Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics

In this paper, we view fluctuating fronts made of particles on a one-dimensional lattice as an extreme value problem. The idea is to denote the configuration for a single front realization at time $t$ by the set of co-ordinates $\{k_i(t)\}\equiv[k_1(t),k_2(t),...,k_{N(t)}(t)]$ of the constituent particles, where $N(t)$ is the total number of particles in that realization at time $t$. When $\{k_i(t)\}$ are arranged in the ascending order of magnitudes, the instantaneous front position can be denoted by the location of the rightmost particle, i.e., by the extremal value $k_f(t)=\text{max}[k_1(t),k_2(t),...,k_{N(t)}(t)]$. Due to interparticle interactions, $\{k_i(t)\}$ at two different times for a single front realization are naturally not independent of each other, and thus the probability distribution $P_{k_f}(t)$ [based on an ensemble of such front realizations] describes extreme value statistics for a set of correlated random variables. In view of the fact that exact results for correlated extreme value statistics are rather rare, here we show that for a fermionic front model in a reaction-diffusion system, $P_{k_f}(t)$ is Gaussian. In a bosonic front model however, we observe small deviations from the Gaussian.Comment: 6 pages, 3 figures, miniscule changes on the previous version, to appear in Phys. Rev.

Probabilistic Phase Space Trajectory Description for Anomalous Polymer Dynamics

It has been recently shown that the phase space trajectories for the anomalous dynamics of a tagged monomer of a polymer --- for single polymeric systems such as phantom Rouse, self-avoiding Rouse, Zimm, reptation, and translocation through a narrow pore in a membrane; as well as for many-polymeric system such as polymer melts in the entangled regime --- is robustly described by the Generalized Langevin Equation (GLE). Here I show that the probability distribution of phase space trajectories for all these classical anomalous dynamics for single polymers is that of a fractional Brownian motion (fBm), while the dynamics for polymer melts between the entangled regime and the eventual diffusive regime exhibits small, but systematic deviations from that of a fBm.Comment: 8 pages, two figures, 3 eps figure files, minor changes, supplementary material included moved to the appendix, references expanded, to appear in J. Phys.: Condens. Matte

Pore-blockade Times for Field-Driven Polymer Translocation

We study pore blockade times for a translocating polymer of length $N$, driven by a field $E$ across the pore in three dimensions. The polymer performs Rouse dynamics, i.e., we consider polymer dynamics in the absence of hydrodynamical interactions. We find that the typical time the pore remains blocked during a translocation event scales as $\sim N^{(1+2\nu)/(1+\nu)}/E$, where $\nu\simeq0.588$ is the Flory exponent for the polymer. In line with our previous work, we show that this scaling behaviour stems from the polymer dynamics at the immediate vicinity of the pore -- in particular, the memory effects in the polymer chain tension imbalance across the pore. This result, along with the numerical results by several other groups, violates the lower bound $\sim N^{1+\nu}/E$ suggested earlier in the literature. We discuss why this lower bound is incorrect and show, based on conservation of energy, that the correct lower bound for the pore-blockade time for field-driven translocation is given by $\eta N^{2\nu}/E$, where $\eta$ is the viscosity of the medium surrounding the polymer.Comment: 14 pages, 6 figures, slightly shorter than the previous version; to appear in J. Phys.: Cond. Ma

Field dependent collision frequency of the two-dimensional driven random Lorentz gas

In the field-driven, thermostatted Lorentz gas the collision frequency increases with the magnitude of the applied field due to long-time correlations. We study this effect with computer simulations and confirm the presence of non-analytic terms in the field dependence of the collision frequency as predicted by kinetic theory.Comment: 6 pages, 2 figures. Submitted to Phys. Rev.

Long-time-tail Effects on Lyapunov Exponents of a Random, Two-dimensional Field-driven Lorentz Gas

We study the Lyapunov exponents for a moving, charged particle in a two-dimensional Lorentz gas with randomly placed, non-overlapping hard disk scatterers placed in a thermostatted electric field, $\vec{E}$. The low density values of the Lyapunov exponents have been calculated with the use of an extended Lorentz-Boltzmann equation. In this paper we develop a method to extend these results to higher density, using the BBGKY hierarchy equations and extending them to include the additional variables needed for calculation of Lyapunov exponents. We then consider the effects of correlated collision sequences, due to the so-called ring events, on the Lyapunov exponents. For small values of the applied electric field, the ring terms lead to non-analytic, field dependent, contributions to both the positive and negative Lyapunov exponents which are of the form ${\tilde{\epsilon}}^{2} \ln\tilde{\epsilon}$, where $\tilde{\epsilon}$ is a dimensionless parameter proportional to the strength of the applied field. We show that these non-analytic terms can be understood as resulting from the change in the collision frequency from its equilibrium value, due to the presence of the thermostatted field, and that the collision frequency also contains such non-analytic terms.Comment: 45 pages, 4 figures, to appear in J. Stat. Phy

Asymptotic Scaling of the Diffusion Coefficient of Fluctuating "Pulled" Fronts

We present a (heuristic) theoretical derivation for the scaling of the diffusion coefficient $D_f$ for fluctuating ``pulled'' fronts. In agreement with earlier numerical simulations, we find that as $N\to\infty$, $D_f$ approaches zero as $1/\ln^3N$, where $N$ is the average number of particles per correlation volume in the stable phase of the front. This behaviour of $D_f$ stems from the shape fluctuations at the very tip of the front, and is independent of the microscopic model.Comment: Some minor algebra corrected, to appear in Rapid Comm., Phys. Rev.

Pathway dependence in redox driven metal–organic gels

Pathway dependence is common in self‐assembly. Here, we show the importance of pathway dependence for redox‐driven gels by constructing a Fe(II)/Fe(III) redox‐based metal‐organic gel system. We show that in situ oxidation of the Fe(II) ions at different rates results in conversion of a Fe(II)‐gel into a Fe(III)‐organic gel, which controls the material properties such as gel stiffness, gel strength, and an unusual swelling behaviour. The rate of formation of Fe(III) ions determines the extent of intermolecular interactions and so whether gelation or precipitation occurs

Chemically fuelled self-regulating gel-to-gel transition

Artificial self‐regulating materials can be prepared by exploiting fuel‐driven pathways. Dynamic covalent bonds are formed and broken reversibly under mild reaction conditions. Herein, we utilise this concept to programme a system that can undergo a fuel‐driven self‐regulated gel‐to‐gel transition. The reaction between the gelator and the fuel resulted in a change in chemical structure of the gelator that initially causes a transition from a solution to gel state by co‐assembly. With time, the intermediate complex collapses, re‐forming the gelator structure. However, the gel does not collapse. This method allows us to prepare gels with improved mechanical strength. Unlike conventional gel‐to‐gel transitions, exploitation of dynamic covalent chemistry provides an opportunity to access materials that cannot be prepared directly under similar final conditions

Controlling hydrogel properties by tuning non-covalent interactions in a charge complementary multicomponent system

Mixing small molecule gelators is a promising route to prepare useful and exciting materials that cannot be accessed from any of the individual components. Here, we describe pH-triggered hydrogelation by mixing of two non-gelling amphiphiles. The intermolecular interactions among the molecules can be tuned either by controlling the degree of ionization of the components or by a preparative pathway, which enables us to control material properties such as gel strength, gel stiffness, thermal stability, and an unusual shrinking/swelling behaviour