Transport under coordinate-dependent diffusion: An Itô Perspective

Abstract

Biological systems at micrometer scales and below often operate in crowded, highly viscous environments characterized by persistent stochastic thermal fluctuations. Diffusion is incessant and plays an important role in transporting these entities in crowded environments. The presence of membranes/pores and multiple biological entities in a constricted space can make the damping/diffusion inhomogeneous. This effect of inhomogeneity is presented by the diffusion becoming coordinate-dependent. Coordinate dependence of damping/diffusion arises near a boundary or interface due to the presence of slow hydrodynamic modes generated by thermal fluctuations of a Brownian particle. Existence of this coordinate/state dependence of damping/diffusion entails new physics for structured mesoscopic objects evolving through an Itô process under uncorrelated white-noise fluctuations of a heat bath. In this thesis, we analyze the consequences of coordinate-dependent diffusion on Brownian systems within the Itô framework. We investigate how spatially varying damping, even in the absence of external driving or correlated noise, can give rise to rich and unexpected transport phenomena. We begin by studying a single Brownian dimer in one dimension with broken damping symmetry and demonstrate that sustained spontaneous directed motion can emerge purely from equilibrium fluctuations. We then extend this framework to a weakly non-equilibrium scenario, where a small external force is applied. In this case, we demonstrate that the symmetry-broken dimer can exploit thermal fluctuations to extract energy from the external drive. In the next part of our work, we move beyond the single-dimer case and examine the transport behavior of many-body systems composed of interacting dimers. We show that collective rectified spontaneous transport can arise purely from equilibrium fluctuations, driven by homogeneous, isotropic fluctuations of the bath in the presence of spatially varying damping/diffusion. Finally, we explore the thermodynamic implications of Itô-type evolution for mesoscopic objects in inhomogeneous media and identify transient non-equilibrium mechanisms such as entropic pulling and the diffusion diode effect. This thesis offers new insights into how passive systems subject to spatial variation in diffusivity/damping in complex environments can generate effective transport without any active input when viewed through the Itô lens.