A model of concentrated polymers described by generalized random walks

A model for concentrated polymer system is proposed. Dynamics of the chain segments is described by a discrete Langevin equation. The Langevin equation has the same structure as the recursion relation for the generalized random walks (GRW). Motions of the chain segments described by the GRW have two different types of jumping probabilities: rapidly varying parts and slowly varying parts. Based on a postulate for the recursion relation, an interplay between the motions arisen from these characteristic behaviors is given explicitly and studied.     

Path integrals for Fokker-Planck equation described by generalized random walks

Path integral representations for Fokker-Planck (FP) equations, described by the random walks (RW) and the generalized random walks (GRW), are given. The GRW is a generalized one from the usual random walks to study non-linear, non-equilibrium processes. The GRW includes some memory effects and couplings through the jumping probabilities. To derive the path integrals of the processes, a transformation of probability, scalings of site (space) and step (time) are performed on the GRW. By a function in the exponent of the path integrals for the FP equation obtained by the RW or the GRW, a Lagrangian giving most probable path is introduced. From the Lagrangian, an effective Hamiltonian is deduced.     

First passage time problems for one-dimensional random walks

This note contains a development of the theory of first passage times for one-dimensional lattice random walks with steps to nearest neighbor only. The starting point is a recursion relation for the densities of first passage times from the set of lattice points. When these densities are unrestricted, the formalism allows us to discuss first passage times of continuous time random walks. When they are negative exponential densities we show that the resulting equation is the adjoint of the master equation. This is the lattice analog of a correspondence well known for systems describable by a Fokker-Planck equation. Finally we discuss first passage problems for persistent random walks in which at each step the random walker continues in the same direction as the preceding step with probability a or reverses direction with probability 1–     

Adsorption of a flexible self-avoiding polymer chain: Exact results on fractal lattices

The large-scale behavior of surface-interacting self-avoiding polymer chains placed on finitely ramified fractal lattices is studied using exact recursion relations. It is shown how to obtain surface susceptibility critical indices and how to modify a scaling relation for these indices in the case of fractal lattices. We present the exact results for critical exponents at the point of adsorption transition for polymer chains situated on a class of Sierpinski gasket-type fractals. We provide numerical evidence for a critical behavior of the type found recently in the case of bulk self-avoiding random walks at the fractal to Euclidean crossover.     

A Relativistic Version of the Ghirardi–Rimini–Weber Model

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times.     

Classical spin in a potential field

We consider an ensemble of restricted discrete random walks in 2+1 dimensions. The restriction on the walks is such as to given particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schrödinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.     

On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

We consider the relation between so called continuous localization models—i.e. non-linear stochastic Schrödinger evolutions—and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.     

Second Order Integrability Conditions for Difference Equations: An Integrable Equation

Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws are discussed. In a generic case, some of these conditions yield nonlocal conservation laws. A new integrable equation satisfying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3 × 3 Lax representation. Finally, via the relation of the symmetries of this equation to the Bogoyavlensky lattice, an integrable asymmetric quad equation and a consistent pair of difference equations are derived.     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

Accessibility in complex networks

This Letter describes a method for the quantification of the diversity of non-linear dynamics in complex networks as a consequence of self-avoiding random walks. The methodology is analyzed in the context of theoretical models and illustrated with respect to the characterization of the accessibility in urban streets.     

The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK _ij =A+B(i+j)+C _ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN _k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN _kl, also determined by a recursion relation. The physical meaning ofN _kl and the recursion relation is given. A method to get explicit expressions forN _k andN _kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.     

An age-dependent approach to random walks on disordered lattices

We derive a new approach for the stochastic transport in random systems, starting from a phenomenological master equation with random transition rates. Our method combines the effective medium approximation with age-dependent dynamics. Within the framework of our approximation, the static disorder may be described by means of a system of age-dependent master equations. For translationally invariant systems which obey certain separability conditions, the approach is equivalent with the continuous time random walk theory. Moreover, for self-avoiding random walks our effective medium approximation is exact. For non self-avoiding random walks, the approximation neglects the correlations between successive transitions leading to closed paths on the lattice.     

Modes of a Gaussian Random Walk

It is demonstrated that a one-dimensional gaussian random walk (GRW) possesses an underlying structure in the form of random oscillatory modes. These modes are not sinusoids, but can be isolated by a well-defined procedure. They have average wavelengths and amplitudes, both of which can be determined by experiments or by theoretical calculations. This paper reports such determinations by both methods and also develops a theory that is ultimately shown to agree with experiments. Both theory and simulations show that the average wavelength and the average amplitude scale with the order of the mode in exactly the same way that the modes of the well-known Weierstrass fractal scale with mode order. This is remarkable since the wave generated by the Weierstrass function, , is fully determined for the variable x whereas the GRW is stochastic. By increasing the size of the steps in the GRW, it is possible to selectively remove the fastest modes, while leaving the remaining modes almost unchanged. For a GRW, the parameters corresponding to a and g in the Weierstrass function are found to be 2.0 and 4.0, respectively. These values are independent of the variance associated with the GRW. Application of the random modes is reserved for a later paper.     

Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces

We derive a fractional Fokker-Planck equation for subdiffusion in a general space- and time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.     

On Continuous-Time Self-Avoiding Random Walk in Dimension Four

We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks.     

Quantum Random Walks and Thermalisation

It is shown how to construct quantum random walks with particles in an arbitrary faithful normal state. A convergence theorem is obtained for such walks, which demonstrates a thermalisation effect: the limit cocycle obeys a quantum stochastic differential equation without gauge terms. Examples are presented which generalise that of Attal and Joye (J Funct Anal 247:253–288, 2007).     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.     

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.     

A generalized simple random walk in one dimension related to the Gaussian polynomials

A generalization of the relation between the simple random walk on a regular lattice and the diffusion equation in a continuous space is described. In one dimension we consider a random walk of a walker with exponentially decreasing mobility with respect to time. It has an exact solution of the conditional probability, that is expressed in terms of the Gaussian polynomials, a generalization of binomial coefficients. Taking a suitable continuum limit we obtain the corresponding transport equation from the recursion relation of the discrete random walk process. The kernel of this differential equation is also directly obtained from that conditional probability by the same continuum limit.