Paradoxal Diffusion in Chemical Space for Nearest-Neighbor Walks over Polymer Chains

Abstract

We consider random walks over polymer chains (modeled as simple random walks or self-avoiding walks) and allow from each polymer site jumps to all Euclidean (not necessarily chemical) neighboring sites. For frozen chain configurations the distribution of displacements (DD) of a walker along the chain shows a paradoxal behavior: The DD's width (interquartile distance) grows with time as L~t a , with a ഠ 0.5, but the DD displays large power-law tails. For annealed configurations the DD is a Lévy distribution and its width is strongly superdiffusive.[S0031-9007 (97)03619-3] PACS numbers: 61.41. + e, 05.40. + j, 05.60. + wRandom walks in random environments occur in many fields of modern statistical physics and have won major attention in the last two decades; see Refs. [1][2][3]. Many situations lead to random walks on random walks (RW) or on self-avoiding random walks (SAW), see Refs. [4][5][6][7][8][9][10][11][12]; in the polymer case such structures depict chains in u solvents (RW) or in good solvents (SAW). The problem arises very naturally in the analysis of energy or exciton transport over polymer chains. For simplicity we view a moving entity (say electron, exciton, or enzyme) which performs steps of length a from one neighboring site (monomer) to another one. A triplet exciton, for instance, can move from one donor to another one close by, not necessarily sequentially placed along the chain. Hence not only steps along the chain but also steps between monomers which are close to each other in (Euclidean) space are involved, even if they are far apart in chemical space (where the distance l ch between two points is defined through the number of monomers n between them along the chain's backbone l ch n 1 1). Viewed in terms of the regular Euclidean distance some properties of random walks on SAW chains (e.g., the spectral dimension and the anomalous diffusion exponent) were evaluated in Refs. [4][5][6][7][8][9][10][11][12]. To our knowledge up to now, no particular attention was paid to the corresponding properties from the viewpoint of the chemical distance; here we show that this viewpoint leads to highly unusual behaviors.We start from the findings for frozen chain configurations in an experimentally relevant 3D situation, where we simulate nearest-neighbor random walks over RW and over SAW chains embedded in a simple cubic lattice. We first generate and save 100 chain configurations. The RW chains are generated as trajectories of a random walker on a simple cubic lattice, and their length is L 30 000. The SAW chains, of length L 10 000, were obtained using the pivot algorithm [13], as described in Ref. [14]. Each of the different realizations was created from an independent initial configuration to which the pivot transformation was applied 2000 times. Then for each configuration we start the walk at a random position and check at each step how many moves are possible from the actual position of the walker. The moves allowed are jumps to all monomers occupying nearest-neighbor positions (either along the ch...

One-dimensional stochastic Lévy-Lorentz gas

“…In this context it is interesting to recall that Sokolov et al [20] have shown that correlations between jumps in a Lévy flight in a chemical space destroy the Lévy statistics of the walk.…”

Small-world networks: Links with long-tailed distributions

“…On the other hand other questions are envisageable over such structures, for example random transport [14]; this requires solving diffusion-type problems, which are mathematically described by the Laplacian on the structure [16] and the corresponding eigenvalues and eigenvectors [17]. Examples of such problems are anomalous transport of charges and of excitations over networks [18]. Most recent SWN-studies center on a one-dimensional chain supplemented with additional links (AL), which connect sites that are arbitrarily far from each other on the underlying lattice.…”

“…Most recent SWN-studies center on a one-dimensional chain supplemented with additional links (AL), which connect sites that are arbitrarily far from each other on the underlying lattice. While this being the simplest SWN envisageable, there are situations in which links between distant sites occur naturally; however, their lengths are then not necessarily uniformly distributed: Considering a polymer chain in solution, monomers which are far apart along the backbone can be quite close to each other in real space, so that for instance energy transfer over the structure may take crosscuts along sites near to each other in space [18]. Now the probability of having such close monomer pairs is related to the return to the origin of random walks, possibly under self-avoiding constraints.…”

Diffusion Processes on Power-Law Small-World Networks

“…It was also recently argued [7] that "wiring-cost" considerations for spatially embedded networks, such as cortical networks [9] or on-chip logic networks [10], can generate such power-law-suppressed link-length distribution. A physical example of such a power-law SW network arises in diffusion on a randomly folded polymer discussed below [11,12].The addition of random long-range links to a regular d-dimensional network, producing a SW network, leads in many cases to a crossover to mean-field-like behavior [13], effectively becoming equivalent to averaging over the long-range links in an annealed fashion. Here we focus on systems where it is not the case, and the contrast between the quenched and annealed systems is strong [5,14].…”

“…One case in which the diffusion equation arises is a macromolecule randomly moving along a polymer chain jumping over adjacent segments with nonzero probability [11]; if the macromolecule motion is fast compared to rearrangements of the links, then the network may be considered as quenched [11,15], while if the macromolecule motion is slow compared to the link rearrangements then the network is annealed [11,12]. The equation describing random walk processes is…”