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Register a new userActive Brownian particle in harmonic trap: exact computation of moments, and re-entrant transition
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We consider an active Brownian particle in a $d$-dimensional harmonic trap, in the presence of translational diffusion. While the Fokker-Planck equation can not in general be solved to obtain a closed form solution of the joint distribution of positions and orientations, as we show, it can be utilized to evaluate the exact time dependence of all moments, using a Laplace transform approach. We present explicit calculation of several such moments at arbitrary times and their evolution to the steady state. In particular we compute the kurtosis of the displacement, a quantity which clearly shows the difference of the active steady state properties from the equilibrium Gaussian form. We find that it increases with activity to asymptotic saturation, but varies non-monotonically with the trap-stiffness, thereby capturing a recently observed active- to- passive re-entrant behavior.
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We consider the motion of an active Brownian particle with speed fluctuations in d-dimensions in the presence of both translational and orientational diffusion. We use an Ornstein-Uhlenbeck process for active speed generation. Using a Laplace transform approach, we describe and use a Fokker-Planck equation-based method to evaluate the exact time dependence of all relevant dynamical moments. We present explicit calculations of such moments and compare our analytical predictions against numerical simulations to demonstrate and analyze several dynamical crossovers. The kurtosis of displacement shows positive or negative deviations from a Gaussian behavior at intermediate times depending on the dominance of speed or orientational fluctuations.
It is well known that path probabilities of Brownian motion correspond to the equilibrium configurational probabilities of flexible Gaussian polymers, while those of active Brownian motion correspond to in-extensible semiflexible polymers. Here we investigate the properties of the equilibrium polymer that corresponds to the trajectories of particles acted on simultaneously by both Brownian as well as active noise. Through this mapping we can see interesting crossovers in mechanical properties of the polymer with changing contour length. The polymer end-to-end distribution exhibits Gaussian behaviour for short lengths, which changes to the form of semiflexible filaments at intermediate lengths, to finally go back to a Gaussian form for long contour lengths. By performing a Laplace transform of the governing Fokker-Planck equation of the active Brownian particle, we discuss a direct method to derive exact expressions for all the moments of the relevant dynamical variables, in arbitrary dimensions. These are verified via numerical simulations and used to describe interesting qualitative features such as, for example, dynamical crossovers. Finally we discuss the kurtosis of the ABPs position which we compute exactly and show that it can be used to differentiate between active Brownian particles and active Ornstein-Uhlenbeck process.
We study the stationary dynamics of an active interacting Brownian particle system. We measure the violations of the fluctuation dissipation theorem, and the corresponding effective temperature, in a locally resolved way. Quite naturally, in the homogeneous phases the diffusive properties and effective temperature are also homogeneous. Instead, in the inhomogeneous phases (close to equilibrium and within the MIPS sector) the particles can be separated in two groups with different diffusion properties and effective temperatures. Notably, at fixed activity strength the effective temperatures in the two phases remain distinct and approximately constant within the MIPS region, with values corresponding to the ones of the whole system at the boundaries of this sector of the phase diagram. We complement the study of the globally averaged properties with the theoretical and numerical characterization of the fluctuation distributions of the single particle diffusion, linear response, and effective temperature in the homogeneous and inhomogeneous phases. We also distinguish the behavior of the (time-delayed) effective temperature from the (instantaneous) kinetic temperature, showing that the former is independent on the friction coefficient.
We study analytically the single-trajectory spectral density (STSD) of an active Brownian motion as exhibited, for example, by the dynamics of a chemically-active Janus colloid. We evaluate the standardly-defined spectral density, i.e. the STSD averaged over a statistical ensemble of trajectories in the limit of an infinitely long observation time $T$, and also go beyond the standard analysis by considering the coefficient of variation $gamma$ of the distribution of the STSD. Moreover, we analyse the finite-$T$ behaviour of the STSD and $gamma$, determine the cross-correlations between spatial components of the STSD, and address the effects of translational diffusion on the functional forms of spectral densities. The exact expressions that we obtain unveil many distinctive features of active Brownian motion compared to its passive counterpart, which allow to distinguish between these two classes based solely on the spectral content of individual trajectories.
As a result of nonequilibrium forces, purely repulsive self-propelled particles undergo macrophase separation between a dense and a dilute phase. We present a thorough study of the ordering kinetics of such motility-induced phase separation (MIPS) in active Brownian particles in two dimensions, and we show that it is generically accompanied by microphase separation. The growth of the dense phase follows a law akin to the one of liquid-gas phase separation. However, it is made of a mosaic of hexatic microdomains whose size does not coarsen indefinitely, leaving behind a network of extended topological defects from which microscopic dilute bubbles arise. The characteristic length of these finite-size structures increases with activity, independently of the choice of initial conditions.
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