Mode-coupling theory for active Brownian particles

Abstract

We present a mode-coupling theory (MCT) for the high-density dynamics of two-dimensional spherical active Brownian particles (ABP). The theory is based on the integration-through-transients (ITT) formalism and hence provides a starting point for the calculation of non-equilibrium averages in active-Brownian particle systems. The ABP are characterized by a self-propulsion velocity v0, and by their translational and rotational diffusion coefficients, Dt and Dr. The theory treats both the translational and the orientational degrees of freedom of ABP explicitly. This allows to study the effect of self-propulsion of both weak and strong persistence of the swimming direction, also at high densities where the persistence length p = v0/Dr is large compared to the typical interaction length scale. While the low-density dynamics of ABP is characterized by a single Péclet number, Pe = v 2 0 /DrDt, close to the glass transition the dynamics is found to depend on Pe and p separately. At fixed density, increasing the self-propulsion velocity causes structural relaxation to sped up, while decreasing the persistence length slows down the relaxation. The theory predicts a nontrivial idealized-glass-transition diagram in the three-dimensional parameter space of density, selfpropulsion velocity and rotational diffusivity. The active-MCT glass is a nonergodic state where correlations of initial density fluctuations never fully decay, but also an infinite memory of initial orientational fluctuations is retained in the positions.

Static structure of active Brownian hard disks

“…Our data can be used as a reference for future theoretical studies which construct approximative closures for the structure based on the Smoluchowski equation [8,35,36] or for mode-coupling-like dynamical theories that either need static structure factor data as an input [36,29,37] or can in principle approximate it based on the equilibrium one [31]. As a remark, however, the extension of theoretical frameworks to calculate S(q) and related structural quantities that characterize the non-equilibrium steady state of active particles is not obvious, even for radially-symmetric pair potentials.…”

“…There are indications that the high-density dynamics of ABP on the contrary depends on both these parameters separately [31,18,30], because at high densities the average distance between ineracting particles is easily shorter than the persistence length associated to the swimming, p = v 0 /D r . It is therefore interesting to see the effect that varying both v 0 and D r has on the stationary static structure.…”

Projection operators in statistical mechanics: a pedagogical approach

“…This density maximizes the informational entropy with respect to the constraint that our macroscopic information is given by the mean values {a i (t)} of the relevant variables [16]. The normalization Z(t) ensures that Tr(ρ(t)) = 1 (20) and the thermodynamic conjugates {λ j (t)} are chosen in such a way that…”

“…(19) is a standard choice, a relevant density can be any function of the mean values that satisfies Eqs. (20) and (21) [11]. It is helpful, in general, to choose the relevant density in such a way that it is a good approximation for the actual microscopic density, in particular if they coincide for t = 0.…”

Mode-coupling theory for the steady-state dynamics of active Brownian particles

“…The difference between the theories of Farage and Brader [29] and of Liluashvili et al [27] and both our theories and the approach of Feng and Hou [30] is that the former theories follow the philosophy of the "integrationthrough-transients" approach to the dynamics of colloidal systems under shear developed by Fuchs and Cates [32]. In this approach one assumes that the system was in an equilibrium state in the infinitely distant past and then the drive, in this case the activity, was turned on.…”