title: Effective Equilibrium, Power Functional, and Interface Structure for Phase-Separating Active Brownian Particles
Creators: name: Krinninger, Philip
Publisher: name: University of Bayreuth
Year: 2019
Subjects: subject: Statistische Mechanik
subject: Brownsche Dynamik
subject: Aktive Systeme
subject: Powerfunktionaltheorie
subject: 500
subject: 530

description: In this thesis we approach the physics of active Brownian particles (ABP) and particularly the emergence of motility induced-phase separation (MIPS) by (i) an effective equilibrium description for small activities [1], (ii) a formally exact power functional theory [2, 3], and (iii) a computer simulation study of the free interface between coexisting phases [4]. Active Brownian particles are modeled as spherical particles that obey Brownian motion described by an overdamped Langevin equation of motion. Activity is thereby induced by a self-propulsion force. This force acts along the built-in orientation of each individual particle, and the time dependence
of the orientation is given by an additional Langevin equation that describes free rotational diffusion. This intrinsic out-of-equilibrium system shows a wide variety of phenomena, where phase separation in absence of explicit interparticle attraction between the particles is one of the most
spectacular open problems.

In the effective equilibrium approach the active system is mapped onto a system of passive Brownian particles that interact via a modified effective interparticle interaction [1]. This is achieved by integrating out the orientations. The resulting Langevin equation contains colored
noise. From this equation of motion an approximated Fokker-Planck equation is constructed. In this Fokker-Planck equation an activity-dependent effective interaction force between the particles is identified. In the case of pairwise interaction, the effective interaction can be represented as an activity-dependent effective pair interaction potential. For purely repulsive interaction potentials, an attractive tail develops above an activity threshold. The strength of this attraction increases even further with increasing activity, eventually leading to bulk phase separation. Furthermore, passively attractive interactions are considered, namely the Lennard-Jones potential. In this case, the attractive minimum of the potential weakens at first when activity is increased and suppression of phase separation is observed. Increasing the activity further, the attractive minimum deepens again and a reentrance of phase separation emerges. As the activity determines only the form of the effective interaction and the many-body dynamics resemble the passive dynamics, common methods of liquid state theory can be applied to active systems.
We use them for instance to calculate spinodals and the pair correlation function for the active system. An important part of the work is the validation via computer simulations, where the orientations are not integrated out, i.e., the full many-body problem of ABP is considered. The result is that in all situations presented we find a good match between theory and the simulation.

Despite the success of the effective equilibrium description in many situations, possible applications are rather limited to low activity cases due to the approximations made in the construction of the Fokker-Planck equation. In order to overcome these limitations we develop a power func-
tional theory (PFT) for active particles [2, 3]. Therein, the orientations are considered as additional degrees of freedom and enter the many-body dynamics as such. Starting from the many-body description, i.e., the Smoluchowski equation, power functional theory offers an exact
formalism to determine the correct physical time evolution of the one-body density, the translation, and rotational current distributions from a microscopically defined power functional. We apply this framework to steady states, where the coupling between the translational and rota-
tional dynamics is realized by the self-propulsion force that determines the magnitude of the activity, and we give a simple approximation for the dissipative parts of the power functional that applies to bulk states. Furthermore, we show that in steady state the value of the power
functional is determined by the negative value of half of the external power and is thus trivially related to the swimming of the particles. For soft repulsive spheres we perform Brownian dynamics computer simulations in bulk and develop sampling strategies for the power functional.
Comparing our theory with the numerics shows a good agreement.

Going beyond the bulk properties of ABP that undergo MIPS a detailed study of the interface between the phases is required [4]. The reason is that the dissipation functional presented for the bulk studies [2, 3] denotes only the friction (drag) induced by particle interactions in bulk. For inhomogeneous situations further superadiabatic contributions have to be considered and thus the dissipation functional has to be generalized in order to cover these contributions. As the corresponding manuscript is in preparation [4], results from computer simulations are presented in Sec. 4. In agreement with the literature we find that the interface is polarized and
hence the orientations of the particles are not evenly distributed there (as it would be in bulk). By analyzing the results for the one-body distribution of the density and the current we give an interpretation of the particles’ behavior at the interface. Further insights are provided by the explicit analysis of the orientation dependence of the one-body distributions including an angular Fourier decomposition which is used to describe the anisotropies that emerge at the interface. Our findings serve as a reference point for an extended power functional theory that can describe the physics of the active particles in bulk and at the interface.