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At least 181 records · Page 10Linked to original sources
PubMed2004

Phase transition in random walks with long-range correlations.

Motivated by recent results in the theory of correlated sequences, we analyze the dynamics of random walks with long-term memory (binary chains with long-range correlations). In our model, the probability for a unit bit in a binary string depends on the fraction of unities preceding it. We show that the system undergoes a dynamical phase transition from normal diffusion, in which the variance D(L) scales as the string's length L, into a superdiffusion phase ( D(L) approximately Lalpha,alpha>1), when the correlation strength exceeds a critical value. We demonstrate the generality of our results with respect to alternative models, and discuss their applicability to various data, such as coarse-grained DNA sequences, written texts, and financial data.

Journal Article↗
PubMed2005

Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations.

The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

Journal Article↗
PubMed1992

Estimating genomic distance from DNA sequence location in cell nuclei by a random walk model.

The folding of chromatin in interphase cell nuclei was studied by fluorescent in situ sequences chromatin according to a random walk model. This model provides the basis for calculating the spacing of sequences along the linear DNA molecule from interphase distance measurements. An interphase mapping strategy based on this model was tested with 13 probes from a 4-megabase pair (Mbp) region of chromosome 4 containing the Huntington disease locus. The results confirmed the locations of the probes and showed that the remaining gap in the published maps of this region is negligible in size. Interphase distance measurements should facilitate construction of chromosome maps with an average marker density of one per 100 kbp, approximately ten times greater than that achieved by hybridization to metaphase chromosome. achieved by hybridization to metaphase chromosomes.

Cell Nucleus↗
PubMed1980

Asymptotic form for random walk survival probabilities on three-dimensional lattices with traps.

The problem of calculating statistics of time-to-trapping of a random walker on a trap-filled lattice is of interest in solid state physics. Several authors have suggested approximate methods for calculating the average survival probabilities. Here, an exact asymptotic form for the probability that an n step random walk visits S(n) distinct sites is used to ascertain the validity of a simple approximation suggested by Rosenstock. For trap concentrations below 0.05, the relative error in using Rosenstock's approximation is less than 10%.

Journal Article↗
PubMed2000

Scaling transformation of random walk distributions in a lattice.

We use a decimation procedure in order to obtain the dynamical renormalization group transformation (RGT) properties of random walk distribution in a 1+1 lattice. We obtain an equation similar to the Chapman-Kolmogorov equation. First we show the existence of invariants through the RGT. We also show the existence of functions which are semi-invariants through the RGT. Second, we show as well that the distribution R(q)(x)=[1+b(q-1)x(2)](1/(1-q)) (q>1), which is an exact solution of a nonlinear Fokker-Planck equation, is a semi-invariant for RGT. We obtain the map q(')=f(q) from the RGT and we show that this map has two fixed points: q=1, attractor, and q=2, repellor, which are the Gaussian and the Lorentzian, respectively. We show the connections between these result and the Levy flights.

Journal Article↗
PubMed2004

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation.

A detailed study is presented for a large class of uncoupled continuous-time random walks. The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.

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PubMed1988

Number of visits to a state in a random walk, before absorption, and related topics.

Equations are derived for the probability of n visits to a given state during the course of a random walk on a finite diagram that starts from a specified state and ends with absorption. By deriving the mean number of visits in two different ways, certain conjectures or theorems are encountered that connect properties of different but related diagrams in an interesting way. Other subjects included are (i) number of one-way transitions between two states before absorption; (ii) time dependence of the rate of cycle completions before absorption; and (iii) the relation of this work to the "return process" of Karlin and Taylor.

Absorption↗
PubMed2000

Molecular switching with nonexponential relaxation patterns: a random walk approach.

The transition from an initial, locally stable configuration to a globally stable state in molecular switches is investigated in terms of a random walk model, effectively taking the reaction pathway through a potentially rugged energy landscape into account. Exponential and nonexponential scenarios are discussed and the implications on measurable quantities are explored.

Journal Article↗
PubMed2005

Multipoint correlation functions for continuous-time random walk models of anomalous diffusion.

Recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous-time random walk (CTRW) in an external potential. Two- and three-point correlation functions are calculated for waiting-time distributions with an anomalous power-law profile t(-alpha-1), 0 < alpha < 1, on intermediate time scales with a crossover to an exponential long time decay. Comparison of the CTRW with the Brownian harmonic oscillator model (Gaussian process) illustrates how higher-order correlation functions may be used to distinguish between dynamical models that have the same two-point correlation function.

Colloids↗
PubMed2006

Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations.

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term. Results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term. We have also developed further extensions of the CTRW model to include more general reaction dynamics.

Journal Article↗
PubMedDate not supplied

Application of continuous time random walk theory to tracer test measurements in fractured and heterogeneous porous media.

We examine a set of analytical solutions based on the continuous time random walk (CTRW) approach, which can be evaluated numerically and used to analyze breakthrough data from tracer tests. Practical application of these solutions, with discussion of the physical meaning of the relevant model parameters, is emphasized. The CTRW theory accounts for the often observed non-Fickian (or scale-dependent) dispersion behavior that cannot be properly quantified by using the advection-dispersion equation. The solutions given here, valid for a wide range of dispersive behaviors of conservative tracers, and useful for both characterization and prediction, have been integrated into a library of external functions for use with the GRACE graphical display and analysis package. Example applications of these solutions are presented. The library and graphics software are freely accessible from a Web site.

Chemical Phenomena↗
PubMed2002

Continuous-time random walks and traveling fronts.

We present a geometric approach to the problem of propagating fronts into an unstable state, valid for an arbitrary continuous-time random walk with a Fisher-Kolmogorov-Petrovski-Piskunov growth/reaction rate. We derive an integral Hamilton-Jacobi type equation for the action functional determining the position of reaction front and its speed. Our method does not rely on the explicit derivation of a differential equation for the density of particles. In particular, we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non-Markovian random processes.

Journal Article↗
PubMed2004

Comparison of random-walk density functional theory to simulation for bead-spring homopolymer melts.

Density profiles for a homopolymer melt near a surface are calculated using a random-walk polymeric density functional theory, and compared to results from molecular dynamics simulations. All interactions are of a Lennard-Jones form, for both monomer-monomer interactions and surface-monomer interactions, rather than the hard core interactions which have been most investigated in the literature. For repulsive systems, the theory somewhat overpredicts the density oscillations near a surface. Nevertheless, near quantitative agreement with simulation can be obtained with an empirical scaling of the direct correlation function. Use of the random phase approximation to treat attractive interactions between polymer chains gives reasonable agreement with simulation of dense liquids near neutral and attractive surfaces.

Journal Article↗
PubMed2004

Quantitative phase diagrams of branching and annihilating random walks.

We demonstrate the full power of nonperturbative renormalization group methods for nonequilibrium situations by calculating the quantitative phase diagrams of simple branching and annihilating random walks and checking these results against careful numerical simulations. Specifically, we show, for the [see text] case, that an absorbing phase transition exists in dimensions d=1 to 6 and argue that mean-field theory is restored not in d=3, as suggested by previous analyses, but only in the limit d--> infinity.

Journal Article↗
PubMed2005

The subtle nature of financial random walks.

We first review the most important "stylized facts" of financial time series, that turn out to be, to a large extent, universal. We then recall how the multifractal random walk of Bacry, Muzy, and Delour generalizes the standard model of financial price changes and accounts in an elegant way for many of their empirical properties. In a second part, we provide empirical evidence for a very subtle compensation mechanism that underlies the random nature of price changes. This compensation drives the market close to a critical point, that may explain the sensitivity of financial markets to small perturbations, and their propensity to enter bubbles and crashes. We argue that the resulting unpredictability of price changes is very far from the neoclassical view that markets are informationally efficient.

Journal Article↗
PubMed2001

Efficient, multiple-range random walk algorithm to calculate the density of states.

We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy, and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape.

Journal Article↗
PubMed2001

Critical branching-annihilating random walk of two species.

The effect of blocking between different species occurring in one dimension is investigated here numerically in the case of particles following branching and annihilating random walk. It is shown that two-dimensional simulations confirm the field theoretical results with logarithmic corrections. In one dimension, however, if particles exhibit hard core interaction I confirm the very recent predictions of Kwon et al. [Phys. Rev. Lett. 85, 1682 (2000)] that there are two different universality classes depending on the spatial symmetry of the offspring production characterized by beta(S)=0.5 and beta(A)=2. Elaborate analysis of simulation data shows that the order parameter exponent beta does not depend on initial conditions or on diffusion rates of species but strong correction to scaling is observed. By systematic numerical simulations the critical point properties have been explored and initial condition dependence of the dynamical exponents Z and alpha is shown. In the case of a random initial state the particle-density decay at the critical point follows the t(-1/4) law with logarithmic corrections with two offsprings.

Journal Article↗