \varphi =2.23$. Color encodes the density averaged over a disk of radius 5. Starting from a random initial condition, the snapshot is taken after a time $t=49800$.

Figure 5

Plot of the normalized current $J/{\tau }^2$ induced by a ratchet potential $\mathrm{\Phi }\left(r\right)=sin(\pi r/2)+sin\left(\pi r\right)$ as a function of the inverse of the persistence time ${\tau }^{-1}$. The blue dots correspond to numerical simulations with error bars given by the standard deviation; the red line is our analytical prediction in the small $\tau$ limit obtained from Eq. (72). In the inset, we plot $J$ as a function of $\tau$.

Figure 6

Snapshots of AOUPs interacting via the short-range soft-core potential (85), and confined in a harmonic potential $U\left(\mathbf{r}\right)=U_0{(r/a_0)}^2$ with finite range $a_0$. The range of the potential is represented by the gray circle. The color of each particle refers to the associated instantaneous value of the entropy production rate, expressed in units of $1/{\tau }_{\text{r}}=\varepsilon /a^2$. We observe that particles form a dense compact cluster centered at the bottom of the harmonic trap with radial symmetry, in contact with a dilute bath of particles. The interface between the dense and dilute phases fluctuates, and the relative size of the dense phase increases with the number of particles from (a) to (d). Number of particles: (a) $N=5625$, (b) $N=6750$, (c) $N=7875$, (d) $N=9000$. Other parameters: $L=150, D=1, U_0=2, \varepsilon =10, a=1, \tau =10, a_0=60$.

Figure 7

Density and local entropy production rate profiles as functions of the distance from the center of the harmonic trap. Parameters are the same as in Fig. 6.