We review some advances in the theory of homogeneous, isotropic turbulence. Our emphasis is on the new insights that have been gained from recent numerical studies of the three-dimensional Navier Stokes equation and simpler shell models for turbulence. In particular, we examine the status of multiscaling corrections to Kolmogorov scaling, extended self similarity, generalized extended self similarity, and non-Gaussian probability distributions for velocity differences and related quantities. We recount our recent proposal of a wave-vector-space version of generalized extended self similarity and show how it allows us to explore an intriguing and apparently universal crossover from inertial- to dissipation-range asymptotics.

We present an insightful ‘derivation’ of the Langevin equation and the fluctuation dissipation theorem in the specific context of a heavier particle moving through an ideal gas of much lighter particles. The Newton’s law of motion (mx = F) for the heavy particle reduces to a Langevin equation (valid on a coarser time-scale) with the assumption that the lighter gas particles follow a Boltzmann velocity distribution. Starting from the kinematics of the random collisions we show that (1) the average force 〈F〉 ∞ −x and (2) the correlation function of the fluctuating forceη = F — 〈F〉 is related to the strength of the average force.

During the last phase of cell division in bacteria, a polymeric ring forms at the division site. The ring, made of intracellular proteins, anchors to the cell wall and starts to contract. That initiates a dividing septum to close in, like the shutter of a camera, eventually guillotining the cell into two daughters. All through, the ring remains at the leading edge of the septum and seems to power its closure. It is not understood why does the ring contract. We propose a theoretical model to explain this. It is worth mentioning that a similar contraction phenomenon occurs for the actin ring in eukaryotes, but there it is due to motor proteins, which however, are absent in bacteria.

Intrinsic curvature of biopolymers is emerging as an essential feature in various biological phenomena. Examples of polymers with intrinsic curvature are microtubule in eukaryotic cells or FtsZ filaments in prokaryotic cells. We consider the general model for polymers with intrinsic curvature. We aim to study both equilibrium and dynamic properties of such polymers. Here we report preliminary results on the equilibrium distribution function $P({\mathbf{R}})$ of the end-to-end distance ${\mathbf{R}}$. We employ transfer matrix method for this study.