Two-dimensional random walk and characteristics of helix-coil transition in DNA

On the basis of two-dimensional random walk model, a model of melting of double-stranded DNA is proposed. The expressions for the free energy, the helicity degree and the correlation function of nitrogenous base pair formation, remoted along the chain, are obtained. The temperature dependence of thermodynamic parameters of the model is analyzed. It is shown that the temperature dependence of the helicity degree is qualitatively similar to those obtained in experiments. It is established that the correlation function decays exponentially with increasing distance along the chain between the base pairs.     

A theory of coupled random walk process

A theory of the coupled random walk (CRW) process which has been proposed by one of the present authors, is developed further. It is shown that the Fokker-Planck equation obtained from the CRW process can be cast into a form of a kinetic equation similar to the Boltzmann equation of the gas theory. To this end, the memory effects are taken into account through the jumping probabilities between the modes. As a special case, the distribution function for the steady state is considered.     

A novel identity from random walk theory

A novel expansion of binomial coefficients in terms of trigonometric functions has been obtained by comparing expressions for the time evolution of the probability distribution for a random walker on a ring obtained by separate combinatoric and eigenvalue approaches.     

Path-length dependent flux from random walk theory

Calculation of the flux Ψ(R, s) due to a point isotropic source in an infinite medium, as a function of the distance R and actual pathlength s is treated as a random walk problem. In the case of isotropic scattering, this method is shown to lead to a standard transport-theoretical result, for the integrated flux. However, for Ψ(R, s) far from the source, the random walk theory predicts a modified maxwellian form.     

Phase transition in a four-dimensional random walk with application to medical statistics

A random walk in a piecewise homogeneous medium can exhibit a variety of asymptotic behaviors. In particular, it may lodge strictly in one region or divide in probability among several. This will depend upon the parameters describing (a) the walk, (b) the interregion boundary, and (c) the initial location of the walk. We analyze from this point of view a special four-dimensional walk on an integer lattice with two homogeneous regions separated by a hyperplane of codimension 1. The walk represents a continuing sequence of clinical trials of two drugs of unknown success probabilities and the two regions represent the Bayes-derived criterion as to which drug to try next. The demarcation in the parameter space of success probabilities and initial coordinates between one- and two-region asymptotics is mapped out analytically in several special cases and supporting numerical evidence given in the general case.     

An improved guided random walk algorithm for quantum field theory computations

We introduce an improved guided random walk algorithm for evaluating vacuum matrix elements in hamiltonian field theories. This algorithm does not require the creation of absorption of walks, unlike related random walk techniques which have appeared in the literature. Applications to a scalar field theory in (1 + 1) dimensions and to an abelian gauge theory in (3 + 1) dimensions are described. The algorithm appears to be very fast; we have used it to generate independent abelian gauge field configurations on an 8³ spatial lattice at a rate of about one per IBM 3081 CPU second.     

A long-time tail for random walk in random scenery

Consider a simple random walk on ^d whose sites are colored black or white independently with probabilityq, resp. 1–q. Walk and coloring are independent. Letn _k be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and let _k =E(n _k )–q ^–1. Our main result is a proof that (*) lim _k k ^d/2 _k = (1 –q)q ^{d/2 – 2}(d/2) ^d/2. Since it is known thatq ^{– 1} _k =E(n ₁ n _{k + 1} B) –E(n ₁ B)E(n _{k + 1} B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations of _k (1k100) ind=1, 2, and 3 show that there is an oscillatory behavior of _k for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay of _k cannot be faster than (*).     

Generalised master equation of random walk in a random medium

A generalised formulation of continuous-time random-walks is introduced to study excitation transport in disordered systems containing permanent traps (the localised states). Its exact equivalence with the generalised master equation is established. The exact generalised transport equation obtained has been shown to reduce under special conditions to other random walk equations known in the literature.     

Persistence in random walk in composite media

We consider a class of inhomogeneous media known as composite media that is often encountered in experimental sciences, and investigate the persistence probability of a random walker in such a system. Analytical and numerical results for the crossover time scales are obtained for a composite system with two homogeneous components and three homogeneous components respectively.     

A theory of 1/f current noise based on a random walk model

Asymptotic solutions for the Montroll-Weiss continuous-time random walk that are appropriate to the conduction of carriers through a resistive medium are utilized to show that, when carrier drift occurs by virtue of an applied electric field, a 1/f type of spectral density may be exhibited in the current noise. When the applied field is removed the spectral density is given by Nyquist's theorem.     

The 3-dimensional random walk with applications to overstretched DNA and the protein titin

We study the three-dimensional persistent random walk with drift. Then we develop a thermodynamic model that is based on this random walk without assuming the Boltzmann-Gibbs form for the equilibrium distribution. The simplicity of the model allows us to perform all calculations in closed form. We show that, despite its simplicity, the model can be used to describe different polymer stretching experiments. We study the reversible overstretching transition of DNA and the static force-extension relation of the protein titin.     

Scale-free avalanches in the multifractal random walk

Avalanches, or Avalanche-like, events are often observed in the dynamical behaviour of many complex systems which span from solar flaring to the Earth's crust dynamics and from traffic flows to financial markets. Self-organized criticality (SOC) is one of the most popular theories able to explain this intermittent charge/discharge behaviour. Despite a large amount of theoretical work, empirical tests for SOC are still in their infancy. In the present paper we address the common problem of revealing SOC from a simple time series without having much information about the underlying system. As a working example we use a modified version of the multifractal random walk originally proposed as a model for the stock market dynamics. The study reveals, despite the lack of the typical ingredients of SOC, an avalanche-like dynamics similar to that of many physical systems. While, on one hand, the results confirm the relevance of cascade models in representing turbulent-like phenomena, on the other, they also raise the question about the current state of reliability of SOC inference from time series analysis.     

Random walk in a random multiplicative environment

We investigate the dynamics of a random walk in a random multiplicative medium. This results in a random, but correlated, multiplicative process for the spatial distribution of random walkers. We show how the details of these correlations determine the asymptotic properties of the walk, i.e., the central limit theorem does not apply to these multiplicative processes. We also study a periodic source-trap medium in which a unit cell contains one source, followed byL–1 traps. We calculate the asymptotic behavior of the number of particles, and determine the conditions for which there is growth or decay in this average number. Finally, we discuss the asymptotic behavior of a random walk in the presence of randomly distributed, partially-absoprbing traps. For this case, a temporal regime of purely exponential decay of the density can occur, before the asymptotic stretched exponential decay, exp(–at ^1/3), sets in.     

Loop-erased self-avoiding random walk in two and three dimensions

If(n) is the position of the self-avoiding random walk in ^d obtained by erasing loops from simple random walk, then it is proved that the mean square displacementE(¦(n)¦²) grows at least as fast as the Flory predictions for the usual SAW, i.e., at least as fast asn ^3/2 ford=2 andn ^6/5 ford=3. In particular, if the mean square displacement of the usual SAW grows liken ^1.18... ind=3, as expected, then the loop-erased process is in a different universality class.