Two-dimensional Copolymers and Multifractality: Comparing Perturbative Expansions, MC Simulations, and Exact Results

We analyze the scaling laws for a set of two different species of long flexible polymer chains joined together at one of their extremities (copolymer stars) in space dimension D=2. We use a formerly constructed field-theoretic description and compare our perturbative results for the scaling exponents with recent conjectures for exact conformal scaling dimensions derived by a conformal invariance technique in the context of D=2 quantum gravity. A simple MC simulation brings about reasonable agreement with both approaches. We analyse the remarkable multifractal properties of the spectrum of scaling exponents.Comment: 5 page

Fractal transit networks: self-avoiding walks and L\'evy flights

Using data on the Berlin public transport network, the present study extends previous observations of fractality within public transport routes by showing that also the distribution of inter-station distances along routes displays non-trivial power law behaviour. This indicates that the routes may in part also be described as L\'evy-flights. The latter property may result from the fact that the routes are planned to adapt to fluctuating demand densities throughout the served area. We also relate this to optimization properties of L\'evy flights.Comment: 7 pages, 3 figures, style files included. Submitted to the topical issue 'From Brownian motion to self-avoiding walks and L\'evy flights' of the journal 'EPJ - Special Topics

Copolymer Networks and Stars: Scaling Exponents

We explore and calculate the rich scaling behavior of copolymer networks in solution by renormalization group methods. We establish a field theoretic description in terms of composite operators. Our 3rd order resummation of the spectrum of scaling dimensions brings about remarkable features: The special convexity properties of the spectra allow for a multifractal interpretation while preserving stability of the theory. This behavior could not be found for power of field operators of usual $\phi^4$ field theory. The 2D limit of the mutually avoiding walk star apparently corresponds to results of a conformal Kac series. Such a classification seems not possible for the 2D limit of other copolymer stars. We furthermore provide a consistency check of two complementary renormalization schemes: epsilon expansion and renormalization at fixed dimension, calculating a large collection of independent exponents in both approaches.Comment: 30 pages, revtex, figures: 5 latex, 1 postscript. New Figs. Text improved. Citations adde

Disorder effects on the static scattering function of star branched polymers

We present an analysis of the impact of structural disorder on the static scattering function of f-armed star branched polymers in d dimensions. To this end, we consider the model of a star polymer immersed in a good solvent in the presence of structural defects, correlated at large distances r according to a power law \sim r^{-a}. In particular, we are interested in the ratio g(f) of the radii of gyration of star and linear polymers of the same molecular weight, which is a universal experimentally measurable quantity. We apply a direct polymer renormalization approach and evaluate the results within the double \varepsilon=4-d, \delta=4-a-expansion. We find an increase of g(f) with an increasing \delta. Therefore, an increase of disorder correlations leads to an increase of the size measure of a star relative to linear polymers of the same molecular weight.Comment: 17 pages, 7 figure

Scaling in public transport networks

We analyse the statistical properties of public transport networks. These networks are defined by a set of public transport routes (bus lines) and the stations serviced by these. For larger networks these appear to possess a scale-free structure, as it is demonstrated e.g. by the Zipf law distribution of the number of routes servicing a given station or for the distribution of the number of stations which can be visited from the chosen one without changing the means of transport. Moreover, a rather particular feature of the public transport network is that many routes service common subsets of stations. We discuss the possibility of new scaling laws that govern intrinsic features of such subsets.Comment: 9 pages, 4 figure

Shapes of macromolecules in good solvents: field theoretical renormalization group approach

In this paper, we show how the method of field theoretical renormalization group may be used to analyze universal shape properties of long polymer chains in porous environment. So far such analytical calculations were primarily focussed on the scaling exponents that govern conformational properties of polymer macromolecules. However, there are other observables that along with the scaling exponents are universal (i.e. independent of the chemical structure of macromolecules and of the solvent) and may be analyzed within the renormalization group approach. Here, we address the question of shape which is acquired by the long flexible polymer macromolecule when it is immersed in a solvent in the presence of a porous environment. This question is of relevance for understanding of the behavior of macromolecules in colloidal solutions, near microporous membranes, and in cellular environment. To this end, we consider a previously suggested model of polymers in d-dimensions [V. Blavats'ka, C. von Ferber, Yu. Holovatch, Phys. Rev. E, 2001, 64, 041102] in an environment with structural obstacles, characterized by a pair correlation function h(r), that decays with distance r according to a power law: h(r) \sim r-a. We apply the field-theoretical renormalization group approach and estimate the size ratio / and the asphericity ratio \hat{A}_d up to the first order of a double \epsilon=4-d, \delta=4-a expansion.Comment: 20 pages, 5 figure