Sabatier Agrawal Machado Advances in Fractional Calculus Episode 5 pps

Let us look on the spherical harmonics Y l,m , given in 40 and the associated Legendre polynomials d d x x x m m m m with 4.2 Spin-1/2-particles in fractional description angular momen

2 2

2

2 , 2 2 2

2

2

sin

1sin

j

L

These commutation rules tell us, that the magnitude of the angular

momentum can always be exactly determined, but only one of its components

can be measured at the same time with arbitrary accuracy The other two remain

indeterminable Usually, L z is taken as the measurable component For a

stationary state, the part of the wave function L which describes the angular

momentum has to be an eigenfunction of the angular momentum operators, thus

it has to satisfy the two equations:

m l

l m l

)!

(4

)12()!

(),(

with the associated Legendre polynomials satisfy the relations:

),()1(),

, 2

m l m

),()

)1(l !

l

for L2 and L z The magnitude of the angular momentum

)1

(l

l

as well as its z-component L z are quantised with l and m as quantum numbers

The latter can be changed applying the ladder operators

SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS

150

y

x iL L

to create (+) or annihilate ( ) a magnetic quantum according to

L L

Since L x and L yare both Hermitian (which implies ) and |L lm|2> 0, we have

the well known restrictions l < m < l between the so-called angular l and

magnetic m quantum numbers

Since the discovery of the electron spin, 1925, this quantum state keeps some

mystery, since it cannot be described like an angular momentum, but shows

experimental evidences to be something like an angular momentum due to the

facts that spin

acts experimentally like something rotating in space having an intrinsic

couples with the normal orbital angular momentum to the resulting total

Since all fundamental particles have spin 1/2, this quantum phenomenon is

very important The usual description of spin 1/2 is based on the Pauli matrices,

avoiding any space-like imagination People who are used to work with

freedom Indeed, this can easily be done Let us look on the spherical harmonics

Y l,m( , ) given in (40) and the associated Legendre polynomials

)(d

d

x x x

m

m m m

with

4.2 Spin-1/2-particles in fractional description

angular momentum

has the physical dimension of an angular momentum

angular momentum like an angular momentum

is conserved as part of angular momentum

exhibits a magnetic moment expected from circulating currents

and some other things remembering on angular momentum

fractional order calculus will ask why physicists do not try to describe the spin

similar to the well-known angular momentum, but only with a further degree of

Krempl

but do not change the magnitude:

151

l l

l l

x l x

d

d

!2

(49)

derivatives in the expressions:

)(d

d

2 / 1

2 / 1 4

1 2 2

/ 1 2 /

x x x

1d

d

2 / 1

2 / 1 2

/

x x

We can merge these two equations to:

2 / 1 2 2 / 1 2 / 1

2 / 1 2 / 1 4

1 2 2

/ 1 2 /

d

d1

2

x x

x x

and need only to assume the existence of the fractional order derivatives, but

the two independent solutions (+ or ):

2expsincot1),(

~

2 / 1 , 2 / 1 2 /

satisfying the operator equations:

2 / 1 2 2

/ 1 , 2 2 / 1

)12

1(2

/ 1 2

/ 1

~2

for the spin operator S which is in complete analogy to the orbital angular

momentum L It might be noticed that usage of the generalised Legendre

polynomials will also yield such solutions The imaginary factor ( i) is due to

the selected branches of the roots and can be omitted, because if is an

eigenfunction, then c with arbitrary constant c is also an eigenfunction of the

differential operator

However, the physical interpretation of the wave function asks for a

normalisation Thus, the integral of its norm | * | over the whole space has to

to look on solution (53) Omitting for simplicity the unitary factor ( i) we have

and extend them to fractional order l = 1/2, m = 1/2 i.e., to evaluate the

semi-fortunately, do not need to evaluate them by fractional calculus Thus we obtain

be 1 But what is the “whole” space in our case ? To answer this question we have

SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS

152

differ monumentally from their analogues in the orbital angular momentum

functions

We immediately see, that there is a periodicity of 4 in the phase This means

that a spin-1/2-particle has to turn twice in the space to return in its initial state

This has intrigued physicists since the first experimental evidences of this

phenomenon [12,13] The interpretation of these experiments was taken into

doubt by many physicists, but recently confirmed again with modified

experiments avoiding the reasons for criticism [14] Today, this 4 -periodicity of

spin-1/2-particles becomes generally accepted If any theorist would have

introduced the fractional description of the spin just after his discovery in the

twenties of the last century, his model would have been made ridiculous due to

wave function is the probability for the particle to occupy the volume element of

the space, this space has to contain all possible configurations This means that

3integral for from 0 to 4 This means that our spin wave functions S for spin

1/2 have to look like:

2expsincot2

12 /

These functions form an orthonormal basis for all spin wave functions over the

3

2 The complete wave function of spin-1/2-particles is the product of the

wave function (r) describing the location of the particle in the 3, and the spin

wave function S over the spin space 3 2

this 4 -periodicity Perhaps, this was one of the reasons, why one did not believe

in a description similar to the orbital angular momentum Nowadays, “dynamical

picture

phases” or similar concepts [15] are proposed to save the “classical” spinor

2 We have to extend the

Krempl

Wave function of spin-1/2-particles

Let us now return to our question about the “whole space” Since the norm of the

twice our civilian space, i.e., the “whole” space is

(c) Ladder operators

the phases /2, which do not affect the magnitude of these functions, but which

(b)

153

y

x iS S

create (+) or annihilate ( ) a magnetic spin quantum according to

S s

which can be proved analogue to the proof for L previously given

The nice aspect of the fractional description of spin-1/2-particles is beside the

direct evidence of their 4 -periodicity the possibility of an interpretation in

the exponential term, which is also the sole complex part of the spin wave

of a particle with the mass mP is proportional to the gradient of the phase of its

Pr m

!

which tells us that we have a rotation around the chosen z-axes Spin +1/2

L z L

Since this fractional description of spin-1/2-particles allows its interpretation

proven if a rigorous application of this fractional description of the spin can

yield all the other observable results like the standard spinor description

(d) Interpretation of the spin

space In this picture, the spin-up and spin-down states differ only by the sign of

function yielding the phase = /2 In quantum mechanics, the “mean velocity”

Applying this to the spin wave functions (56) we get in polar coordinates:

corresponds to a right-handed rotation, spin 1/2 to a left-handed rotation

in the real space, it is complementary to Pauli’s spinor picture It has to be

SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS

These examples show that semi-integrals and semi-derivatives are appropriate

to describe natural phenomena Today, the application of semi-integrals in

154

connection with Abel-type integral equations pervades all natural and technical sciences, as well as modern medicine The fractional description of spin-1/2-particles can perhaps contribute to enlighten the mystery of the spin

5 Krempl PW (1974) The Abel-type integral transformation with the kernel (t2

-x2)−1/2 and its application to density distributions of particle beams CERN MPS/Int.BR/74-1, pp 1–31

6 Krempl PW (2005) Some Applications of Derivatives and Integrals in Physics Proc ENOC 05, Eindhoven, ID 11-363, 10 pp

Semi-7 Deans SR (1996) Radon and Abel Transforms in Poularikas AD (ed.), The

Transforms and Applications Handbook CRC Press, Boca Raton, pp 631–

11 Krempl PW (1975) Beamscope CERN PSB/Machine Experiment News 126b

12 Rauch H, Zeilinger A, Badurek G, Wilfing A (1975) Verification of coherent spinor rotation of fermions Phys Lett., 54A(6):425–427

13 Werner SA, Colella R, Overhauser AW, Eagen CF (1975) Observation of the phase shift of a neutron due to precession in a magnetic field Phys Rev Lett., 35(16):1053–1055

14 Ioffe A, Mezei F (2001) 4π-symmetry of the neutron wave function under space rotation, Physica B, 297:303–306

15 Hasegawa Y, Badurek G (1999) Noncommuting spinor rotation due to balaced geometrical and dynamical phases Phys Rev A, 59(3):4614–4622

Part 2

Classical Mechanics and Particle Physics

Raoul R Nigmatullin1 and Juan J Trujillo2

1

2

Abstract

averaged collective motion in the mesoscale region In other words, it means that

after a proper statistical average the microscopic dynamics is converted into a

relaxation that is widely used for description of relaxation phenomena in disordered media It is shown that the generalized stretched-exponential function describes the

integer integral and derivatives with real and complex exponents and their possible generalizations can be applicable for description of different relaxation or diffusion processes in the intermediate (mesoscale) region

Key words

VERSUS A RIEMANN–LIOUVILLE INTEGRAL TYPE

MESOSCOPIC FRACTIONAL KINETIC EQUATIONS

Tenerife Spain; E-mail: JTrujill@ull.es

Kazan, Tatarstan, Russian Federation; E-mail: nigmat@knet.ru

in the most cases the original of the memory function recovers the Riemann–

fractal-branched processes one can derive the stretched exponential law of

relaxation phenomena is also discussed These kinetic equations containing non

Generalized Riemann–Liouville fractional integral, universal decoupling procedure

Theoretical Physics Department, Kazan State University, Kremlevskaya 15, 420008, Departamento de Análisis Matemático, University of La Laguna, 38271, La Laguna.

It is proved that kinetic equations containing noninteger integrals and deriva- tives are appeared in the result of reduction of a set of micromotions to some

collective complex dynamics in the mesoscopic regime A fractal medium con- taining strongly correlated relaxation units has been considered It is shown that

Liouville fractional integral For a strongly correlated fractal medium a genera-

lization of the Riemann–Liouville fractional integral is obtained For the

averaged collective motion in the fractal-branched complex systems The appli- cation of the fractional kinetic equations for description of the dielectric

© 2007 Springer

155

in Physics and Engineering, 155 – 167

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

1 Introduction

integration/differentiation operators based on the given structure of a disordered

mechanics is absent So, there is a barest necessity to derive kinetic equations with

the statistical mechanics, based on the consideration of an infinite chain of equations for a set of correlation functions It becomes evident that equations with fractional derivatives can play a crucial role in description of kinetic and transfer phenomena in the mesoscale region From our point of view this necessary fractional calculus

In present time the interest in application of the mathematical apparatus of the fractional calculus in different branches of techniques and natural sciences is considerably increased Here one can remind the applications of the fractional calculus in

constitutive relations and other properties of various engineering materials such as viscoelastic polymers, foam, gel, and animal tissues, and their engineering and

Detailed references can be found in the recent review, in the proceedings of the The first attempt to understand the result of averaging of a smooth function over the given fractal (Cantor) set has been undertaken in [15] In the note and later in paper some doubts were raised to the reliability of the previously obtained result this paper (RRN) to reconsider the former result, and the detailed study of this problem showed that the doubts had some grounds and were directly linked with the relatively delicate procedure of averaging a smooth function over fractal sets, in particular, on Cantor set and its generalizations

integer operators with real fractional exponent [1–7] But in papers related to

Recently much attention has been paid to existence of equations containing non

integration or differentiation are realized on an “intuitive” level in the form of some

medium with the usage of the modern methods of nonequilibrium statistical noninteger operators of differentiation and integration from the first principles of

mathematical instrument should lie in deep understating of the “physics” of the

1 Fractional control of engineering systems

dynamic systems

3 Analytical and numerical tools and techniques

scientific applications

measurements and verifications

6 Bioengineering and biomedical applications

conference and in papers [2, 4, 8–14]

[15–17] The criticism expr essed in these publications forced one of the authors of

consideration of the fractional equations containing noninteger operators of

postulates/suppositions imposed on a structure or model considered At the pre-

sent time a systematic deduction of kinetic equations containing noninteger

Nigmatullin and Trujillo

2 Advancement of calculus of variations and optimal control to fractional

4 Fundamental explorations of the mechanical, electrical, and thermal

5 Fundamental understanding of wave and diffusion phenomenon, their

In order to dissipate these doubts and realize mathematically correct averaging procedure over fractal sets it was necessary to carry out a special study Complete investigation has been given in the book [18], where the correct averaging procedure was considered in detail One can prove that the previous result [15] is

correct for random fractals, for regular fractals the procedure of averaging of a

smooth function over fractal sets leads to the memory function expressed in terms further generalization for the modified Cantor sets has been realized in papers of

to a conclusion that the physical meaning of the fractional integral with real

exponent has been understood Temporal fractional integral can be interpreted as a conservation of part of states localized on a self-similar (fractal) object if the

associated with Cantor set or its generalizations, occupying an intermediate position between the classical Euclidean point and continuous line But the meaning of

fractional integral with real fractional exponent is not complete in the light of with the complex fractal dimensions is discussed These interesting ideas forced one

of the authors of this paper (RRN) to reconsider their previous results obtained in [18] and give a possibility to understand the geometrical/physical meaning of

mathematical operator with the complex fractional exponent [4] So the basic

question, which we are going to solve and discuss in this paper, can be formulated

as follows:

the mesoscale region from a kinetic equation with memory?

We are going to show that details of the averaging procedure developed in [18] will help us to find the proper answer for the question formulated above

present the basic equations of statistical mechanics containing a memory function

Liouville integral The general solutions containing log-periodic function help to imaginary part of the complex fractional exponent

of the Riemann–Liouville integral containing the complex power-law exponent The

papers [8, 23–26], where the correct understanding of different self-similar objects

Is it possible to suggest a “universal” decoupling procedure for a memory function in order to obtain noninteger operator with real or complex exponent in

medium In this section we show also how it is possible to generalize the Riemann–

understand the geometrical/physical meaning of noninteger operator containing an

physical system considered has at least two parts of different states One part

is distributed inside of a fractal set (the conserved part of states) and another part

of states is located outside of the fractal set (the lost part of states) That’s

why it is easy to understand the fractional integral of one-half order, when for

its understanding the consideration of a fractal object is not necessary Half of

states are lost automatically in diffusion process with semi-infinite boundary tions [22] From the geometrical point of view the temporal fractional integral is

condi-In section 3 we derive the memory function for a strongly correlated fractal The following content of this article obeys the next structure In the section 2 we

MESOSCOPIC FRACTIONAL KINETIC EQUATIONS

R Hilfer in the recent book [1] Independent analysis of above-cited papers could lead Prof Fu-Yao Ren with coauthors [19–21] Another approach leading to the fractional integral and related to coarse graining time averaging is considered by

structures, which, in turn, help to derive the stretched-exponential law of relaxation

In many branches of physics the relationship between physical values are related by means of memory function For example, in the theory of linear response the deviation from the mean physical value evoked by the applied external field can be expressed as [27]

value F(t1) defines an amplitude of the external field, entering into the perturbation Hamiltonian

Here B corresponds to a quantum-mechanical operator, which determines the

interaction of the many-body Hamiltonian with external field For example, in the

case of interaction with electric field B coincides with the operator of total polarization P, for magnetic field the operator B corresponds to the magnetization operator M and etc.

formalism [28] then the relationship between the autocorrelation function of the

second order M1(t) with correlation functions of higher orders can be written as

Here 1 and 1 are some characteristic parameters, K(t) is the correlation

function of the next order, which plays a role of a memory function for the initial

1and similar variant of this equation was derived in papers [29]

Based on the Zubarev kinetic formalism one can derive the diffusion equation with memory [27,30,31]

2 1 , ,

and n(r, t) coincides with local density of electric dipoles or spins

RL integral In section 6 we determine the strong-correlated fractal-branched The basic results are summarized in final section 7

2 Different Kinetic Equations with Memory

If one can try to consider the dynamics of the system in the Mori–Zwanzig

correlation function M (t) An analog of Eq (3) with specific memory function

In sections 4 and 5 we consider the basic mathematical properties of the generalized

Nigmatullin and Trujillo

159Other equations with memory in the framework of the generalized Zubarev

some assumption related with the calculation of a memory function We are going

to suggest rather general decoupling procedure for calculation of the memory function based on a self-similar structure of the medium considered

has been considered recently in [33] The special procedure for recognition of the

For the case of strongly correlated clusters one can suppose that the memory

function forms a self-similar structure combining these clusters in the form of a

product Such formation is possible, for example, in the case of percolation

In this section we are going to show that evaluation of expression (7) does not

depend on the concrete form of the microscopic function f (z) One can notice that the product satisfies to the following exact equation

3 Memory Function for a Strongly Correlated Fractal Medium

In this paper we want to derive a structure of a kernel K (t) for a strongly correlated

[34–36] For further purpose it is useful to use the Laplace/Fourier transform applied to Eq (1), (3), and (4) in order to have a possibility to consider the kernel

K(t) separately.

suppositions made in the previous section 3 the Laplace image of the memory

f

Without loss of a generality we suppose that the Laplace-image of the function

(z) depending on a complex variable z (the variable z defines the dimensionless Laplace parameter z = s with respect to some characteristic time ) and describing

order to derive possible “fractional” kinetic equations it is necessary to impose

fractal medium The structure of the kernel for a weakly correlated fractal medium

“fractional” kinetics from dielectric spectroscopy data has been suggested in papers

MESOSCOPIC FRACTIONAL KINETIC EQUATIONS

describing the interaction process of a dipole with thermostat The reason for such division is that the minimum value of the function ( )f t LT f s( )(we are using the

same notation f for original f(t) and Laplace image f(s) and suppose that it does not evoke further misunderstanding) in the first case is f(t = 0) 0 and moreover f(t)

F (t = 0) = 0 and the microscopic function f (t) has at least one maximum and so

may tend to zero monotonically or nonmonotonically as t

For K = P the process of interaction with thermostat has delta-like function (t)

collision character

Taking into account the asymptotic decompositions (10) at < 1 in the limit N

the last relationship for the fixed N is reduced to the scaling functional equation of

P K N

K P K P P

z

2 0

N Mx

b z

a b a b (13)

The limiting parameters entering into the last expression are defined by expression (9)

At P = K that physically corresponds the (t)-like collisions with thermostat the

solution of the last functional Eq (11) has the form

tends to zero as t monotonically In the exchange case, however, the value

Nigmatullin and Trujillo

with K P + 1 and the polynomial in denominator has only negative and

complex-conjugated roots

0 1 0

ln(1/ )

ln(1/ )

b A b

A

For P K one can obtain the general solution of the scaling equation by the

method of a free constant variation Taking the natural logarithm from the both part

the values 0 The properties of the kernel (22) need a special mathematical

examination and considered in the next section For b > 1 it is possible to check by direct calculations that the solution for P (z) can be written in the same form (20)

with constants determined by expression (21)

of Eq (11) we have

The solution of the functional Eq (16) we are presenting in the form

ln( ), C , are free variation constants, which are determined from Eq (15)

From Eq (18) one can obtain for < 1

dz (22)

generalizes the conventional definition [37] of the Riemann–Liouville integral for

MESOSCOPIC FRACTIONAL KINETIC EQUATIONS

(14) 161

,

(K f t)( ), which depends on two parameters

and acts on a smooth and arbitrary function f(t) as:

2 ln( ) (ln ( ))

(K f t)( ) L e s s ( ) * ( )t f t , (23) where the asterisk

0

( ) * ( ) ( ) ( ) ,

t

f t g t f t x g x dx (24) determines the Laplace convolution operation This GFIO can be written in the

0

Q u d u

and differential equation for the function Y(s) = exp[- ln(s)- ln2s] that it is obtained

easily in s-complex plane

4 The Analytical Form of the Kernel K(t)

fractional integral operator (GFIO)

following implicit form in time domain:

(E = 0.5772156649… is the Euler constant) and initial conditions Y(0) = 0 or

Y(1) = 1 one can obtain from (27) the desired Volterra equation (26)

Nigmatullin and Trujillo

In this section it is convenient to give another definition of the generalized

fractional operator, which proof hold from the definition (23)

n

The physical meaning of this function is related to the fact that microscopic

relaxation function f(z) has additional branching in the self-similar volumes V n

n0b n The variable z in (29) can coincide with a dimensionless frequency variable or

a temporal variable t, respectively The evaluation of the last expression depends essentially on the asymptotic behavior of the function f(z) and from the interval of

1

By analogy with expression (11) one can show that the sum S N (z) figuring in

expression (29) satisfies to the relationship

5

The new fractional Riemann–Liouville type operator which we have introduced

It recovers the fractional Riemann–Liouville integral operator

is the fractional Riemann–Liouville derivative

6 Consideration of Relaxation Processes in the Fractal-Branched Structures

Basic Properties of the Generalized Riemann–Liouville Integral

location of the scaling parameters and b We suppose that these scaling para-

meters satisfy to the following inequality

MESOSCOPIC FRACTIONAL KINETIC EQUATIONS

( ) Aexp( ) A exp( 2 )

In the last relationship we combined the exponential and power-law asymptotic

in order to consider them together At = P – K 0 and r = 0 it describes the exponential asymptotic In the limit N one can obtain the following scaling

equation for the limiting value of the sum S(z)

(37)

The further investigations show that for the case b < 1, < 1 the value of the constant c1 = 0 in expansion (32a); the power-law exponents and can be

power-law asymptotic similar to (8); the case = 0 and r 0 corresponds to the

period ln( ) In the one-mode approximation (OMA) this function can be presented

in the form

Nigmatullin and Trujillo

simultaneously negative satisfying to condition > 0 The damping constant can

accept positive or negative values

The case (b = 1, < 1) generalizes expression (20) obtained above

discovered and mathematically confirmed the reduction phenomenon, when a set of

micromotions is averaged and transformed again into a collective motion It is

interesting to note that different partial cases (for some concrete forms of f(t),

exponential dependence in time domain has been considered by many authors in

Approach developed in this paper helps to understand the general decoupling procedure applied to a memory function that can lead to equations containing non-integer integrals and derivatives with real or complex power-law exponents These equations naturally explain temporal irreversibility phenomena which can be

a many-body system lost many microscopic states and only part of states in the form of collective motions are conserved on the following level of intermediate scales and expressed in the form of the fractional integral This approach opens new possibilities for analysis of different kinetic equations with remnant memory, when the RL-operators can be modified by a damping constant defined by (21) or new convolution term appearing in the Laplace image appearing in (20) The temperature dependence of the power-law exponents and < >, which can enter into the corresponding kinetic equation merits a special examination

Expression (38) generalizes the well-known Kohlrausch–Williams–Watts

describing different types of micromotions) leading to the “pure” section 8 published in the Proceedings of the International Symposium [41] These

relaxation law suggested many years ago for description of nonexponential relaxation phenomena in many disordered systems [39, 40] As before, we

nonexponential functions have been applied for description of relaxation pheno- mena of statistical defects in condensed media, in glasses etc

appeared in linear systems with “remnant” memory For linear systems the

so-called “partial” irreversibility is appeared in the result of reduction procedure, when

MESOSCOPIC FRACTIONAL KINETIC EQUATIONS

166 Nigmatullin and Trujillo

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