Vladimir V. Palyulin

Levy flights do not always optimize random blind search for sparse targets.

Significance Has natural selection led to adaptations of Lévy flight foraging, as stated on the respective Wikipedia page? Random walks with scale-free jump length distributions were indeed shown to optimize the search for sparse targets as supported by extensive movement data of many animal species and humans. Here we demonstrate that small variations of the search conditions strongly modify these claims: In the presence of a bias, underwater currents for sea predators or winds for airborne searchers, a Lévy searcher easily overshoots the target, and Brownian strategies become advantageous. Even in the absence of a bias, there exist conditions for which a Brownian strategy may effect faster target localization. Our results show clear limitations for the universality of Lévy flight foraging. It is generally believed that random search processes based on scale-free, Lévy stable jump length distributions (Lévy flights) optimize the search for sparse targets. Here we show that this popular search advantage is less universal than commonly assumed. We study the efficiency of a minimalist search model based on Lévy flights in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space) based on two different optimization criteria with respect to minimal search time and search reliability (cumulative arrival probability). Although Lévy flights turn out to be efficient search processes when the target is far from the starting point, or when relative to the starting point the target is upstream, we show that for close targets and for downstream target positioning regular Brownian motion turns out to be the advantageous search strategy. Contrary to claims that Lévy flights with a critical exponent α = 1 are optimal for the search of sparse targets in different settings, based on our optimization parameters the optimal α may range in the entire interval (1, 2) and especially include Brownian motion as the overall most efficient search strategy.

Microphase separation in melts of double comb copolymers

Within the random-phase approximation, the conditions providing instability of a spatially homogeneous state of melts of double comb copolymers relative to microphase separation have been studied. As compared with the case of diblock copolymer melt connectivity between branch points of linear diblock copolymers and a backbone (densely grafted comb copolymer) is expected to assist the transition from the spatially homogeneous state to the microphase separated state; in other words, this factor decreases the incompatibility parameter at the transition point. Two characteristic types of spinodal behavior as a function of the number of repeating units have been found: one type is observed when the fragment of backbone between neighboring branch points is long; another type of behavior takes place when this distance is small. In one of the cases under study, the period of microstructure abruptly changes with variation of the interaction parameters of the system.

Nematic ordering of rigid rod polyelectrolytes induced by electrostatic interactions: Effect of discrete charge distribution along the chain

Similar to the Debye-Hückel plasma, charged groups in solutions of rigid rod polyelectrolytes attract each other. We derive expression for the correlation free energy of electrostatic attraction of the rods within the random phase approximation. In this theory, we explicitly take into account positions of charged groups on the chains and examine both charge and polymer concentration fluctuations. The correlation free energies and the osmotic pressures are calculated for isotropic and completely ordered nematic phase. The results of the discrete model are compared with results of a continuous model. The discrete model gives rise to a stronger attraction between the charged groups both in the isotropic and nematic phases and to a stronger orienting action of the electrostatic forces.

Surface induced self-organization of comb-like macromolecules

Summary We present a review of the theoretical and experimental evidence for the peculiar properties of comb copolymers, demonstrating the uniqueness of these materials among other polymer architectures. These special properties include an increase in stiffness upon increasing side-chain length, the spontaneous curvature of adsorbed combs, rod–globule transition, and specific intramolecular self-assembly. We also propose a theory of chemically heterogeneous surface nanopattern formation in ultrathin films of comblike macromolecules containing two different types (A and B) of incompatible side chains (so-called binary combs). Side chains of the binary combs are strongly adsorbed on a surface and segregated with respect to the backbone. The thickness of surface domains formed by the B side chains is controlled by the interaction with the substrate. We predict the stability of direct and inverse disc-, torus- and stripelike nanostructures. Phase diagrams of the film are constructed.

Space-fractional Fokker?Planck equation and optimization of random search processes in the presence of an external bias

Based on the space-fractional Fokker–Planck equation with a δ-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker–Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index α = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for α may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.

Comparison of pure and combined search strategies for single and multiple targets

Abstract We address the generic problem of random search for a point-like target on a line. Using the measures of search reliability and efficiency to quantify the random search quality, we compare Brownian search with Lévy search based on long-tailed jump length distributions. We then compare these results with a search process combined of two different long-tailed jump length distributions. Moreover, we study the case of multiple targets located by a Lévy searcher.

Interpretation of the Vibrational Spectra of Glassy Polymers Using Coarse-Grained Simulations

The structure and vibrational density of states (VDOS) of polymer glasses are investigated using numerical simulations based on the classical Kremer–Grest bead–spring model. We focus on the roles of chain length and bending stiffness, the latter being set by imposing three-body angular potentials along chain backbones. Upon increasing the chain length and bending stiffness, structural reorganization leads to volumetric expansion of the material and buildup of internal stresses. The VDOS has two dominant bands: a low-frequency one corresponding to inter- and intrachain nonbonding interactions and a high-frequency one corresponding principally to vibrations of bonded beads that constitute skeletal chain backbones. Upon increasing the steepness of the angular potential, vibrational modes associated with chain bending gradually move from the low-frequency to the high-frequency band. This redistribution of modes is reflected in a reduction of the so-called Boson peak upon increasing chain stiffness. Remarkably...

Speeding up the first-passage for subdiffusion by introducing a finite potential barrier

We show that for a subdiffusive continuous time random walk with scale-free waiting time distribution the first-passage dynamics on a finite interval can be optimized by introduction of a piecewise linear potential barrier. Analytical results for the survival probability and first-passage density based on the fractional Fokker?Planck equation are shown to agree well with Monte Carlo simulations results. As an application we discuss an improved design for efficient translocation of gradient copolymers compared to homopolymer translocation in a quasi-equilibrium approximation.

How a finite potential barrier decreases the mean first-passage time

We consider the mean first-passage time of a random walker moving in a potential landscape on a finite interval, the starting and end points being at different potentials. From analytical calculations and Monte Carlo simulations we demonstrate that the mean first-passage time for a piecewise linear curve between these two points is minimized by the introduction of a potential barrier. Due to thermal fluctuations, this barrier may be crossed. It turns out that the corresponding expense for this activation is less severe than the gain from an increased slope towards the end point. In particular, the resulting mean first-passage time is shorter than for a linear potential drop between the two points.