From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series

doi:10.1016/j.chaos.2008.06.027

Chaos, Solitons & Fractals

Volume 41, Issue 4, 30 August 2009, Pages 1590-1604

https://doi.org/10.1016/j.chaos.2008.06.027 Get rights and content

Abstract

By considering a coarse-grained space as a space in which the point is not infinitely thin, but rather has a thickness, one can arrive at an equivalence, on the modeling standpoint, between coarse-grained space and fractal space. Then, using fractional analysis (slightly different from the standard formal fractional calculus), one obtains a velocity conversion formula which converts problems in fractal space to problems in fractal time, therefore one can apply the corresponding fractional Lagrangian theory (previously proposed by the author). The corresponding fractal Schrödinger’s equation then appears as a direct consequence of the usual correspondence rules. In this framework, the fractal generalization of the Minkowskian pseudo-geodesic is straightforward.

Introduction

The motivation of the present article comes from some remarks related to three topics which are respectively, the increasing presence of fractal in physics, the wish (we do not say the need) to have an analytic mechanics to deal with dynamical systems defined in coarse-grained spaces and fractal space, and the fact that fractional calculus appears to be intimately related to fractal spaces and self-similar functions.

Many authors who applied fractional calculus to physics did it in a rather formal way, by formally substituting fractional derivative for derivative everywhere without clearly expliciting the physical motivation for this approach, and very often the formalism so involved did not put in evidence sufficiently clearly the physical meaning of the problem. We believe that this is mainly due to the fact that one is used to define fractional derivative by integral (?), what hides the intrinsic properties of the increments of the function under consideration.

We think that a good framework of fractional derivative, which would be fully efficient in modeling physical science problems, should refer to increments of the function which we are dealing with. The velocity of a particle is the quotient of two differentials, and in the same way the fractional velocity of this particle should be also defined as the quotient of two differentials.

The most natural and direct way to question the classical framework of physics is to remark that in the space of our real world, the generic point is not infinitely small (or thin) but rather has a thickness. This remark can be taken for granted very easily. In contrast, with respect to time at first glance the two assumptions are at stake: time may be continuous or on the contrary the clock of our universe would evolve by quantum of time.

Coarse-grained and thus fractal spaces are basic in El Naschie’s work [11], [12], [13], [14], [15], [16]. Fractals are basic in Nottale’s work [50], [51], [52], [53], and transparent in Ord’s paper [55]. See also [2], [6], [49]. In these works fractals are introduced as a tool to revisit the foundation of physics as natural science.

The problem of defined a theory of fractional Lagrangian mechanics has attracted many researcher [3], [36], [45], [46], [47], [48], [57], but to the best of our knowledge, these theories or approaches deal with dynamical systems which are mainly fractal in time. And in a large number of papers, one supposes merely that the system velocity under consideration is formally defined by fractional derivative either in Riemann–Liouville sense [37], [40], [41] or in Djrbashian–Caputo sense [5], [8]. It is hard to find in these works the very origin of the use of fractional derivative as a tool for modeling. As a result, we have several models of Lagrangian mechanics fractal in time [45], [46], [47], and one of them is the model that we proposed recently and which is defined in terms of action function involving integral with respect to (dt)^α [33].

Fractional derivative has been introduced formally by Liouville by using an integral involving the function under consideration, and in most cases this fractional calculus is more or less a formal calculus which is very often converted into Laplace’s transform. This was true until when Kolwantar and Gangal [38], [39], starting from the Hölder exponent of functions defined on Cantor’s sets, arrived at the definition of fractional derivative. Later, we arrived at the same identification, but by using the opposite way. We started from a fractional derivative defined as the limit of fractional difference, which allowed us to obtain the generalized fractional Taylor’s series, and we so found that the Hölder exponent is exactly the order of the fractional derivative of the function under consideration.

As a last, but not least remark, we shall notice that non-differentiability and randomness are mutually related in their nature, in such a way that studies in fractals on the one hand and fractional Brownian motion on the other hand are often parallel in the same paper. Indeed, as pointed out by Nottale [50], a function which is everywhere continuous but is nowhere differentiable cannot be duplicated, and as a result necessarily exhibits random-like or pseudo-random feature. This may explain the huge amount of literature which extends the theory of stochastic differential equation [22], [59] to stochastic dynamics driven by fractional Brownian motion [7], [9], [21], [42], [43], [44].

As a matter of fact, the main problem with fractional derivative as defined by integral, is that it is more or less formal so that its exact meaning, from the physical standpoint, is not transparent. And it is exactly why many mathematicians discard it at first glance, and consider it as being merely a formal calculus. This drawback can be circumvented if instead one uses the definition expressed in term of finite difference, and which, as a consequence, provides the Leibnitz formula. The article can be thought of as the continuation of the reference [33] and it is organised as follows. For the convenience of the reader, we first bear in mind the essential of the fractional modified Riemann–Liouville derivative and the fractional Taylor’s series, as well as some useful formulae which one can so obtain (Sections 2 A short background on fractional analysis, 3 Background on Taylor’s series of fractional order, 4 Basic formulas for fractional derivative and integral). Then one considers the problem of modeling velocity in coarse-grained space by the quotient (dx)^α/dt and by this way, via fractional analysis, we come across fractional derivative. This velocity appears as being the fractional derivative of time with respect to space, and by using some inverse function fractional formulae, we converted it into the fractional derivative of space with respect to time (Section 5). Then, by duplicating previous results [33], one derives the basic of the Lagrangian mechanics (Section 6). Section 7 illustrates the matter in the case of a single particle. And then, after considering the wave function as a probability density of fractional order (Section 8), we shall show how the classical correspondence rule will allow us to obtain the corresponding fractal Schrödinger’s equation.

Kolwankar and Gangal [38], [39] considered a function defined on a fractal set, and after introducing its derivative dx/(dt)^α in terms of Hölder exponent α, they arrive at fractional derivative and fractional (local) Rolle–Taylor’s formula. In our approach, we worked in the opposite way. Firstly, irrespective of any fractal set, we start from the expression of the fractional derivative as the limit of a fractional difference involving an infinite number of terms, and therefore we obtain the fractional generalized Taylor’s series, whereby we come across the Hölder’s exponent.

In order to deal with functions which are not differentiable, Ben Adda and Cresson [4] introduced a so-called quantum derivative, different from the Nottale’s scale-derivative, which also provides a (local) Rolle–Taylor’s formula. Here, we shall use a different modeling based on fractional derivative.

Quite recently, Eid et al. [10] consider a Schrödinger equation in α-dimensional space with a Coulomb potential, and obtained its solution. This equation is defined by the authors and has to be taken for granted by the reader. Here, we shall try to bring some rationales to support the derivation of the fractional Schrödinger equation.

Section snippets

Fractional derivative via fractional difference

Definition 2.1

Let $f : R \to R, x \to f (x)$ , denote a continuous (but not necessarily differentiable) function, and let h > 0 denote a constant discretization span. Define the forward operator FW(h) by the equality (the symbol ≔ means that the left side is defined by the right one) $FW (h) f (x) ≔ f (x + h),$ then the fractional difference of order α, 0 < α < 1, of f(x) is defined by the expression $Δ^{α} f (x) ≔ (FW - 1)^{α} f (x) = \sum_{k = 0}^{\infty} (- 1)^{k} (\begin{matrix} α \\ k \end{matrix}) f [x + (α - k) h]$ and its fractional derivative is defined as the limit [] $f^{(α)} (x) ≔ \lim_{h ↓ 0} \frac{Δ^{α} [f (x) - f (0)]}{h^{α}} .$

This definition

Basic formula for one-variable functions

A generalized Taylor expansion of fractional order which applies to non-differentiable functions (F-Taylor series in the following) reads as follows [33], [34].

Proposition 3.1

Assume that the continuous function $f : R \to R, x \to f (x)$ has fractional derivative of order kα, for any positive integer k and α, 0 < α ⩽ 1, then the following equality holds, which is $f (x + h) = \sum_{k = 0}^{\infty} \frac{h^{α k}}{(α k)!} f^{(α k)} (x), 0 < α ⩽ 1,$ where f^(αk)(x) is the derivative of order αk of f(x), and with the notation $Γ (1 + α k) = : (α k)!,$ where Γ(.) denotes the Euler gamma

Fractional derivative of compounded functions

The Eq. (3.1) provides the useful differential relation $d^{α} f ≅ Γ (1 + α) d f, 0 < α < 1,$ or in terms of fractional difference, Δ^αf ≅ α! Δf, which is basic in our approach..

Corollary 4.1

The following equalities hold, which are $D^{α} x^{γ} = Γ (γ + 1) Γ^{- 1} (γ + 1 - α) x^{γ - α}, γ > 0,$ or, what amounts to the same (we set α = n + θ). $\begin{matrix} D^{n + θ} x^{γ} = Γ (γ + 1) Γ^{- 1} (γ + 1 - n - θ) x^{γ - n - θ}, 0 < θ < 1, \\ (u (x) v (x))^{(α)} = u^{(α)} (x) v (x) + u (x) v^{(α)} (x), \end{matrix}$ $(f [u (x)])^{(α)} = f_{u}^{'} (u) u^{(α)} (x)$ $= f_{u}^{(α)} (u) (u_{x}^{'})^{α} .$ u(x) and v(x) are not necessarily differentiable in (4.3), u(x) is differentiable in (4.4), (4.5), and f(u) is

The basic modeling assumption

Assume that we are considering a mechanical point with the mass m, which is moving in a one-dimensional coarse-grained space defined by the space coordinate x(t), where t denotes the time.

On a modeling standpoint, we shall assume that in a coarse-grained space, the point is not infinitely thin but rather has a thickness. So, if dx and (dx)_c refer respectively to the size of the thin point and the size of the coarse-grained space point, then we should have dx < (dx)_c, and this inequality suggests

Definition of the framework

As it is customary in physics, one can explicitly introduce the freedom degree of the system by expressing x in terms of n generalized co-ordinates q₁, q_n, … , ,q_n and write x = x(q₁, q₂, … , q_n, t).

Fractional velocity in coarse-grained space

We shall assume that there is some coarse-grained phenomenon which causes that the trajectory q(t) exhibits some roughness. As a result, the increment of the displacement is not dq = o(dt), but rather is dq = o((dt)^α), 0 < α < 1. In this framework, in quite a natural way, we shall refer to the quotient dq/(dt)^α,

Application to the dynamics of a single particle

In the present section, we consider the special case of a particle defined by the Lagrangian (we come back to the x state variable) $L = (1 / 2) {mu}_{α}^{2} - V (x) .$ Eq. (6.6) direct yields the fundamental equation $m ρ (α) \frac{d^{α}}{d t^{α}} ((xt)^{α - 1} u_{α}) - {mu}_{α} \frac{\partial u_{α}}{d x} = - \frac{\partial V}{\partial x} .$ Remark that when u_α reduces to $\dot{x}$ , then one comes across the fundamental equation of dynamics. A more detailed expression for (7.2) can be obtained as follows. We have successively $\begin{matrix} \partial_{x} L + \partial_{x} V = {mu}_{α} \partial_{x} u_{α}, \\ \partial_{u} L = {mu}_{α}, \\ \partial_{x} u_{α} = (α - 1) ρ x^{α - 2} t^{α - 1} x^{(α)}, \\ \partial u_{α} / \partial x^{(α)} = ρ x^{α - 1} t^{α - 1}, \end{matrix}$ therefore, on

Background on probability density of fractional order

Before to work on the Schrödinger’s equation as we intend to do it, we would like to show that the wave function itself could be considered with a fractional point of view, if one takes for granted the concept of probability density of fractional order.

Definition 8.1

Let X denote a real-valued random variable defined on the interval [a, b] and let p_α(x), p_α(x) ⩾ 0 denote a positive function also defined on [a, b]. X is referred to as a random variable of fractional order α, 0 < α < 1, with the probability p_α(x),

Background on the basic of quantum mechanics

Historically, quantum mechanics has been constructed as a generalization of the Lagrangian mechanics which explains some observations in laboratory.

The Hamilton’s function associated with a particle in a force field defined by the potential function v(x, t) is $H = (1 / 2 m) p^{2} + V (x, t),$ with p = mv. In order to formally derive the corresponding Schrödinger’s equation, it is customary to use the correspondence rule $p \to - i ℏ \nabla, p^{2} \to - ℏ^{2} Δ, V (x, t) \to V \times$ and we so obtain the equation of the wave function ψ(x, t) in the form $i$

System with fractal time

In quite a natural way, we are led to the question of how to meaningfully generalize the pseudo-metric ds² = c²dt² − dx², in the presence of fractals. And we believe that this should be rather simple if we refer to the relative meaning of (dx)^α and dx (respectively (dt)^α and dt) in the framework of our fractional calculus via fractional differences.

In the first case, we assume that it is the time which is fractal of order β, 0 < β < 1, to yield the increment (dt)^β instead of dt.

In such a case, we shall

Concluding remarks

In the present article, we have shown that, if one selects formulae defined in terms of finite increments (instead of integrals) to define fractional derivative, then the latter appears to be quite a suitable tool to construct a theory of Lagrangian mechanics for particle moving in coarse-graining space and fractal space.

The initial problem defined in a coarse-grained space has been considered as equivalent to a ((dx)^α, dt) problem defined in a fractal space, and then converted in a (dx, (dt)^α)

References (60)

V.V. Anh et al.
Scaling laws for fractional diffusion–wave equations with singular initial data
Stat Probabil Lett
(2000)
F. Ben Adda et al.
Quantum derivatives and the Schrödinger equation
Chaos, Solitons & Fractals
(2004)
M.S. El Naschie
A review of E infinity theory and the mass spectrum of high energy particle physics
Chaos, Solitons & Fractals
(2004)
M.S. El Naschie
Non-linear dynamics and infinite dimensional topology in high energy particle physics
Chaos, Solitons & Fractals
(2003)
M.S. El Naschie
Gravitational instanton in Hilbert space and the mass of high energy elementary particles
Chaos, Solitons & Fractals
(2004)
M.S. El Naschie
On Penrose view of transfinite sets and computability and fractal character of E-infinite spacetime
Chaos, Solitons & Fractals
(2005)
M.S. El Naschie
Elementary prerequisites for E-infinity (Recommended background readings in nonlinear dynamics, geometry and topology)
Chaos, Solitons & Fractals
(2006)
M.S. El Naschie
Intermediate prerequisites for E-infinity (Further recommended background readings in nonlinear dynamics, geometry and topology)
Chaos, Solitons & Fractals
(2006)
G. Jumarie
Fractional Brownian motion with complex variance via random walk in the complex plane. Applications
Chaos, Solitons & Fractals
(2000)
G. Jumarie
Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker–Planck equations
Chaos, Solitons & Fractals
(2004)

G. Jumarie

On the representation of fractional Brownian motion as an integral with respect to (dt)^α

Appl Math Lett

(2005)

G. Jumarie

On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion

Appl Math Lett

(2005)

G. Jumarie

Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDF, Taylor’s series of non-differentiable functions

Chaos, Solitons & Fractals

(2007)

G. Jumarie

Probability calculus of fractional order and fractional Taylor’s series application to Fokker–Planck equation and information of non-random functions

Chaos, Solitons & Fractals

(2009)

L. Nottale

Scale relativity and fractal space-time. Applications to quantum physics, cosmology and chaotic systems

Chaos, Solitons & Fractals

(1996)

L. Nottale

The scale-relativity programme

Chaos, Solitons & Fractals

(1999)

E.M. Rabei et al.

The Hamiltonian formalism with fractional derivatives

J Math Anal Appl

(2007)

N.T. Shawagfeh

Analytical approximate solutions for nonlinear fractional differential equations

Appl Math Comput

(2002)

E. Bakai

Fractional Fokker–Planck equation, solutions and applications

Phys Rev E

(2001)

D. Baleanu et al.

Fractional Hamilton formalism within Caputo’s derivative

Czech J Phys

(2006)

M. Caputo

Linear model of dissipation whose Q is almost frequency dependent II

Geophys J Roy Ast Soc

(1967)

R. Carrol

On quantum potential

Appl Anal

(2005)

L. Decreusefond et al.

Stochastic analysis of the fractional Brownian motion

Potential Anal

(1999)

M.M. Djrbashian et al.

Fractional derivative and the Cauchy problem for differential equations of fractional order (in Russian)

Izv Acad Nauk Armjanskoi SSR

(1968)

T.E. Duncan et al.

Stochastic calculus for fractional Brownian motion, I. Theory

SIAM J Control Optim

(2000)

R. Eid et al.

On fractional Schrödinger equation in α-dimensional fractional space

Nonlinear Anal, Real World Appl

(2008)

A. El-Sayed

Fractional order diffusion–wave equation

Int J Theor Phys

(1996)

B.R. Frieden

Physics from Fisher Information

(2000)

G. Grössing

Quantum cybernetics

(1957)

A. Hanyga

Multidimensional solutions of time-fractional diffusion–wave equations

Proc Roy Soc London, A

(2002)

Cited by (27)

A novel method for numerical simulation of sand motion model in beach formation based on fractional Taylor–Jumarie series expansion and piecewise interpolation technique
2019, Applied Mathematics and Computation
In this paper, for the first time, fractional Taylor–Jumarie series expansion is used to solve a glass of time-fractional delay partial differential equation by piecewise interpolation reproducing kernel method (RKM), this class of equations describe sand motion model in beach formation. The aim of this work is to obtain more accurate numerical solution by fractional Taylor–Jumarie series expansion and piecewise interpolation technique. Three numerical experiments are provided to show the advantage of this method, the results show the characteristics of sand motion. The research in this paper is the theoretical basis for further sand motion study.
On chain rule for fractional derivatives
2016, Communications in Nonlinear Science and Numerical Simulation
For some types of fractional derivatives, the chain rule is suggested in the form $D_{x}^{α} f (g (x)) = {(D_{g}^{1} f (g))}_{g = g (x)} D_{x}^{α} g (x)$ . We prove that performing of this chain rule for fractional derivative $D_{x}^{α}$ of order α means that this derivative is differential operator of the first order ( $α = 1$ ). By proving three statements, we demonstrate that the modified Riemann–Liouville fractional derivatives cannot be considered as derivatives of non-integer order if the suggested chain rule holds.
Numerical solution for fractional variational problems using the Jacobi polynomials
2015, Applied Mathematical Modelling
On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric
2015, Physica A: Statistical Mechanics and its Applications
Over the recent decades, diverse formalisms have emerged that are adopted to approach complex systems. Amongst those, we may quote the $q$ -calculus in Tsallis’ version of Non-Extensive Statistics with its undeniable success whenever applied to a wide class of different systems; Kaniadakis’ approach, based on the compatibility between relativity and thermodynamics; Fractional Calculus (FC), that deals with the dynamics of anomalous transport and other natural phenomena, and also some local versions of FC that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations.
The question we might ask is whether or not there are common aspects that connect these alternative approaches. In this short communication, we discuss a possible relationship between $q$ -deformed algebras in two different contexts of Statistical Mechanics, namely, Tsallis’ framework and Kaniadakis’ scenario, with local form of fractional-derivative operators defined in fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. This connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics, such as porous or granular media. Possible connections with other alternative definitions of FC are also contemplated. Insights on complexity connected to concepts like coarse-grained space–time and physics in general are pointed out.
Physics in space-time with scale-dependent metrics
2013, Physics Letters, Section A: General, Atomic and Solid State Physics
We construct three-dimensional space $R_{γ}^{3}$ with the scale-dependent metric and the corresponding Minkowski space–time $M_{γ, β}^{4}$ with the scale-dependent fractal ( $D_{H}$ ) and spectral ( $D_{S}$ ) dimensions. The local derivatives based on scale-dependent metrics are defined and differential vector calculus in $R_{γ}^{3}$ is developed. We state that $M_{γ, β}^{4}$ provides a unified phenomenological framework for dimensional flow observed in quite different models of quantum gravity. Nevertheless, the main attention is focused on the special case of flat space–time $M_{1 / 3, 1}^{4}$ with the scale-dependent Cantor-dust-like distribution of admissible states, such that $D_{H}$ increases from $D_{H} = 2$ on the scale $≪ ℓ_{0}$ to $D_{H} = 4$ in the infrared limit $≫ ℓ_{0}$ , where $ℓ_{0}$ is the characteristic length (e.g. the Planck length, or characteristic size of multi-fractal features in heterogeneous medium), whereas $D_{S} \equiv 4$ in all scales. Possible applications of approach based on the scale-dependent metric to systems of different nature are briefly discussed.
General necessary conditions for infinite horizon fractional variational problems
2013, Applied Mathematics Letters

View all citing articles on Scopus

View full text

From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series

Abstract

Introduction

Section snippets

Fractional derivative via fractional difference

Basic formula for one-variable functions

Fractional derivative of compounded functions

The basic modeling assumption

Definition of the framework

Fractional velocity in coarse-grained space

Application to the dynamics of a single particle

Background on probability density of fractional order

Background on the basic of quantum mechanics

System with fractal time

Concluding remarks

Stat Probabil Lett

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Appl Math Lett

Appl Math Lett

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

J Math Anal Appl

Appl Math Comput

Fractional Fokker–Planck equation, solutions and applications

Phys Rev E

Fractional Hamilton formalism within Caputo’s derivative

Czech J Phys

Linear model of dissipation whose Q is almost frequency dependent II

Geophys J Roy Ast Soc

On quantum potential

Appl Anal

Stochastic analysis of the fractional Brownian motion

Potential Anal

Fractional derivative and the Cauchy problem for differential equations of fractional order (in Russian)

Izv Acad Nauk Armjanskoi SSR

Stochastic calculus for fractional Brownian motion, I. Theory

SIAM J Control Optim

On fractional Schrödinger equation in α-dimensional fractional space

Nonlinear Anal, Real World Appl

Fractional order diffusion–wave equation

Int J Theor Phys

Physics from Fisher Information

Quantum cybernetics

Multidimensional solutions of time-fractional diffusion–wave equations

Proc Roy Soc London, A