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Monte Carlo simulations of self-avoiding walks on the percolation cluster

Final Activity Report Summary - SAWS ON FRACTAL (Monte Carlo simulations of self-avoiding walks on the percolation cluster)

The configurational properties of long polymer macromolecules possess characteristics which are universal, i.e. independent of the details of the chemical structure of both the polymer itself and the solvent. These properties are perfectly described within a model of a self-avoiding walk (SAW) on a regular lattice.

The question of how linear polymers behave in disordered media has not only academic interest, but is also relevant for understanding transport properties of polymer chains in porous media, such as enhanced oil recovery, gel electrophoresis, gel permeation chromatography, etc. In connection to this problem, new challenges have been recently raised in studying the protein folding problem in a cellular environment. Cells can be described as a very crowded environment since they are composed by many different kinds of biomolecules which may occupy a large fraction of the total volume. This condition leads to a phenomenon called 'volume exclusion'.

The scaling properties of a SAW in disordered media remain a subject which has many principal questions still unsettled, hence our main goal was to contribute to this subject. In the present project, we were interested in the special case where the disordered lattice was exactly at the percolation threshold and a SAW resided on the percolation cluster that had a fractal structure. For numerous years the percolation model served as a paradigm to study structural disorder. In this way, the study of a SAW on percolation clusters was of importance for the description of various phenomena in which the statistical properties of SAWs on a disordered structure played a role.

To get a comprehensive overview, we studied the universal behaviour of SAWs on percolation clusters in two, three and four dimensions by computer simulations, applying the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results brought about the estimates of the so-called critical exponents, governing the scaling laws of configurational properties of polymer macromolecules in porous media.

When studying physical processes on complicated fractal objects, one often encounters the interesting situation of coexistence of a family of singularities, each associated with a set of different fractal dimensions. These peculiarities are usually referred to as multifractality. The multifractal spectrum can be used to provide information on the subtle geometrical properties of a fractal object, which cannot be fully described by its fractal dimensionality. Multifractal properties arise in many different contexts, for example in studies of turbulence in chaotic dynamical systems and strange attractors, human heartbeat dynamics, Anderson localisation transition, etc. Both a SAW and a percolation cluster are among the most frequently encountered examples of fractals in condensed matter physics, thus a SAW on a percolation cluster is a good candidate to possess multifractal behaviour.

Through the application of numerical simulations, we could indeed show that a whole multifractal spectrum of singularities emerged when exploring the peculiarities of the model. We obtained estimates for the set of critical exponents that governed scaling laws of moments of the distribution of percolation cluster sites visited by SAWs. Our results supported recent approximate analytical field-theoretical studies. With the tool box which was developed so far we were ideally equipped to tackle the molecular crowding problem of biological cells in a future project.