We study the role of active coupling on the transport properties of homogeneously charged macromolecules in an infinitely dilute solution. An enzyme becomes actively bound to a segment of the macromolecule, exerting an electrostatic force on it. Eventually, thermal fluctuations cause it to become unbound, introducing active coupling into the system. We study the mean-squared displacement (MSD) and find a new scaling regime compared to the thermal counterpart in the presence of hydrodynamic and segment-segment electrostatic interactions. Furthermore, the study of segment-segment equal-time correlation reveals the swelling of the macromolecule. Further, we derive the concentration equation of the macromolecule with active binding and study how the cooperative diffusivity of the macromolecules get modified by its environment, including the macromolecules itself. It turns out that these active fluctuations enhance the effective diffusivity of the macromolecules. The derived closed-form expression for diffusion constant is pertinent to the accurate interpretation of light scattering data in multi-component systems with binding-unbinding equilibria.

We study the mean first-passage time of a one-dimensional active fluctuating membrane that is stochastically returned to the same flat initial condition at a finite rate. We start with a Fokker-Planck equation to describe the evolution of the membrane coupled with an Ornstein-Uhlenbeck type of active noise. Using the method of characteristics, we solve the equation and obtain the joint distribution of the membrane height and active noise. In order to obtain the mean first-passage time (MFPT), we further obtain a relation between the MFPT and a propagator that includes stochastic resetting. The derived relation is then used to calculate it analytically. Our studies show that the MFPT increases with a larger resetting rate and decreases with a smaller rate, i.e., there is an optimal resetting rate. We compare the results in terms of MFPT of the membrane with active and thermal noises for different membrane properties. The optimal resetting rate is much smaller with active noise compared to thermal. When the resetting rate is much lower than the optimal rate, we demonstrate how the MFPT scales with resetting rates, distance to the target, and the properties of the membranes.

Integrin is an important transmembrane receptor protein which remodels the actin network and anchors the cell membrane towards the extracellular matrix via mechanochemical pathways. The clustering of specific lipids and lipid-anchored proteins, which is essential for a certain type of endocytosis process, is facilitated at integrin-mediated active regions. To study this, we propose a minimal exactly solvable model which includes the interplay of stochastic shuttling between integrin on and off states with the intrinsic dynamics of the membrane. We propose a two-step mechanism in which the integrin induces an aster-like arrangement in the actin network, followed by clustering of lipids in that region. We obtain an analytic expression for the deformation and local membrane velocity, and thereby the evolution of clustering mediated by a single integrin. The deformation evolves nonmonotonically and its dependence on the stochastic shuttling timescales and membrane properties is elucidated. Our estimates of the area of the deformed region and the number of lipids in it indicate strong clustering.

We investigate the role of active coupling on the transport properties of the macromolecules. The active coupling comes due to bound enzymes with a segment of the macromolecule wherein the enzyme exerts an electrostatic force on the segment of the macromolecule, and eventually, it gets unbound due to the thermal fluctuations. This binding and unbinding process generates active fluctuations in the dynamics of the macromolecule. Starting with segment dynamics and correlations for three dynamical models with active coupling, we obtain the cooperative diffusivity for the realistic charged macromolecules with hydrodynamics. First, we construct the three models by incorporating the features of a real polymer systematically, starting from simple Rouse dynamics with active coupling. We further include segment-segment interactions and in addition, hydrodynamic interactions with active coupling. Our obtained scaling form for segment-segment correlations for the models in terms of the size exponent of the polymer indicating that hydrodynamic and segment-segment interactions along with the active coupling lead to new scaling regimes. We finally study the dynamics of a homogeneously charged flexible polymer in an infinitely dilute solution where enzymes and counterions affect the dynamics of the polymers. We analytically investigate how these active fluctuations affect the coupled dynamics of the polymer and counterions. It turns out that these active fluctuations enhance the effective diffusivity of the polymer. The derived closed-form expression for diffusivity is pertinent to accurate interpretation of light scattering data on multi-component systems with binding-unbinding equilibria.

How should dispersal strategies be chosen to increase the likelihood of survival of a species? We obtain the answer for the spatially extended versions of three well-known models of two competing species with unequal diffusivities. Though identical at the mean-field level, the three models exhibit drastically different behaviour leading to different optimal strategies for survival, with or without a selective advantage for one species. With conserved total particle number, dispersal has no effect on survival probability. With a fluctuating number, faster dispersal is advantageous if intra-species competition is present, while moving slower is the optimal strategy for the disadvantaged species if there is no intra-species competition: it is imperative to include fluctuations to properly formulate survival strategies.

We obtain the behavior of skewness of the longitudinal velocity gradient with respect to the Reynolds number of a turbulent fluid from the incompressible Navier?Stokes dynamics. This is achieved in a renormalized scheme incorporating a von Karman interpolation function between the production and inertial ranges together with an interpolation between the inertial and dissipation ranges. The corresponding second- and third-order renormalized moments of the longitudinal velocity gradient and the resulting skewness values are found to depend on the Reynolds number in nontrivial fashions. For intermediate and moderately high Reynolds numbers, the skewness value does not exhibit any scaling behavior with respect to the Reynolds number. Instead, the skewness value remains bounded for any Reynolds number and it approaches a constant value for infinitely high Reynolds numbers. In the same scheme, we also incorporate another interpolation function between the production and inertial ranges coming from energy balance equation and inputs from renormalization-group analysis. We find exact agreement between the asymptotic values of skewness at high Reynolds numbers while the skewness values are in the same neighborhood in the finite Reynolds number (non-universal) regime.

We show that a scaling approach successfully characterizes clustering and intermittency in space and time, in systems of noninteracting particles driven by fluctuating surfaces. We study both the steady state and the approach to it, for passive particles sliding on one-dimensional Edwards-Wilkinson or Kardar-Parisi-Zhang surfaces, with particles moving either along or against the growth direction in the latter case. Extensive numerical simulations are supplemented by analytical results for a sticky slider model in which particles coalesce when they meet. Results for single particle displacement versus time show to what extent particle dynamics is slaved to the surface, while scaling properties of the probability distribution of the separation of two particles have important implications for replica symmetry breaking for a pair of trajectories. For the many-particle system, clustering in steady state is studied via moments of particle number fluctuations in a single stretch, revealing different degrees of spatial multiscaling with different driving. Temporal intermittency in steady state is established by showing that the scaled flatness diverges. Finally we consider the approach to the steady state, and study both the flatness and the evolution of equal-time correlation functions, as in coarsening of phase ordering systems. Our studies give clear evidence for a simple scaling description of the approach to steady state, with a diverging length scale. An investigation of aging properties reveals that flatness is nonmonotonic in time with two distinct branches, and that a scaling description holds for each one.

How does a steady state with strong intermittency develop in time from an initial state which is statistically random? For passive sliders driven by various fluctuating surfaces, we show that the approach involves an indefinitely growing length scale which governs scaling properties. A simple model of sticky sliders suggests scaling forms for the time-dependent flatness and hyperflatness, both measures of intermittency and these are confirmed numerically for passive sliders driven by a Kardar-Parisi-Zhang surface. Aging properties are studied via a two-time flatness. We predict and verify numerically that the time-dependent flatness is, remarkably, a nonmonotonic function of time with different scaling forms at short and long times. The scaling description remains valid when clustering is more diffuse as for passive sliders evolving through Edwards-Wilkinson driving or under antiadvection, although exponents and scaling functions differ substantially.

We apply a renormalized perturbative scheme to the Navier-Stokes equation for an incompressible isotropic turbulent velocity field. This allows us to obtain the renormalized expressions for second-and third-order cumulants of the velocity derivative directly from the corresponding Feynman diagrams. The resulting expressions are integrated numerically by excluding and including the dissipation range assuming Kolmogorov and Pao's phenomenological expressions for the energy spectrum. The ensuing values for skewness are found to be S = -0.647 (when the dissipation range is excluded) and S = -0.682 (when the dissipation is included). These estimated values are compared with various experimental, numerical and theoretical results.

The phenomenon of stochastic growth of a surface on a two-dimensional substrate occurs in Nature in a variety of circumstances and its statistical characterization requires the study of higher order cumulants. Here, we consider the statistical cumulants of height fluctuations governed by the (2 + 1)-dimensional KPZ equation for flat geometry. We follow a diagrammatic scheme to derive the expressions for renormalized cumulants up to fourth order in the stationary state. Assuming a value for the roughness exponent from reliable numerical predictions, we calculate the second, third and fourth cumulants, yielding skewness S = 0.2879 and kurtosis Q = 0.1995. These values agree well with the available numerical estimations.

We study the (1 + 1)-dimensional Villain, Lai, and Das Sarma (VLDS) equation driven by a Gaussian white noise and implement a renormalization scheme without rescaling at one-loop order. Using a diagrammatic method, we calculate the renormalized second and third moments in the large-scale and long-time limits. The ensuing skewness value is S = -0.0441. This (negative) value is consistent with the numerical prediction of Das Sarma et al (1996 Phys. Rev. E 53 359).

We study the fourth order normalized cumulant of height fluctuations governed by $1+1$ dimensional Kardar-Parisi-Zhang (KPZ) equation for a growing surface. Following a diagrammatic renormalization scheme, we evaluate the kurtosis $Q$ from the connected diagrams leading to the value $Q=0.1523$ in the large-scale long-time limit.

We use the (1+1)-dimensional Kardar-Parisi-Zhang equation driven by a Gaussian white noise and employ the dynamic renormalization-group of Yakhot and Orszag without rescaling [J. Sci. Comput. 1, 3 (1986)]. Hence we calculate the second- and third-order moments of height distribution using the diagrammatic method in the large-scale and long-time limits. The moments so calculated lead to the value S=0.3237 for the skewness. This value is comparable with numerical and experimental estimates.