Long-Range Persistence Active Fluctuation Effect on a Colloidal Particle: Dissipation and Entropy Production.

We study the dynamics and thermodynamics of a harmonically trapped colloidal particle driven by active noise with long-range memory. The active force is modeled as a stationary Gaussian process with a power-law decay, allowing us to interpolate between short- and long-time regimes by varying {the power law exponent $\alpha$}. In the overdamped setting, we derive exact solutions for the particle's position statistics and two-time correlations, and characterize how active noise affects its relaxation spectrum. An effective temperature emerges naturally from the steady-state fluctuation-dissipation ratio, capturing the nonequilibrium character of the active bath even in the presence of thermal fluctuations. We then consider the purely active regime, where the thermal noise is switched off and the system evolves under active driving alone. In this setting, we construct the stochastic entropy balance at the trajectory level and identify a consistent definition of medium entropy using a time-dependent active temperature derived from the noise correlation function. We confirm that the total entropy production satisfies an integral fluctuation theorem, and demonstrate how the power law exponent $\alpha$ controls the degree of irreversibility: smaller $\alpha$ enhances time correlations and increases entropy production.