A linear non-Markovian process driven by additive exponentially correlated Gaussian quadratic noise is considered. An exact master equation for a probability distribution of the process is presented. A Kramers-Moyal-type expansion of the master equation is studied. Some limiting cases of the expansion are investigated. It is demonstrated that any finite truncation of the expansion fails. An alternative and correct Fokker-Planck modeling is constructed.

Tribological surgical adjuvants constitute a therapeutic discipline made possible by surgical advances in the treatment of damaged articular cartilage beyond palliative care. The purpose of this study is to analyze interactions between hyaluronic acid and phospholipid molecules, and the formation of geometric forms, that play a role in the facilitated lubrication of synovial joint organ systems. The analysis includes an evaluation of the pathologic state to detail conditions that may be encountered by adjuvants during surgical convalescence. The synovial fluid changes in pH, hyaluronic acid polydispersity, and phospholipid concentration associated with osteoarthritis are presented as features that influence the lubricating properties of adjuvant candidates. Molecular dynamic simulation studies are presented, and the Rouse model is deployed, to rationalize low molecular weight hyaluronic acid behavior in an osteoarthritic environment of increased pH and phospholipid concentration. The results indicate that the hyaluronic acid radius of gyration time evolution is both pH- and phospholipid concentration-dependent. Specifically, dipalmitoylphosphatidylcholine induces hydrophobic interactions in the system, causing low molecular weight hyaluronic acid to shrink and at high concentration be absorbed into phospholipid vesicles. Low molecular weight hyaluronic acid appears to be insufficient for use as a tribological surgical adjuvant because an increased pH and phospholipid concentration induces decreased crosslinking that prevents the formation of supramolecular lubricating forms. Dipalmitoylphosphatidylcholine remains an adjuvant candidate for certain clinical situations. The need to reconcile osteoarthritic phenotypes is a prerequisite that should serve as a framework for future adjuvant design and subsequent tribological testing.

Diffusion of fractal clusters of dimension ${\mathit{d}}_{\mathit{f}}$ in a three-dimensional space is investigated. The diffusion process is assumed to be modeled by a standard parabolic diffusion equation but with a random diffusion coefficient. The motivation for this assumption is provided by two pieces of evidence: (1) the cluster diffusion coefficients depend on the clusters’ masses, sizes, and shapes; (2) the masses of clusters change stochastically in time due to random attachment or detachment of particles. Two models of the growing process are considered: (a) a Poisson process; (b) a simple birth-and-death process with linear rules. The mean square displacement of the cluster mass centers is analyzed and its anomalous behavior is demonstrated as a function of the fractal cluster dimension.