  Active Matter models commonly consider particles with overdamped dynamics
subject to a force (speed) with constant modulus and random direction. Some
models include also random noise in particle displacement (Wiener process)
resulting in a diffusive motion at short time scales. On the other hand,
Ornstein-Uhlenbeck processes consider Langevin dynamics for the particle
velocity and predict a motion that is not diffusive at short time scales.
However, experiments show that migrating cells may present a varying speed as
well as a short-time diffusive behavior. While Ornstein-Uhlenbeck processes can
describe the varying speed, Active Mater models can explain the short-time
diffusive behavior. Isotropic models cannot explain both: short-time diffusion
renders instantaneous velocity ill-defined, hence impeding dynamical equations
that consider velocity time-derivatives. On the other hand, both models apply
for migrating biological cells and must, in some limit, yield the same
observable predictions. Here we propose and analytically solve an Anisotropic
Ornstein-Uhlenbeck process that considers polarized particles, with a Langevin
dynamics for the particle movement in the polarization direction while
following a Wiener process for displacement in the orthogonal direction. Our
characterization provides a theoretically robust way to compare movement in
dimensionless simulations to movement in dimensionful experiments, besides
proposing a procedure to deal with inevitable finite precision effects in
experiments or simulations.
