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Il Nuovo Cimento B

The quantum-fractional-dimensional Brownian motion: A path integral study

Luiz C. L. Botelho

Published Apr 1, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1393/ncb/i2003-10027-2

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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 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.
CONTENTS: Preface Bosonic Path Integrals Nelson's Stochastic Mechanics by Means of Feynman Path Integrals Feymman path-integrals for Brownian Classical Motions (1+1)-dimensional Burger turbulence in the framework of Hopf-Cole Transformation and Feymman Trajectories path-integrals Feynman-Wiener Path Integral Representations for the Quantum Harmonic Oscillator with Time-Dependent Stochastic Frequency: Quantum Brownian Motions and Navier-Stokes Weakly Turbulence Some studies on functional integrals representations for fluid motion with random conditions Statistics turbulence suppression by a strong magnetic field: A Feynman path integral argument A non-linear diffusion equation, Some existence and uniqueness results on weak solutions for random initial conditions and its path integral representation -- a mathematical methods approach The Exact Evalution of the Response Function for Wave Propagation in Random Media: A Functional Integral Approach On the Abstract Method of Integration for Wave Equation and some Applications for Wave Physics Feynman Path-Integral Studies for Quantum Particles Propagators Interacting with Environments The Quantum Fractional Dimensional Brownian Motion: A Path-Integral Study Ohm's Law: A Langevin Path-Integral Study at Classical and Quantum W.K.B. Regime On the Statistical of An Ideal Gas with Mass Fluctuations and the Boltzman Ergodic Theorem: The Combined Boltzman-Tsallis Statistics The Suppression of the Quantum Phenomena by an External Strong White-Noise Magnetic Field: A Path Integral Argument Index.
We first point out that nonrelativistic quantum mechanics formulated at imaginary time s = it can formally be recognized as the Fokker-Planck description of a classical Brownian motion, which occurs (in general) in a medium where particles are absorbed and emitted. We next offer a new formulation of quantum dynamics, which is basically the Langevin treatment of that Brownian motion. More explicitly, we derive a noise average representation of the transition probability W ( X", s"|X', s' ) in terms of the solutions to a Langevin equation with a Gaussian white noise. Upon analytic continuation back to real time W ( X", s"|X', s' ) becomes the propagator of the original Schrödinger equation, namely we have ⟨ X", t"|X', t' ⟩ = W ( X", it"|X', it' ). The main advantage of this approach is to permit a systematic application to quantum-dynamical problems of the mathematical techniques developed in the theory of classical stochastic processes, even when vector potentials and/or time-dependent potentials are present.
A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Levy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.
We give a new description of quantum Brownian motion in terms of stochastic pure states. The corresponding path integral propagator allows us to establish a direct connection to the classical Langevin equation, in the Schrodinger picture. We show that in the quantum domain, one is naturally led to consider two stochastic processes driving the Brownian dynamics, one of them representing thermal fluctuations, as in the classical case. The second process reflects growing entanglement between the Brownian particle and its environment, and is therefore a truly quantum noise process. Technically, our result rests on the representation of the full propagator of the Brownian particle and its environment in a coherent state basis. For open system dynamics that may be described by a master equation, such stochastic schemes have already been proven to be efficient Monte Carlo methods. Here, we give a new stochastic scheme that covers low temperatures and strong friction, the latter limit being the case Einstein originally investigated.
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.