Diffusion on lattices with traps: A self-consistent calculation

The behavior of random walkers on a lattice with randomly distributed traps is investigated. Specifically, we use the coherent potential approximation to solve the master equation for nearest-neighbor hopping. The results of this calculation are compared with recent CPA calculations for the diffusion equation with randomly placed spherical traps in a continuum.     

Survival probabilities in three-dimensional random walks

We evaluate the survival probability for random walkers on lattices doped with traps. We consider nearest-neighborstep walks and walks mediated bi multipolar interactions, for which the probability of steps of length r is proportional to r^?s. The decay law due to trapping is calculated for the diamond, simple cubic and body- and face-centered cubic lattices. We establish validity domains of approximate expressions.     

Revisiting random walks in fractal media: on the occurrence of time discrete scale invariance

This paper addresses the kinetic behavior of random walks in fractal media. We perform extensive numerical simulations of both single and annihilating random walkers on several Sierpinski carpets, in order to study the time behavior of three observables: the average number of distinct sites visited by a single walker, the mean-square displacement from the origin, and the density of annihilating random walkers. We found that the time behavior of those observables is given by a power law modulated by soft logarithmic-periodic oscillations. We conjecture that logarithmic-periodic oscillations are a manifestation of a time domain discrete scale iNvariance (DSI) that occurs as a consequence of the spatial DSI of the substrate. Our conjecture implies that the logarithmic periods of oscillations in space and time domains are linked by a dynamic exponent z, through z=log(tau)/log(b(1)), where tau and b(1) are the fundamental scaling ratios of the DSI symmetry in the time and space domains, respectively. We use this relationship in order to compute z for different observables and fractals. Furthermore, we check the values obtained with independent measurements provided by the power-law behavior of the mean-square displacement with time [R(2)(t) proportional variant t(2/z)]. The very good agreement obtained between both computations of the z exponent gives strong support to the idea of an intimate interplay between spatial and time symmetry properties that we expect will have a quite general scope. We expect that the application of the outlined concepts in the field of dynamic processes in fractal media will stimulate further research.     

Density and energy relaxation in an open one-dimensional system

A new master equation to mimic the dynamics of a collection of interacting random walkers in an open system is proposed and solved numerically. In this model, the random walkers interact through excluded volume interaction (single-file system); and the total number of walkers in the lattice can fluctuate because of exchange with a bath. In addition, the movement of the random walkers is biased by an external perturbation. Two models for the latter are considered: (1) an inverse potential (V proportional, variant 1/r), where r is the distance between the center of the perturbation and the random walker and (2) an inverse of sixth power potential (V proportional, 1/r6). The calculated density of the walkers and the total energy show interesting dynamics. When the size of the system is comparable to the range of the perturbing field, the energy relaxation is found to be highly nonexponential. In this range, the system can show stretched exponential (e-(t/taus)beta) and even logarithmic time dependence of energy relaxation over a limited range of time. Introduction of density exchange in the lattice markedly weakens this nonexponentiality of the relaxation function, irrespective of the nature of perturbation.     

Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.     

Osmotic pressure and polymer size in semidilute polymer solutions under good-solvent conditions

We consider the lattice Domb-Joyce model at a value of the coupling for which scaling corrections approximately vanish and determine the universal scaling functions associated with the osmotic pressure and the polymer size for semidilute polymer solutions (c/c( *)相似文献   

Parallel search of long circular strands: modeling, analysis, and optimization

We introduce and explore a model of an ensemble of agents searching, in parallel, a long circular strand for a target site. The agents performing the search combine local-scanning (conducted by a one-dimensional motion along the strand) and random relocations on the strand. The agent-ensemble search-durations are analyzed, their limiting probability distributions are obtained in closed-form, and the optimal relocation strategies are derived. The results encompass the cases of parallel and massively parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation durations. The results are applicable to a wide spectrum of local-scans, including linear motions, Brownian motions, subdiffusive motions, fractional Brownian motions, and fractional Lévy motions.     

Self avoiding walk trees and laces

This paper is divided into two sections. In the first section we describe the use depth first search trees and breadth first search trees in understanding self avoiding walks and restricted random walks. While we derive our results for Euclidean lattices the method is totally general and can be used for other lattices as well. In the second section derive an expression for the number of laces in the lace expansion of a memory four restricted random walk.     

Calculation of the entropy of random coil polymers with the hypothetical scanning Monte Carlo method

Hypothetical scanning Monte Carlo (HSMC) is a method for calculating the absolute entropy S and free energy F from a given MC trajectory developed recently and applied to liquid argon, TIP3P water, and peptides. In this paper HSMC is extended to random coil polymers by applying it to self-avoiding walks on a square lattice--a simple but difficult model due to strong excluded volume interactions. With HSMC the probability of a given chain is obtained as a product of transition probabilities calculated for each bond by MC simulations and a counting formula. This probability is exact in the sense that it is based on all the interactions of the system and the only approximation is due to finite sampling. The method provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation. HSMC is independent of existing techniques and thus constitutes an independent research tool. The HSMC results are compared to those obtained by other methods, and its application to complex lattice chain models is discussed; we emphasize its ability to treat any type of boundary conditions for which a reference state (with known free energy) might be difficult to define for a thermodynamic integration process. Finally, we stress that the capability of HSMC to extract the absolute entropy from a given sample is important for studying relaxation processes, such as protein folding.     

Polymers confined between two parallel plane walls

Single three-dimensional polymers confined to a slab, i.e., to the region between two parallel plane walls, are studied by Monte Carlo simulations. They are described by N-step walks on a simple cubic lattice confined to the region 1< or = z < or = D. The simulations cover both regions DRF (where RF approximately Nnu is the Flory radius, with nu approximately 0.587), as well as the cross-over region in between. Chain lengths are up to N=80 000, slab widths up to D=120. In order to test the analysis program and to check for finite size corrections, we actually studied three different models: (a) ordinary random walks (mimicking Theta polymers); (b) self-avoiding walks; and (c) Domb-Joyce walks with the self-repulsion tuned to the point where finite size corrections for free (unrestricted) chains are minimal. For the simulations we employ the pruned-enriched-Rosenbluth method with Markovian anticipation. In addition to the partition sum (which gives us a direct estimate of the forces exerted onto the walls), we measure the density profiles of monomers and of end points transverse to the slab, and the radial extent of the chain parallel to the walls. All scaling laws and some of the universal amplitude ratios are compared to theoretical predictions.     

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.     

Wavelet-accelerated Monte Carlo sampling of polymer chains

We introduce a new method for coarse-graining polymer chains, based on the wavelet transform, a multiresolution data analysis technique. This method, which assigns a cluster of particles to a coarse-grained bead located at the center of mass of the cluster, reduces the complexity of the problem significantly by dividing the simulation into several stages, each with a small fraction of the number of beads in the overall chain. At each stage, we compute the distributions of coarse-grained internal coordinates as well as potential functions required for subsequent simulation stages. We show that, with this wavelet-accelerated Monte Carlo method, coarse-grained Gaussian and self-avoiding random walks can reproduce results obtained from atomistic simulations to a high degree of accuracy in orders of magnitude less time. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 897–910, 2005     

Application of second-order Møller-Plesset perturbation theory with resolution-of-identity approximation to periodic systems

Efficient periodic boundary condition (PBC) calculations by the second-order M?ller-Plesset perturbation (MP2) method based on crystal orbital formalism are developed by introducing the resolution-of-identity (RI) approximation of four-center two-electron repulsion integrals (ERIs). The formulation and implementation of the PBC RI-MP2 method are presented. In this method, the mixed auxiliary basis functions of the combination of Poisson and Gaussian type functions are used to circumvent the slow convergence of the lattice sum of the long-range ERIs. Test calculations of one-dimensional periodic trans-polyacetylene show that the PBC RI-MP2 method greatly reduces the computational times as well as memory and disk sizes, without the loss of accuracy, compared to the conventional PBC MP2 method.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

Adsorption of comb copolymers on weakly attractive solid surfaces

In this work continuum and lattice Monte Carlo simulation methods are used to study the adsorption of linear and comb polymers on flat surfaces. Selected polymer segments, located at the tips of the side chains in comb polymers or equally spaced along the linear polymers, are attracted to each other and to the surface via square-well potentials. The rest of the polymer segments are modeled as tangent hard spheres in the continuum model and as self-avoiding random walks in the lattice model. Results are presented in terms of segment-density profiles, distribution functions, and radii of gyration of the adsorbed polymers. At infinite dilution the presence of short side chains promotes the adsorption of polymers favoring both a decrease in the depletion-layer thickness and a spreading of the polymer molecule on the surface. The presence of long side chains favors the adsorption of polymers on the surface, but does not permit the spreading of the polymers. At finite concentration linear polymers and comb polymers with long side chains readily adsorb on the solid surface, while comb polymers with short side chains are unlikely to adsorb. The simple models of comb copolymers with short side chains used here show properties similar to those of associating polymers and of globular proteins in aqueous solutions, and can be used as a first approximation to investigate the mechanism of adsorption of proteins onto hydrophobic surfaces.     

Unbiased expectation values from diffusion quantum Monte Carlo simulations with a fixed number of walkers

We append forward walking to a diffusion Monte Carlo algorithm which maintains a fixed number of walkers. This removes the importance sampling bias of expectation values of operators which do not commute with the Hamiltonian. We demonstrate the effectiveness of this approach by employing three importance sampling functions for the hydrogen atom ground state, two very crude. We estimate moments of the electron-nuclear distance, static polarizabilities, and high-order hyperpolarizabilites up to the fourth power in the electric field, where no use is made of the finite field approximation. The results agree with the analytical values, with a statistical error which increases substantially with decreasing overlap of the guiding function with the exact wave function.     

Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics

Recently, we have introduced a new method, metadynamics, which is able to sample rarely occurring transitions and to reconstruct the free energy as a function of several variables with a controlled accuracy. This method has been successfully applied in many different fields, ranging from chemistry to biophysics and ligand docking and from material science to crystal structure prediction. We present an important development that speeds up metadynamics calculations by orders of magnitude and renders the algorithm much more robust. We use multiple interacting simulations, walkers, for exploring and reconstructing the same free energy surface. Each walker contributes to the history-dependent potential that, in metadynamics, is an estimate of the free energy. We show that the error on the reconstructed free energy does not depend on the number of walkers, leading to a fully linear scaling algorithm even on inexpensive loosely coupled clusters of PCs. In addition, we show that the accuracy and stability of the method are much improved by combining it with a weighted histogram analysis. We check the validity of our new method on a realistic application.