Time Fractional Formalism: Classical and Quantum Phenomena

Transcription

1 Time Fractional Formalism: Classical and Quantum Phenomena Hosein Nasrolahpour * Abstract In this review, we present some undamental classical and quantum phenomena in view o time ractional ormalism. Time ractional ormalism is a very useul tool in describing systems with memory and delay. We hope that this study can provide a deeper understanding o the physical interpretations o ractional derivative. Keywords: Fractional calculus; Fractional classical mechanics; Fractional classical electromagnetism; Fractional quantum mechanics. This work is dedicated to the soul o my ather 1- Introduction Fractional calculus is a very useul tool in describing the evolution o systems with memory, which typically are dissipative and to complex systems. Complex systems include very broad and general class o systems and materials. For instance, glasses, biopolymers, biological cells, porous materials, amorphous semiconductors and liquid crystals can be considered as complex systems. Scaling laws and sel-similar behavior are supposed to be undamental eatures o complex systems. In recent decades the ractional calculus and in particular the ractional dierential equations has attracted interest o researches in several areas including mathematics, physics, chemistry, biology, engineering and economics [1-4].Applications o ractional calculus in the ield o physics have gained considerable popularity and many important results were obtained during the last years. Some o the areas o these applications include: classical mechanics [8-11], classical electromagnetism [3-38], special relativity [39, 4], non-relativistic quantum mechanics [43-5] and relativistic quantum mechanics and ield theory [51-58]. Despite these various applications, there are some important challenges. For example physical interpretation or the ractional derivative is not completely clariied yet. In this review, we aim to present some aspects o physical interpretation or the ractional derivative by studying the behavior o undamental classical and quantum phenomena within the ramework o time ractional ormalism. In the ollowing, ractional calculus is briely reviewed in Sec.. The ractional relaxation and oscillation process are discussed in Sec. 3. Time ractional Maxwell s equations are presented in sec. 4. In Sec. 5 time ractional Schrödinger equation and time ractional Pauli equation are given. Finally in Sec. 6, we will present our summary and discussion. - Mathematical tools: Fractional calculus Although the application o Fractional calculus has attracted interest o researches in recent decades, it has a long history when the derivative o order.5 has been described by Leibniz in a letter to L'Hospital in Fractional calculus is the calculus o derivatives and integrals with arbitrary (real or even complex order, which uniy and generalize the notions o integer order dierentiation and n-old integration, which have ound many applications in recent studies to model a variety processes rom classical to quantum physics. In the ollowing, we briely revisit essentials o ractional calculus..1. The Caputo ractional derivative operator The commonest way to obtain a ractional dierential equation or describing the evolution o a typical system is to generalize the ordinary derivative in the standard dierential equation into the ractional derivative. Fractional dierential equation can be include or instance derivative o order.5,, and * Correspondence: Hosein Nasrolahpour, hnasrolahpour@gmail.com

2 so on. Since the age o Leibniz various types o ractional derivatives have been proposed. In act, the deinition o the ractional order derivative is not unique and there exists several deinitions including, Grünwald Letnikov, Riemann-Liouville, Weyl, Riesz and Caputo or ractional order derivative. Fractional dierential equations deined in terms o Caputo derivatives require standard boundary (initial conditions. Also the Caputo ractional derivative satisies the relevant property o being zero when applied to a constant. For these reasons, in this paper we preer to use the Caputo ractional derivative. The let (orward Caputo ractional derivative o a time dependent unction ( t is deined by t c 1 n 1 ( n t ( ( ( ( n D t t d, t ( n Where, n is an integer number and is the order o the derivative such that n-1< <n and ( denotes the n-th derivative o the unction (. For example when is between and 1, we have (1 c 1 ( Dt ( t ( t d (1 t 1 ( As we can see rom the above equations Caputo derivative implies a memory eects by means o a convolution between the integer order derivative and a power o time. Also the Laplace transorm to Caputo's ractional derivative gives n 1 c m 1 ( m t m L{ D ( t } s F ( s s ( (3 where, F ( s is the Laplace transorm o ( t... The Mittag-Leler unction During the recent years the Mittag-Leler (ML unction has caused extensive interest among physicist due to its role played in describing realistic physical systems with memory and delay. It was originally introduced by G.M. Mittag-Leler in 19[5]. The ML unction is such a one-parameter unction deined by the series expansion as E k z ( z (1 k z C, k (4 And its general two-parameter representations is deined as E, k z ( z ( k z C, β C, k (5 where C is the set o complex numbers and ( denotes the Gamma unction. This unction is in act a generalization o the exponential unction. For example, or the special case o 1, the ML unction z Eq. (4 reduces to the exponential unction E 1 ( z e. Furthermore, since the ML unction generalizes the exponential unction, the Euler identity or an i exponential unction with a complex argument (i.e., e cos( i sin( can also be written or the ML unction in a similar manner. So we have E ( i cos ( i sin ( (6

3 Where sin ( and cos ( are sine and cosine ML unctions respectively and deined as n n 1 ( 1 ( sin (, ((n 1 1 n cos ( n n n ( 1 ( (n 1 (7 Also, it is notable that although exponential unction possesses the semigroup property (i.e., a( z 1 z az 1 az e e e the unction E ( az does not possess the semigroup property in general [6] (this property leads to important results in ractional quantum mechanics [48]. Mittag-Leler unction, as a generalized exponential unction, naturally arises in the solutions o ordinary dierential equations o arbitrary (non-integer order. Thereore the Laplace transorm or ML unction will be very useul in solving ractional dierential equations: L t E t m 1 ( m { ( }, m! s ( s m 1 (8 Where s Classical mechanics: ractional relaxation and ractional oscillation The undamental processes in physics are described by equations or the time evolution o a quantity X ( t in the orm: dx ( t (9 LX ( t dt where L can be both linear or nonlinear operator. For instance there are many relaxation phenomena in nature whose relaxation unction obeys the simple approximate equation dx ( t (1 x ( t dt We can write the above equation as dx ( t 1 x ( t (11 dt The solution o the above equation is the normalized exponential Debye-relaxation unction (i.e. t x ( t e, with relaxation time. However, there are some experimental evidences that relaxation in several complex disordered systems deviates rom the classical exponential Debye pattern [1-4]. Nowadays, it has proved that the ractional relaxation equation can be a successul mathematical construct that relects the main eatures o evolution o such systems. The commonest way to obtain a ractional dierential equation or describing the evolution o a typical system is to generalize the ordinary derivative in the standard dierential equation into the ractional derivative d 1 d (1 1 dt dt where d denotes the Caputo s derivative operator o order and, is a new parameter representing dt the ractional time components in the system[3] and its dimension is the second. In the case 1the expression transorms into ordinary time derivative operator 1 d d ( dt dt

4 Thereore we can easily arrive at the ractional relaxation equation by changing the irst order derivative in the Eq. (1 to a derivative o an arbitrary order: d x ( t x ( t 1 (14 1 dt with the solution: 1 t (15 x ( t x ( E ( ( It is showed that this solution and this model or the relaxation processes can be successully adopted to interpret experimental data on relaxation in several complex disordered systems. The second example is the simple harmonic oscillator.the harmonic oscillator, given by the well-known second order linear dierential equation with constant coeicients d x (16 m kx dt is a cornerstone o classical mechanics [7]. We can obtain the dierential equation o a simple ractional oscillator [5-31] by changing the second derivative in the harmonic oscillator equation to a derivative o an arbitrary order (Eq. (1: m d x kx (1 1 dt We can write the above equation also as d x (1 k x dt m where the parameter deined by (1 k and, so we can rewrite the ractional dierential equation o the system as m d x x dt The solution o the above equation reads: x ( t x ( E ( t x ( te ( t, Now i we choose x ( 1and x ( as the initial condition, the solution becomes x ( t E ( t ( We can easily see that as 1, above equations gives (3 E ( t E ( t cosh( t cosh( i t cos( t As we can see rom Eq. (, the displacement o the ractional oscillator is essentially described by the Mittag Leler unction E ( t or our considered initial conditions. It is showed by numerical calculations that the displacement o the ractional oscillator varies as a unction o time and how this time variation depends on the parameter [5]. Also it is proved that, i is less than 1 the displacement shows the behavior o a damped harmonic oscillator. As a result, in consistent with the case o simple harmonic oscillator, the total energy o simple ractional oscillator will not be a constant. What is surprising is that the damping o ractional oscillator is intrinsic to the equation o motion and not introduced by additional orces as in the case o a damped harmonic oscillator. Up to now, the source o this intrinsic damping is not clearly understood. However, there are some attempts in this regard. For example an interesting ormulation o the notion o intrinsic damping orce has been proposed in Res. [9, 3]. (17 (18 (19 ( (1

5 4- Classical electromagnetism: A plane wave with time decaying amplitude In classical electromagnetism, behavior o electric ields ( E, magnetic ields ( B and their relation to their sources charge density ( ( r, t, and current density ( j ( r, t, is described by the ollowing Maxwell s equations: 4. E ( r, t (4. B (5 1 B E (6 c t 4 E (7 B j ( r, t c c t Where and are electric permittivity and magnetic permeability, respectively. Now, introducing the potentials, vector A ( x i, t and scalar ( x i, t B A (8 1 A (9 E c t and using the Lorenz gauge condition we obtain the ollowing decoupled dierential equations or the potentials A ( r, t 4 A ( r, t j ( r, t (3 c t c ( r, t 4 (31 ( r, t ( r, t c t 1 where c v. v is the velocity o the light in the medium. Furthermore, or a particle with charge q in the presence o electric and magnetic ield we can write the Lorentz orce as FL q ( E v B (3 In terms o scalar and vector potentials, Eq. (8, 9, we may write the Lorentz orce as 1 A FL q ( v ( A (33 c t As we saw in previous section, in classical mechanics, the ractional ormalism leads to relaxation and oscillation processes that exhibit memory and delay. This ractional nonlocal ormalism is also applicable on materials and media that have electromagnetic memory properties. So the generalized ractional Maxwell s equations can give us new models that can be used in these complex systems. Up to now, several dierent versions o ractional electromagnetism based on the dierent approaches to ractional vector calculus have been investigated [33-38]. However, in this paper we study a new approach on this area [3]. The idea is in act, to write the ordinary dierential wave equations in the ractional orm with respect tot. 4 (34. E ( r, t. B ( B E 1 1 c (36 t 4 1 E B j ( r, t 1 (37 1 c c t And the Eq. (8, 9 become

6 B A (38 1 A E 1 (39 1 c t And the Lorentz orce Eq. (33 becomes 1 A FL q( v ( A 1 (4 1 c t Then, applying the Lorentz gauge condition we obtain the corresponding time ractional wave equations or the potentials 1 A( r, t 4 A ( r, t j ( r, t (41 (1 c t c 1 ( r, t 4 ( r, t ( r, t (4 (1 c t I,, and, j, we have the homogeneous ractional dierential equations 1 A ( r, t A ( r, t (43 (1 c t 1 ( r, t (44 ( r, t (1 c t We are interested in the analysis o the electromagnetic ields in the medium starting rom the equations. Now, we can write the ractional equations in the ollowing compact orm z ( x, t 1 z ( x, t (45 (1 x c t where z ( x, t represents both A ( r, t and ( r, t. We consider a polarized electromagnetic wave, then Ax, A y, Az. A particular solution o this equation may be ound in the orm (46 ikx z ( x, t z e u ( t where k is the wave vector in the x direction and z is a constant. Substituting into Eq. (45 we obtain d u ( t (47 u ( t dt Where (1 (1 (48 v k and is the undamental requency o the electromagnetic wave. The solution o this equation may be u ( t E ( t (49 Substituting this expression in Eq. (46 we have a particular solution o the equation as ikx z ( x, t z e E ( t (5 We can easily see that in the case 1, the solution to the equation is ( z ( x, t Re( z i t kx e (51 which deines a periodic, with undamental period T, monochromatic wave in the, x, direction and in time, t.this result is very well known rom the ordinary electromagnetic waves theory. However or the arbitrary case o ( 1 the solution is periodic only respect to x and it is not periodic with respect to t.the solution represents a plane wave with time decaying amplitude. 1 For example or the case we have ( 1( t u t E t e (5

7 Thereore the solution is t z ( x, t ( z e e ikx Then, or this case the solution is periodic only respect to x and it is not periodic with respect tot. In act the solution represents a plane wave with time decaying amplitude. 5- Quantum mechanics: Time ractional Schrödinger and Pauli equation Nowadays, application o the ractional calculus to quantum processes is a new and ast developing part o quantum physics which studies nonlocal quantum phenomena. Nonlocal eects may occur in space and time. In the time domain the extension rom a local to a nonlocal description becomes maniest as a memory eect. Thereore the underlying undamental processes become o non-markovian type.in the realm o non-relativistic quantum mechanics [41, 4], Schrödinger equation represents a undamental equation to study many quantum processes (54 V ( x i m x t Recently the time ractional Schrödinger equation, which has a Caputo ractional time derivative, was considered by Naber [46], in order to describe non-markovian evolution in quantum mechanics. The general idea to obtain the time ractional Schrödinger equation is to keep the position and momentum operators in the usual orm and replacing i ( i c Dt or i ( i c C Dt, where D t t t denotes the Caputo s derivative operator o order and M c T P P is a scaled Planck constant. Also the parameters M P and T P are Planck mass and Planck time, respectively, are deined as (53 T P G 5 c M, P c G (55 where G and c are the gravitational constant and the speed o light in vacuum, respectively. Naber gives some arguments in avour o the latter case and many authors ollow him therein [47, 48].However one can consider the ormer one as a possible case or studying time ractional Schrödinger equation. For instance the wave unction and the probability density or a ree particle within this type o time ractional Schrödinger equation c V ( x ( i D t 1 (56 m x have been studied in Re. [49]. As we mentioned above, one can introduce the time ractional Schrödinger equation to describe non- Markovian evolution in quantum realm. Now we generalize the time ractional Schrödinger equation Eq. (56 and obtain the ollowing time ractional Pauli equation [5], 1 ˆ e ˆ B c [ ( P A e. B ] ( i Dt 1 m c One can use this equation to discuss the electron spin precession problem in a homogeneous constant magnetic ield [5]. Here we assume that, the electron is ixed at a certain location and its spin is the only degree o reedom. Also, let the magnetic ield consist o a constant ield B in the Z direction (i.e. B B ˆ k.thereore, that part o the time ractional Pauli equation Eq. (57, which contains the spin yields ( i c D t L z ˆ (57 (58

8 eb Where L are the so-called Larmor requency and ˆ is the well-known Pauli matrices or a mc spin 1 particles.science the Hamiltonian o our system is a matrix, the spin unction in arbitrary time (t must be written as a column matrix o two components and can be derived as below, χ (t = a(t b(t = cos E e ( i(ω t e sin E (i(ω t (59 Where and are arbitrary phase constants. Now, by use o Eq. (59 we able to calculate the probability or spin-up, P, and spin-down, P, at t : P a( t cos ( [ E ( i ( Lt E ( i ( Lt ] P b( t sin ( [ E ( i ( Lt E ( i ( Lt ]. (6 (61 We can explicitly see that as 1, above equations gives P 1 P P But or the arbitrary case o ( 1, we have P P P cos (( Lt sin (( Lt tot tot Where is obtained in terms o the sine and cosine ML unctions Eq. (7. It is clearly seen that the total probability o upness and downness o electron's spin varies as a unction o time and also it depends on the parameter. 6- Summary and discussion Fractional calculus is a very useul tool in describing the evolution o systems with memory, which typically are dissipative and to complex systems. In recent decades it has attracted interest o researches in several areas o science. Specially, in the ield o physics applications o ractional calculus have gained considerable popularity [3, 4] (and the reerences therein. In spite o these various applications, there are some important challenges. For example physical interpretation or the ractional derivative is not completely clariied yet. In this review, we present some undamental classical and quantum phenomena in the ramework o time ractional ormalism in order to provide a deeper understanding o the physical interpretations o ractional derivative. We have seen that, a simple ractional oscillator behaves like a damped harmonic oscillator. What is surprising is that the damping is intrinsic to the equation o motion and not introduced by additional orces as in the case o a damped harmonic oscillator. Also, in the case o ractional electromagnetism we see that behavior o electromagnetic waves is not same as the standard ones. In act we see that the ractional Maxwell's equations lead to the plan wave with time decaying amplitude (Eq. (5, 53.It is showed that amplitude o this plane wave varies as a unction o time and this time variation depends explicitly on the parameter (the order o the ractional derivative. Finally we see that total probability o upness and downness o electron's spin Eq. (6 is not equal to unity and it depends on t and the parameter, as well. The interpretation o this time dependent probability is an open area o research. It is worth noticing that an expansion method has been proposed [8, 3] to discuss the dynamics in the media where the order o the ractional derivative is close to an integer number. It will be o interest to consider above mentioned phenomena within this scheme. We hope to report on these issues in the uture. (6

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