Active Ornstein-Uhlenbeck Particles Research Articles

Statistical field theory and effective action method for scalar active matter

We employ Statistical Field Theory techniques for coarse-graining the steady-state properties of Active Ornstein-Uhlenbeck particles. The computation is carried on in the framework of the Unified Colored Noise approximation that allows an effective equilibrium picture. We thus develop a mean-field theory that allows to describe in a unified framework the phenomenology of scalar Active Matter. In particular, we are able to describe through spontaneous symmetry breaking mechanism two peculiar features of Active Systems that are (i) The accumulation of active particles at the boundaries of a confining container, and (ii) Motility-Induced Phase Separation (MIPS). \textcolor{black}{We develop a mean-field theory for steric interacting active particles undergoing to MIPS and for Active Lennard-Jones (ALJ) fluids.} \textcolor{black}{Within this framework}, we discuss the universality class of MIPS and ALJ \textcolor{black}{showing that it falls into Ising universality class.} We \textcolor{black}{thus} compute analytically the critical line $T_c(\tau)$ for both models. In the case of MIPS, $T_c(\tau)$ gives rise to a reentrant phase diagram compatible with an inverse transition from liquid to gas as the strength of the noise decreases. \textcolor{black}{However, in the case of particles interacting through anisotropic potentials, } the field theory acquires a $\varphi^3$ term that, \textcolor{black}{in general, cannot be canceled performing the expansion around the critical point.} In this case, the \textcolor{black}{Ising} critical point might \textcolor{black}{be replaced} by a first-order phase transition \textcolor{black}{region}.

Active Ornstein-Uhlenbeck particles.

Active Ornstein-Uhlenbeck particles (AOUPs) are overdamped particles in an interaction potential subject to external Ornstein-Uhlenbeck noises. They can be transformed into a system of underdamped particles under additional velocity dependent forces and subject to white noise forces. There has been some discussion in the literature on whether AOUPs can be in equilibrium for particular interaction potentials and how far from equilibrium they are in the limit of small persistence time. By using a theorem on the time reversed form of the AOUP Langevin-Ito equations, I prove that they have an equilibrium probability density invariant under time reversal if and only if their smooth interaction potential has zero third derivatives. In the limit of small persistence Ornstein-Uhlenbeck time τ, a Chapman-Enskog expansion of the Fokker-Planck equation shows that the probability density has a local equilibrium solution in the particle momenta modulated by a reduced probability density that varies slowly with the position. The reduced probability density satisfies a continuity equation in which the probability current has an asymptotic expansion in powers of τ. Keeping up to O(τ) terms, this equation is a diffusion equation, which has an equilibrium stationary solution with zero current. However, O(τ^{2}) terms contain fifth- and sixth-order spatial derivatives and the continuity equation no longer has a zero current stationary solution. The expansion of the overall stationary solution now contains odd terms in the momenta, which clearly shows that it is not an equilibrium.

Self-propelled particle in a nonconvex external potential: Persistent limit in one dimension.

Equilibrium mapping techniques for nonaligning self-propelled particles have made it possible to predict the density profile of an active ideal gas in a wide variety of external potentials. However, they fail when the self-propulsion is very persistent and the potential is nonconvex, which is precisely when the most uniquely active phenomena occur. Here, we show how to predict the density profile of a 1D active Ornstein-Uhlenbeck particle in an arbitrary external potential in the persistent limit and discuss the consequences of the potential's nonconvexity on the structure of the solution, including the central role of the potential's inflection points and the nonlocal dependence of the density profile on the potential.

The entropy production of Ornstein–Uhlenbeck active particles: a path integral method for correlations

By employing a path integral formulation, we obtain the entropy production rate for a system of active Ornstein–Uhlenbeck particles (AOUP) both in the presence and in the absence of thermal noise. The present treatment clarifies some contraddictions concerning the definition of the entropy production rate in the AOUP model, recently appeared in the literature. We derive explicit formulas for three different cases: overdamped Brownian particle, AOUP with and without thermal noise. In addition, we show that it is not necessary to introduce additional hypotheses concerning the parity of auxiliary variables under time reversal transformation. Our results agree with those based on a previous mesoscopic approach.

Active Brownian ring polymers.

The conformational and dynamical properties of semiflexible active Brownian ring polymers are investigated analytically. A ring is described by the Gaussian semiflexible polymer model accounting for the finite contour length. Activity is implemented by a Gaussian, non-Markovian stochastic process resembling either an external nonthermal force or a local self-propulsion velocity as for an active Ornstein-Uhlenbeck particle. Specifically, the fluctuation spectrum of normal-mode amplitudes is analyzed. At elevated activities, flexible (tension) modes dominate over bending modes even for semiflexible rings, corresponding to enhanced conformational fluctuations. The fluctuation spectrum exhibits a crossover from a quadratic to a quartic dependence on the mode number with increasing mode number, originating from intramolecular tension, but the relaxation behavior is either dominated by intra-polymer processes or the active stochastic process. A further increase in activity enhances fluctuations at large length scales at the expense of reduced fluctuations at small scales. Conformationally, the mean square ring diameter exhibits swelling qualitatively comparable to liner polymers. The ring's diffusive dynamics is enhanced, and the mean square displacement shows distinct activity-determined regimes, consecutively, a ballistic, a subdiffusive, and a diffusive regime. The subdiffusive regime disappears gradually with increasing activity.

Linear response and correlation of a self-propelled particle in the presence of external fields

We study the non-equilibrium properties of non interacting active Ornstein–Uhlenbeck particles (AOUP) subject to an external nonuniform field using a Fokker–Planck approach with a focus on the linear response and time-correlation functions. In particular, we compare different methods to compute these functions including the unified colored noise approximation (UCNA). The AOUP model, described by the position of the particle and the active force acting on it, is usually mapped into a Markovian process, describing the motion of a fictitious passive particle in terms of its position and velocity, where the effect of the activity is transferred into a position-dependent friction. We show that the form of the response function of the AOUP depends on whether we put the perturbation on the position and keep unperturbed the active force in the original variables or perturb the position and maintain unperturbed the velocity in the transformed variables. Indeed, as a result of the change of variables the perturbation on the position becomes a perturbation both on the position and on the fictitious velocity. We test these predictions by considering the response for three types of convex potentials: quadratic, quartic and double-well potential. Moreover, by comparing the response of the AOUP model with the corresponding response of the UCNA model we conclude that although the stationary properties are fairly well approximated by the UCNA, the non equilibrium properties are not, an effect which is not negligible when the persistence time is large.

Confined active Brownian particles: theoretical description of propulsion-induced accumulation

The stationary-state distribution function of confined active Brownian particles (ABPs) is analyzed by computer simulations and analytical calculations. We consider a radial harmonic as well as an anharmonic confinement potential. In the simulations, the ABP is propelled with a prescribed velocity along a body-fixed direction, which is changing in a diffusive manner. For the analytical approach, the Cartesian components of the propulsion velocity are assumed to change independently; active Ornstein–Uhlenbeck particle (AOUP). This results in very different velocity distribution functions. The analytical solution of the Fokker–Planck equation for an AOUP in a harmonic potential is presented and a conditional distribution function is provided for the radial particle distribution at a given magnitude of the propulsion velocity. This conditional probability distribution facilitates the description of the coupling of the spatial coordinate and propulsion, which yields activity-induced accumulation of particles. For the anharmonic potential, a probability distribution function is derived within the unified colored noise approximation. The comparison of the simulation results with theoretical predictions yields good agreement for large rotational diffusion coefficients, e.g. due to tumbling, even for large propulsion velocities (Péclet numbers). However, we find significant deviations already for moderate Péclet number, when the rotational diffusion coefficient is on the order of the thermal one.