Stochastic dynamics in soft matter: Non-uniform activity and fluctuating fields

In this thesis we investigate the stochastic dynamics of particles in contact with active environments or fluctuating correlated media at equilibrium, offering a comprehensive analysis that encompasses both equilibrium and non-equilibrium systems. In the first part, we focus on assemblies of active particles subject to a spatially varying degree of activity. We demonstrate that the interplay between the inter-particle interactions and a non-homogeneous activity leads to unexpected migration properties. In particular, we show that interacting active particles can be directed and localized within specific spatial regions, and we highlight potential applications in the design of autonomous systems able to migrate towards specific target zones. In the second part of the thesis we analyze the equilibrium behavior of particles moving within a fluctuating medium that acquires significant spatio-temporal correlations. Specifically, we first study the extent to which the field-induced forces caused by a fluctuating correlated medium affect the conformational and the dynamical properties of a polymer chain. Secondly, we investigate the self-diffusion coefficient of an odd-diffusive tracer, whose dynamics is characterized by probability fluxes perpendicular to the density gradient, coupled to a Gaussian-core fluid, showing that it can be enhanced by the interaction with the medium. By combining these studies, the thesis advances our understanding of the behavior of particles in correlated and active media, offering new insights into the transport properties and collective behaviors of collodal systems.