Field Theory of free Active Ornstein-Uhlenbeck Particles

Abstract

We derive a Doi-Peliti field theory for free active Ornstein-Uhlenbeck particles, or, equivalently, free inertial Brownian particles, and present a way to diagonalise the Gaussian part of the action and calculate the propagator. Unlike previous coarse-grained approaches this formulation correctly tracks particle identity and yet can easily be expanded to include potentials and arbitrary reactions.

Run-and-tumble motion in a harmonic potential: field theory and entropy production

“…with eigenvalue λ n = −n/L 2 , where H n (x) is the nth Hermite polynomial in the 'probabilists' convention', see appendix B [35,[42][43][44]. The functions u n (x) are similar to the so-called Hermite functions exp(−x 2 /(4L 2 )) H n (x/L).…”

“…Our work can be extended to studying Active Ornstein-Uhlenbeck Particles [44], whose self-propulsion is modelled by an Ornstein-Uhlenbeck process, and therefore share important features with an RnT particle in a harmonic potential. In particular, the field decomposition is also based on Hermite polynomials.…”

Active Ornstein–Uhlenbeck model for self-propelled particles with inertia

“…Columns: (i) force-free AOUP, cf section 3.1, (ii) constant external force F, cf section 3.2, (iii) harmonic external potential with constant k (α(t) is illustrated here for a spring with k > 0), cf section 3.3, (iv) two harmonically coupled AOUPs with equal mass m but different diffusion coefficient D and persistence time τ (α(t) is illustrated here for the center-of-mass coordinate R), cf section 3.4, and (v) with time-dependent mass m(t) of constant slope ṁ (α(t) is illustrated here for ṁ < 0), cf section 3.5. memory decays exponentially in time, leading to a persistence in the particle motion which mimicks the activity. This model, originally proposed by Ornstein and Uhlenbeck to study velocity distributions of passive particles [11] and subsequently exploited for various other physical and mathematical problems [12][13][14][15], has by now become a basic reference for active motion [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Although the AOUP model does not resolve the orientational degrees of freedom, it admits some characteristic features of activity, like persistent motion, surface accumulation and, most prominently, motility-induced phase separation (MIPS) [16,31].…”