The diusion of colloids inside an active system-e.g. within a living cell or the dynamics of active particles itself (e.g. self-propelled particles) can be modeled through overdamped Langevin equation which contains an additional noise term apart from the usual white Gaussian noise, originating from the thermal environment. The second noise is referred to as 'active noise' as it arises from activity such as chemical reactions. The probability distribution function (PDF or the propagator) in space-time along with moments provides essential information for understanding their dynamical behavior. Here we employ the phase-space path integral method to obtain the propagator, thereby moments and PDF for some possible models for such noise. At first, we discuss the diusion of a free particle driven by active noise. We consider four dierent possible models for active noise, to capture the possible traits of such systems. We show that the PDF for systems driven by noises other than Gaussian noise largely deviates from normal distribution at short to intermediate time scales as a manifestation of out-of-equilibrium state, albeit converges to Gaussian distribution after a long time as a consequence of the central limit theorem. We extend our work to the case of a particle trapped in a harmonic potential and show that the system attains steady state at long time limit. Also, at short time scales, the nature of distribution is dierent for dierent noises, e.g. for particle driven by dichotomous noise, the probability is mostly concentrated near the boundaries whereas a long exponential tail is observed for a particle driven by Poissonian white noise.

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In this topical review, we give an overview of the structure and dynamics of a single polymer chain in active baths, Gaussian or non-Gaussian. The review begins with the discussion of single flexible or semiflexible linear polymer chains subjected to two noises, thermal and active. The active noise has either Gaussian or non-Gaussian distribution but has a memory, accounting for the persistent motion of the active bath particles. This finite persistence makes the reconfiguration dynamics of the chain slow as compared to the purely thermal case and the chain swells. The active noise, also results superdiffusive or ballistic motion of the tagged monomer. We present all the calculations in details but mainly focus on the analytically exact or almost exact results on the topic, as obtained from our group in recent years. In addition, we briefly mention important works of other groups and include some of our new results. The review concludes with pointing out the implications of polymer chains in active bath in biologically relevant context and its future directions.

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