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Diffusion of Active Particles With Stochastic Torques Modeled as $α$-Stable Noise

We investigate the stochastic dynamics of an active particle moving at a constant speed under the influence of a fluctuating torque. In our model the angular velocity is generated by a constant torque and random fluctuations described as a Lévy-stable noise. Two situations are investigated. First, we study white Lévy noise where the constant speed and the angular noise generate a persistent motion characterized by the persistence time $τ_D$. At this time scale the crossover from ballistic to normal diffusive behavior is observed. The corresponding diffusion coefficient can be obtained analytically for the whole class of symmetric $α$-stable noises. As typical for models with noise-driven angular dynamics, the diffusion coefficient depends non-monotonously on the angular noise intensity. As second example, we study angular noise as described by an Ornstein-Uhlenbeck process with correlation time $τ_c$ driven by the Cauchy white noise. We discuss the asymptotic diffusive properties of this model and obtain the same analytical expression for the diffusion coefficient as in the first case which is thus independent on $τ_c$. Remarkably, for $τ_c>τ_D$ the crossover from a non-Gaussian to a Gaussian distribution of displacements takes place at a time $τ_G$ which can be considerably larger than the persistence time $τ_D$.