Reconstructing Complex Fluid Properties from the Behavior of Fluctuating Immersed Particles

Complex fluids have long been characterized by two functions that summarize the fluid's elastic and viscous properties, respectively called the storage (G'(ω)) and loss (G''(ω)) moduli. A fundamental observation in this field, which is called passive microrheology, is that information about these bulk fluid properties can be inferred from the path statistics of immersed, fluctuating microparticles. In this work, we perform a systematic study of the multistep protocol that forms the foundation of this field. Particle velocities are assumed to be well described by the Generalized Langevin Equation, a stochastic integro-differential equation uniquely characterized by a memory kernel Gᵣ(t), which is hypothesized to be inherited from the surrounding fluid. We investigate the covariation between a particle's velocity process and the non-Markovian fluctuations that force it, and we establish rigorous justification for a key relationship between a particle's Mean Squared Displacement and its memory kernel Gᵣ(t). With this foundation in hand, by way of a tunable four-parameter family of functions that can serve as particle memory functions, we analyze errors and uncertainties intrinsic in passive microrheology techniques. We show that, despite the fact that certain parameters are essentially unidentifiable on their own, the protocol is remarkably effective in reconstructing G'(ω) and G''(ω) in a range that corresponds to the experimentally observable regime.