Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Multiscale derivation of an augmented Smoluchowski equation

Smoluchowski and Fokker-Planck equations for the stochastic dynamics of order parameters have been derived previously. The question of the validity of the truncated perturbation series and the initial data for which these equations exist remains unexplored. To address these questions, we take a simple example, a nanoparticle in a host medium. A perturbation parameter ε, the ratio of the mass of a typical atom to that of the nanoparticle, is introduced and the Liouville equation is solved to O(ε²). Via a general kinematic equation for the reduced probability W of the location of the center-of-mass of the nanoparticle, the O(ε²) solution of the Liouville equation yields an equation for W to O(ε³). An augmented Smoluchowski equation for W is obtained from the O(ε²) analysis of the Liouville equation for a particular class of initial data. However, for a less restricted assumption, analysis of the Liouville equation to higher order is required to obtain closure.     

Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion

A generalised random walk scheme for random walks in an arbitrary external potential field is investigated. From this concept which accounts for the symmetry breaking of homogeneity through the external field, a generalised master equation is constructed. For long-tailed transfer distance or waiting time distributions we show that this generalised master equation is the genesis of apparently different fractional Fokker-Planck equations discussed in literature. On this basis, we introduce a generalisation of the Kramers-Moyal expansion for broad jump length distributions that combines multiples of both ordinary and fractional spatial derivatives. However, it is shown that the nature of the drift term is not changed through the existence of anomalous transport statistics, and thus to first order, an external potential Φ(x) feeds back on the probability density function W through the classical term ∝/ x (x)W(x, t), i.e., even for Lévy flights, there exists a linear infinitesimal generator that accounts for the response to an external field. Received 30 June 2000 and Received in final form 12 November 2000     

Angular distribution calculation based on the exciton model taking account of the influence of the Fermi motion and the Pauli principle

An exact closed form solution to the time-integrated master equation of the exciton model is applied to the calculation of the angular distribution for both the preequilibrium and equilibrium decays of the neutron-induced reaction. The distribution probability of two-nucleon collision fromΩ toΩ′ based on the Fermi gas model and the influence of the Fermi motion and the Pauli principle on the shape of the angular distribution are studied in detail. We have concluded that the influence of these effects on the shape of the angular distribution is rather significant for reactions in the energy range of several tens of MeV. As an example to compare with the experiments we have calculated the neutron-induced reaction⁹³Nb(n,n′) atE _n =15 MeV. It seems that the most significant improvement of the present approach is the rise of the backward direction of the double differential cross section for the higher energy emitted neutrons.     

A fluctuation problem in particle transport theory II

A balance equation is formulated for the probability that a particle injected into an infinite, amorphous medium will have suffered N collisions and have given rise to n new particles in a given energy range at time t. The method of regeneration points has been employed and this leads, in the case of two particle production, to a non-linear, integro-differential equation for the probability generating function. This equation is solved for the case of foreign particles slowing down, in which case it becomes linear and results are obtained which include the effects of electronic stopping and absorption, thus generalizing the work in part I. In the cascade problem, a single particle gives rise to two new particles in every collision and it is shown, for a simple hard-sphere model with 1/v scattering and absorption, how the non-linear equation may be solved. The probability for the number of particles and the number of collisions suffered to absorption is obtained in the case of zero absorption, the probability law is shown to obey a Furry distribution. The limitations of the method described in part I for dealing with cascades are highlighted.     

Transport and percolation in disordered systems—A self-consistent time-local approach

A new self-consistent equation for the transport of excitations in disordered systems, which forms the basis for a new class of time-domain coherent potential approximations, is developed. As an example, we calculate the probability of remaining in the original site G₀(t) as well as the second moment of the distribution of excitations r²(t) for a random mixture of donors which satisfy a master equation with short-range transition rates. A percolation-type transition is observed and its characteristics are analyzed both above and below the transition point.Alfred P. Sloan fellow, Camille and Henry Dreyfus teacher-scholar.     

Anomalous effects ine + e − annihilation into boson pairs I.e + e −→W + W −

We present a phenomenological analysis of the various kinds of anomalous effects which can apper either in the leptonic or in the bosonic sector of the reactionse ⁺ e ^?→V V′(V V′=W ^±,Z ⁰ or γ). A general descritpin of invariant amplitudes and helicity amplitudes is given. Applications are made with non standar couplings, new intermediate states, residuale ⁺ e ^? VV′ contact interactions. In this first part results are given fore ⁺ e ^?→W ⁺ W ^? in view of LEP 200 and future linear colliders in the TeV range.     

Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of Lévy-stable type and admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function ρ(x, t). Our main goal is to demonstrate a compatibility of a direct solution method (an explicit, albeit numerically assisted, integration of the master equation) with an indirect pathwise procedure, recently proposed in [Physica A 392, 3485, (2013)] as a valid tool for a dynamical analysis of non-Langevin jump-type processes. The path-wise method heavily relies on an accumulation of large sample path data, that are generated by means of a properly tailored Gillespie’s algorithm. Their statistical analysis in turn allows to infer the dynamics of ρ(x, t). However, no consistency check has been completed so far to demonstrate that both methods are fully compatible and indeed provide a solution of the same dynamical problem. Presently we remove this gap, with a focus on potential deficiencies (various cutoffs, including those upon the jump size) of approximations involved in simulation routines and solutions protocols.     

Time evolution of the degree distribution of model A of random attachment growing networks

For random growing networks, Barabás and Albert proposed a kind of model in Barabás et al. [Physica A 272 (1999) 173], i.e. model A. In this paper, for model A, we give the differential format of master equation of degree distribution and obtain its analytical solution. The obtained result P(k, t) is the time evolution of degree distribution. P(k, t) is composed of two terms. At given finite time, one term decays exponentially, the other reflects size effect. At infinite time, the degree distribution is the same as that of Barabás and Albert. In this paper, we also discuss the normalization of degree distribution P(k, t) in detail.     

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp <Emphasis Type="BoldItalic">(−φ(η))</Emphasis>-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (?φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.     

Microscopic calculation of the surface absorption in the optical potential

We calculated the contributions of the two-particle one-hole (2p-1h) and the one-particle two-hole (1p-1h) excitations to the imaginary partW(E, R) of then-⁴⁰Ca optical potential. The bound single particle states and energies of the⁴⁰Ca nucleus are calculated quantum mechanically by solving the Schrödinger equation with a Woods-Saxon potential. For the excited states in the continuum we use the Thomas-Fermi approximation. Different forms of contact residual interactions have been tested. A combination of aδ-force and a smeared SDI can fit the phenomenologicalW(E, R).     

Impact parameter dependence of the probability for removal of the 2s and 2p atomic electrons by nuclei

The probability W(x) of inner shell ionization by nuclei as a function of the impact parameter x is expressed through the ionization matrix element M_if. Values of W(x) for removal of the 2s and 2p electrons are calculated with the first Born M_if and compared with the SCA result.     

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation (u/q _i )²+V(q)=E consists of finding new curvilinear coordinates x _i (q) in which the transformed equation admits a complete separated solution u(x)=u _(i)(x _i ;). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.     

Quantum state engineering in ion-traps via adiabatic passage

We propose two relatively robust schemes to generate entangled W states of three (or generally N) ions in ion trap systems by using adiabatic passage technique and appropriately designed ion-field couplings in a single step. In the first scheme, we apply the N-pod fractional stimulated Raman adiabatic passage (F-STIRAP) technique to generate W state of N ions using two Gaussian laser pulses. We also show that the W state of N ? 1 ions can be created via a simple N-pod standard STIRAP by two laser pulses. In the second scheme, we generate the entangled state of N ions via ??-pulse technique by a single laser pulse. We also study the population transfer of the system by numerical solutions of the master equation, considering the effect of decoherence channels due to laser intensity fluctuations and dissipation in the phonon modes.     

A semi-classical model of multi-step direct and compound nuclear reactions

A semi-classical model of multi-step direct and compound nuclear reactions is proposed to describe the angular distributions of particles emitted from the inelastic scattering induced by a nucleon with an energy of several tens of MeV. The energy-angle correlation is exactly taken into account for the first few steps of the collision process (multi-step direct process) and the generalized master equation is employed for the following stages of collision process, using the energy-averaged kernelG(Ω → Ω′) (multicompound process). The calculations for¹⁹⁷Au(p, p′),¹²⁰Sn(p, p′) and⁹³Nb(n, n′) show that the model can rather nicely reproduce the experimental data of double-differential cross sections.     

Robustness of Greenberger-Horne-Zeilinger and W states for teleportation in external environments

By solving analytically a master equation in the Lindblad form, we study quantum teleportation of the one-qubit state under the influence of different surrounding environments, and compared the robustness between Greenberger-Horne-Zeilinger (GHZ) and W states in terms of their teleportation capacity. The results revealed that when subject to zero temperature environment, the GHZ state is always more robust than the W state, while the reverse situation occurs when the channel is subject to infinite temperature or dephasing environment.     

Embedding theorem for weighted Sobolev classes with weights that are functions of the distance to some h-set

The paper continues the first part (Russ. J. Math. Phys. 20 (3), 360–373). Let Ω be a John domain, let Γ ? ?Ω be an h-set, and let g and υ be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class W _p,g ^r (Ω) is continuously embedded in the space L _q,v (Ω). Moreover, bounds for the approximation of functions in W _p,g ^r (Ω) by polynomials of degree not exceeding r ? 1 in L _q,v ( $\tilde \Omega $ ) are found, where $\tilde \Omega $ is a subdomain generated by a subtree of the tree T defining the structure of Ω.     

The algebra of differential operators on the circle and W KP (q)

Radul has recently introduced a map from the Lie algebra of differential operators on the circle of W _n . In this Letter, we extend this map to W _KP ^(q) , a recently introduced one-parameter deformation of W_KP - the second Hamiltonian structure of the KP hierarchy. We use this to give a short proof that W is the algebra of additional symmetries of the KP equation.     

A matrix integral solution to two-dimensionalW p-gravity

Thep ^th Gel'fand-Dickey equation and the string equation [L, P]=1 have a common solution τ expressible in terms of an integral overn×n Hermitean matrices (for largen), the integrand being a perturbation of a Gaussian, generalizing Kontsevich's integral beyond the KdV-case; it is equivalent to showing that τ is a vacuum vector for aW _?p ⁺ , generated from the coefficients of the vertex operator. This connection is established via a quadratic identity involving the wave function and the vertex operator, which is a disguised differential version of the Fay identity. The latter is also the key to the spectral theory for the two compatible symplectic structures of KdV in terms of the stress-energy tensor associated with the Virasoro algebra. Given a differential operator $$L = D^p + q_2 (t) D^{p - 2} + \cdots + q_p (t), with D = \frac{\partial }{{dx}},t = (t_1 ,t_2 ,t_3 ,...),x \equiv t_1 ,$$ consider the deformation equations¹ (0.1) $$\begin{gathered} \frac{{\partial L}}{{\partial t_n }} = [(L^{n/p} )_ + ,L] n = 1,2,...,n + - 0(mod p) \hfill \\ (p - reduced KP - equation) \hfill \\ \end{gathered} $$ ofL, for which there exists a differential operatorP (possibly of infinite order) such that (0.2) $$[L,P] = 1 (string equation).$$ In this note, we give a complete solution to this problem. In section 1 we give a brief survey of useful facts about theI-function τ(t), the wave function Ψ(t,z), solution of ?Ψ/?t _n=(L ^n/p) _x Ψ andL ^1/pΨ=zΨ, and the corresponding infinitedimensional planeV ⁰ of formal power series inz (for largez) $$V^0 = span \{ \Psi (t,z) for all t \in \mathbb{C}^\infty \} $$ in Sato's Grassmannian. The three theorems below form the core of the paper; their proof will be given in subseuqent sections, each of which lives on its own right.     

Problèmes mathématiques de l'équation de Boltzmann relativiste

This work deals with relativistic Boltzmann equation and more particulary with integral operator of complete equation and integral operator of linearized equation. These operators depend on the differential cross sectionh(〈p, q〉, cos θ) which is a fonction of energy 〈p, q〉 and of the deviation angle θ. The only hypothesis is thath is a symetric function of cosθ. The second part deals essentially with linearized equation in Special Relativity. We take for the distribution function: $$F\left( {x,p} \right) = a e^{ - \frac{{\lambda p}}{2}} \left( {e^{ - \frac{{\lambda p}}{2}} + \varepsilon f\left( {x,p} \right)} \right)$$ wherea is a constant, λ a constant vector and ? a small constant so that ?² can be neglected. We obtain the equation: $$\frac{{p^\alpha }}{{p^0 }}\frac{{\partial f}}{{\partial x^\alpha }} = - K\left( p \right) \cdot f + G\left( f \right)$$ whereK(p) is a positive function andG an Hilbert-Schmidt operator. Then we resolve the Cauchy's problem by taking the Fourier's transformation off, and in the last part by investigating properties of the resolvent of ?K+G we establish that asx ⁰→+∞ the solution of this problem has for limit the equilibrium distributiona e ^?λp .