Polymer dynamics in time-dependent Matheron-de Marsily flows: an exactly solvable model

Dynamics of generalized Gaussian polymeric structures in random layered flows

We develop a formalism for the dynamics of a flexible branched polymer with arbitrary topology in the presence of random flows. This is achieved by employing the generalized Gaussian structure (GGS) approach and the Matheron-de Marsily model for the random layered flow. The expression for the average square displacement (ASD) of the center of mass of the GGS is obtained in such flow. The averaging is done over both the thermal noise and the external random flow. Although the formalism is valid for branched polymers with various complex topologies, we mainly focus here on the dynamics of the flexible star and dendrimer. We analyze the effect of the topology (the number and length of branches for stars and the number of generations for dendrimers) on the dynamics under the influence of external flow, which is characterized by their root-mean-square velocity, persistence flow length, and flow exponent α. Our analysis shows two anomalous power-law regimes, viz., subdiffusive (intermediate-time polymer stretching and flow-induced diffusion) and superdiffusive (long-time flow-induced diffusion). The influence of the topology of the GGS is unraveled in the intermediate-time regime, while the long-time regime is only weakly dependent on the topology of the polymer. With the decrease in the value of α, the magnitude of the ASD decreases, while the temporal exponent of the ASD increases in both the time regimes. Also there is an increase in both the magnitude of the ASD and the crossover time (from the subdiffusive to the superdiffusive regime) with an increase in the total mass of the polymeric structure.

From Langevin to generalized Langevin equations for the nonequilibrium Rouse model

We investigate the nature of the effective dynamics and statistical forces obtained after integrating out nonequilibrium degrees of freedom. To be explicit, we consider the Rouse model for the conformational dynamics of an ideal polymer chain subject to steady driving. We compute the effective dynamics for one of the many monomers by integrating out the rest of the chain. The result is a generalized Langevin dynamics for which we give the memory and noise kernels and the effective force, and we discuss the inherited nonequilibrium aspects.

Persistence of a Rouse polymer chain under transverse shear flow

We consider a single Rouse polymer chain in two dimensions in the presence of a transverse shear flow along the x direction and calculate the persistence probability P0(t) that the x coordinate of a bead in the bulk of the chain does not return to its initial position up to time t. We show that the persistence decays at late times as a power law P0(t) approximately t{-theta} with a nontrivial exponent theta. The analytical estimate of theta=0.359... obtained using an independent interval approximation is in excellent agreement with the numerical value theta approximately 0.360+/-0.001.

Persistence of randomly coupled fluctuating interfaces

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h(1) and h(2) , respectively, each growing over a d -dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h(2) , however, is coupled to h(1) via a quenched random velocity field. In the limit d-->0 , our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t(0) -->infinity , the stochastic process h(2) , at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H2 =1- beta(1) /2 , where beta(1) is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be theta(2)(s) =1- H2 = beta(1) /2 . These analytical results are verified by numerical simulations.

Persistence of a particle in the Matheron-de Marsily velocity field

We show that the longitudinal position x(t) of a particle in a (d+1)-dimensional layered random velocity field (the Matheron-de Marsily model) can be identified as a fractional Brownian motion (fBm) characterized by a variable Hurst exponent H(d)=1-d/4 for d<2 and H(d)=1/2 for d>2. The fBm becomes marginal at d=2. Moreover, using the known first-passage properties of fBm we prove analytically that the disorder averaged persistence [the probability of no zero crossing of the process x(t) up to time t], has a power-law decay for large t with an exponent theta=d/4 for d<2 and theta=1/2 for d>/=2 (with logarithmic correction at d=2), results that were earlier derived by Redner based on heuristic arguments and supported by numerical simulations.