Effective equilibrium states in mixtures of active particles driven by colored noise

We consider the steady-state behavior of pairs of active particles having different persistence times and diffusivities. To this purpose we employ the active Ornstein-Uhlenbeck model, where the particles are driven by colored noises with exponential correlation functions whose intensities and correlation times vary from species to species. By extending Fox's theory to many components, we derive by functional calculus an approximate Fokker-Planck equation for the configurational distribution function of the system. After illustrating the predicted distribution in the solvable case of two particles interacting via a harmonic potential, we consider systems of particles repelling through inverse power-law potentials. We compare the analytic predictions to computer simulations for such soft-repulsive interactions in one dimension and show that at linear order in the persistence times the theory is satisfactory. This work provides the toolbox to qualitatively describe many-body phenomena, such as demixing and depletion, by means of effective pair potentials.

Figure 1

Effective potentials in one dimension from OUPs simulation and the Fox theory, Eq. (36), between two active particles with the parameters ${\tau }^{\left(1\right)}$ and ${\mathcal{D}}_{\text{a}}^{\left(1\right)}$ (as labeled) with ${\tau }^{\left(2\right)}=0.1-{\tau }^{\left(1\right)}$ and ${\mathcal{D}}_{\text{a}}^{\left(2\right)}=9.6-{\mathcal{D}}_{\text{a}}^{\left(1\right)}$, such that $\overline{\tau }=0.05$ and ${\overline{\mathcal{D}}}_{\text{a}}=4.8$ (the results are invariant when exchanging the particles, i.e., the labels 1 and 2). Here we consider only particles with at least one pair of equal parameters (${\tau }^{\left(2\right)}={\tau }^{\left(1\right)}$ or ${\mathcal{D}}_{\text{a}}^{\left(2\right)}={\mathcal{D}}_{\text{a}}^{\left(1\right)}$), such that the results only depend on $|\mathrm{\Delta }{\mathcal{D}}_{\text{a}}|$ or $\left|\mathrm{\Delta }\tau \right|$, respectively. The case of all parameters being equal is labeled as (ep). The Fox theory predicts the same result (ep) in all cases. We consider a system (a) without and (b) with thermal noise; note the different scale on the vertical axes. The legends apply to both subfigures and the colored lines reappear in subsequent figures for comparison.

Figure 2

As described in the caption of Fig. 1, but for two particles with fixed ${\tau }^{\left(1\right)}=0.025$ and thus ${\tau }^{\left(2\right)}=0.075$. We qualitatively compare (a) simulations to (b) theory, where $\mathrm{\Delta }{\mathcal{D}}_{\text{a}}$ is chosen much smaller for the theoretical curves, since the changes for different ${\mathcal{D}}_{\text{a}}^{\left(1\right)}$ are more significant and the curves start to diverge for some ${\mathcal{D}}_{\text{a}}^{\left(1\right)}<4.8$. This is a direct consequence of the form of Eq. (36). The colored lines for ${\mathcal{D}}_{\text{a}}^{\left(1\right)}=4.8$ and for all parameters being equal (ep) are the same as in Fig. 1.

Figure 3

As described in the caption of Fig. 1, but for two particles with fixed ${\mathcal{D}}_{\text{a}}^{\left(1\right)}=2.4$ and thus ${\mathcal{D}}_{\text{a}}^{\left(2\right)}=7.2$. As in Fig. 2, we compare (a) simulations to (b) theory, where the curves diverge for some ${\tau }^{\left(1\right)}<0.05$. The colored lines for ${\tau }^{\left(1\right)}=0.05$ and for all parameters being equal (ep) are the same as in Fig. 1.

Figure 4

Comparison between the effective potentials from theory (thick lines) and simulations (dots and thin lines) for (a) a fixed persistence time ${\tau }^{\left(1\right)}={\tau }^{\left(2\right)}=0.05$ but different active diffusivities (as labeled) and (b) an active-passive mixture with fixed ${\mathcal{D}}_{\text{a}}^{\left(1\right)}={\mathcal{D}}_{\text{a}}^{\left(2\right)}=1$ and the passive ${\tau }^{\left(2\right)}=0$ but different active persistence times (as labeled). In this case, the equilibration appears to proceed very slowly and the numerical values only gradually approach 0 for separations larger than shown here. However, even for large values of ${\tau }^{\left(1\right)}$ there is no attractive well, suggesting that the fully equilibrated data should reflect the behavior of two passive particles.