Abstract
The notion of non-integer dimension although developed by mathematicians in late 19th – early 20th centuries has found its wide application in physics only in the last few decades. This application was mainly initiated by the pioneering work of B. Mandelbrot who attracted attention to the fact that “fractal” geometry provides an appropriate framework for the description of a whole range of complex structures that are formed from smaller subunits [1]. Whereas such structures are characterized by appropriate fractal dimensions, their growth and spatial correlations [2] are described by a (non-trivial) spectrum of multifractal (MF) dimensions [3,4]. In this paper, we study the properties of diffusion phenomena in the presence of an absorbing polymer. This provides another example of a multifractal phenomenon in condensed matter physics. To this end we use the model proposed by Cates and Witten [5] and derive the MF spectrum in the frames of a field theoretical formalism using the renormalization group [6,7] method. The MF spectrum is related to the spectrum of exponents governing scaling properties of copolymer stars [8]. We calculate this spectrum to the third order of perturbation
Highlights
The notion of non-integer dimension developed by mathematicians in late 19th – early 20th centuries has found its wide application in physics only in the last few decades
We have calculated the spectral function that describes the scaling of the moments of measure defined by diffusion near an absorbing polymer
These moments were calculated as averages over all configurations of the absorber instead of performing a site average
Summary
The notion of non-integer dimension developed by mathematicians in late 19th – early 20th centuries has found its wide application in physics only in the last few decades. Mandelbrot who attracted attention to the fact that “fractal” geometry provides an appropriate framework for the description of a whole range of complex structures that are formed from smaller subunits [1] Whereas such structures are characterized by appropriate fractal dimensions, their growth and spatial correlations [2] are described by a (non-trivial) spectrum of multifractal (MF) dimensions [3,4]. We study the properties of diffusion phenomena in the presence of an absorbing polymer This provides another example of a multifractal phenomenon in condensed matter physics. The MF spectrum is related to the spectrum of exponents governing scaling properties of copolymer stars [8] We calculate this spectrum to the third order of perturbation c Ch. von Ferber, Yu.Holovatch. Ch. von Ferber, Yu.Holovatch theory and report the numerical values of the quantities that characterize the MF behaviour
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