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Fizika A

SUBDIFFUSION OVER FRACTIONAL QUANTUM PATHS WITHOUT FRACTIONAL DERIVATIVE

Rami Ahmad El‐Nabulsi

Published Jan 1, 2008• 6 citations
OpenAlex

Abstract

The standard Brownian diffusion process governs the scaling of a random walk, where a tracer in a two dimensional spacetime is governed by the variance of the displacement of a particle distribution σ(t) ∝ t , γ = 1. In other words, if we denote by P (x, t) the relative probability of particles at location x ∈ R at time t, the classical diffusion equation ∂P/∂t = 12∂ p/∂x can be solved by means of the Fourier transform which inverts to a normal probability distribution with the mean zero and standard deviation t. In the last decade, it has been realized that the understanding of complex dynamical systems has required the implementation of the subdiffusion (slower spreading rate, i.e. γ 1) anomalous random processes. Anomalous superdiffusion can be modeled using infinite variance particle jumps that lead to space-fractional derivatives in the governing diffusion equation. Anomalous subdiffusion can be modeled using IID infinite mean waiting times between particle jumps, leading to a fractional time derivative in the governing diffusion equation [1]. The aim of this short communication is to prove that the anomalous subdiffusion can be realized in quantum physical phenomena without implementing any kind of time-fractional derivative.

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The standard Brownian diffusion process governs the scaling of a random walk, where a tracer in a two dimensional spacetime is governed by the variance of the displacement of a particle distribution σ(t) ∝ t , γ = 1. In other words, if we denote by P (x, t) the relative probability of particles at location x ∈ R at time t, the classical diffusion equation ∂P/∂t = 12∂ p/∂x can be solved by means of the Fourier transform which inverts to a normal probability distribution with the mean zero and standard deviation t. In the last decade, it has been realized that the understanding of complex dynamical systems has required the implementation of the subdiffusion (slower spreading rate, i.e. γ 1) anomalous random processes. Anomalous superdiffusion can be modeled using infinite variance particle jumps that lead to space-fractional derivatives in the governing diffusion equation. Anomalous subdiffusion can be modeled using IID infinite mean waiting times between particle jumps, leading to a fractional time derivative in the governing diffusion equation [1]. The aim of this short communication is to prove that the anomalous subdiffusion can be realized in quantum physical phenomena without implementing any kind of time-fractional derivative.
Abstract Quantum walks are known to propagate quadratically faster than their classical counterparts and are used to model dynamics in various quantum systems. The spread of the quantum walk in position space shows anomalous diffusion behavior. By controlling the action of quantum coin operation on the corresponding coin degree of freedom of the walker, one can demonstrate control over the diffusion behavior. In this work, we report different forms of coin operations on quantum walks exhibiting anomalous diffusion behavior. Homogeneous and accelerated quantum walks display superdiffusive behavior, whereas uncorrelated static and dynamic disorders in the evolution induce strong and weak localization of the particle indicating subdiffusive and normal diffusive behavior. The role played by the interference effects in the spreading of the walker has remained elusive and our aim in this work is to present the interplay between quantum coherence and mean squared displacement of the walker. We employ two reliable measures of coherence for conclusively establishing the role of quantum interference as the driving force behind the anomalous diffusive behavior in the dynamics of quantum walks.
The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.
First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.
We write in terms of path integrals the effective quantum propagator associated to a quantum particle interacting with a fractional-dimensional reservoir modelled by a scalar and vectorial phonon field in a space-time R v , with v an arbitrary real number. We show, thus, that the classical regime related to this quantum-fractional-dimensional Brownian particle motion is given by the fractional Langevin equation (a fractional time diffusion process). As a consequence, we deduce from first (microscopic) principles fractional-dimensional diffusion phenomena.
The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order.The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [𝑀𝐿 2 /𝑇 2 ].The action 𝑆 is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced.The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order.The present work corrects a number of errors in Naber's work.The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained.The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity.A generalization to the case of a particle moving in a potential is also given.
We develop an analytical theory of anomalous transport in a noninteracting Anderson model with heavy-tailed hopping amplitudes. The broad distribution of hopping amplitudes gives rise to an extended intermediate-time regime with a subdiffusive effective exponent, despite the absence of interactions or genuine many-body effects. By solving the transport equations analytically, we derive the time dependence of the mean-square displacement and identify a continuous crossover from an intermediate anomalous regime to asymptotically diffusive transport. As the localization transition is approached, the spatial extent of the subdiffusive window diverges parametrically, while the crossover to conventional diffusion remains finite in units of Γ0−1Γ_0^{-1}Γ0−1​. This produces an increasingly broad anomalous transport regime in space that can closely resemble Griffiths-type transport observed near the many-body localization transition. Our results demonstrate that rare hopping processes alone provide a microscopic single-particle mechanism for robust transport anomalies, establishing an analytical benchmark for distinguishing interaction-induced effects from disorder-driven dynamics.