Fractional Generalization of Kac Integral

F ractional Generalizati o n of Kac In tegral V asily E. T araso v 1 , 2 , and George M. Zasla vsky 1 , 3 1) Cour ant Institute of Mathematic al Scienc es, New Y ork Univers i ty 251 Mer c er St., New Y ork, NY 10 012, USA 2) Skob eltsyn I n stitute of Nucle ar Physics, Mosc ow State University, Mosc ow 119992 , Russia 3) Dep artment of Physics, New Y ork University, 2-4 Washington Plac e, New Y ork, NY 100 03, USA Abstract Generalizati on of the K ac integral and K ac metho d for p aths m easure b ased on the L ´ evy distribution h as b een used to deriv e fractio nal diffusion equation. Application to nonlinear fractional Ginzburg-Landau equation is discuss ed . 1 In tr o d u ction Kac in tegral [1, 2, 3] app ears as a path-wise presen tatio n of Brownian motion and shortly b ecomes, with F ey nman approach [4], a p o w erful to o l to study differen t pro cesses describ ed b y the w a v e-t yp e or diffusion-t yp e equations. In the basic pap ers [1 , 4], the paths distribution w as based on av eraging ov er the Wiener measure. It is w orthw hile to men tion the Kac commen t that the Wiener measure can b e replaced b y the L´ evy distribution that has infinite second and higher momen ts. There exists a fa irly ric h literature related to functional integrals with generalization o f the Wiener measure (see for example [5, 6]). Recen tly the L ´ evy measure w a s applied to deriv e a fr a ctional generalization of the Schr¨ odinger equation [7, 8] using the F eynman-t yp e approac h and expres sing the L ´ evy measure throug h the F ox function [9] In this pap er, w e derive the fractional generalization of the diffusion equation (FDE) from the path in tegral ov er the L ´ evy measure using the integral equation approach of Kac. 1 2 L ´ evy distributio n Let us consider the transition probability P ( x, t | x ′ , t ′ ) tha t describ es the ev o lut io n of the prob- abilit y densit y ρ ( x, t ) by the equation ρ ( x, t ) = Z + ∞ −∞ dx ′ P ( x, t | x ′ , t ′ ) ρ ( x ′ , t ′ ) , (1) where Z + ∞ −∞ dxρ ( x, t ) = 1 . (2) The function P ( x, t | x ′ , t ′ ) can b e considered as conditional distribution f unction. Then the normalization condition Z + ∞ −∞ dxP ( x, t | x ′ , t ′ ) = 1 (3) holds. Assume that P ( x, t | x ′ , t ′ ) satisfies t he Marko vian (semigroup) condition P ( x, t | x 0 , t 0 ) = Z + ∞ −∞ dx ′ P ( x, t | x ′ , t ′ ) P ( x ′ , t ′ | x 0 , t 0 ) (4) kno wn also as the Chapman-Kolmogorov equation. In phys ical theories, the stabilit y of a family o f probabilit y distributions is an imp ortan t prop ert y whic h basic ally states that if one has a n um b er of random v aria bles that b elong to some family , a n y linear com bination of these v ariables will also b e in this family . The imp ortance of a stable family of probabilit y distributions is t ha t they serv e as ”attractor s” for linear com binatio ns of non-stable random v a r iables. The most noted examples are the normal Gaussian distributions, whic h form one fa mily of stable distributions. By the classical central limit theorem the linear sum of a set of random v ar iables, eac h with a finite v ariance, tends to the normal distribution a s the n umber of v ar ia bles increases . All con tinuous stable distributions can b e sp ecified by the pro p er choice of para meters in the L´ evy sk ew alpha-stable distribution [10] that is defined by L ( x, y , α, β , c ) = 1 2 π Z + ∞ −∞ dp e − ipx U ( p, y , α, β , c ) , (5) where U ( p, y , α, β , c ) = ex p  iy p − | cp | α [1 − iβ sig n ( p )Φ( α , p )]  , (6) 2 and Φ( α, p ) =      tan( π α / 2) , 0 < α ≤ 2 , α 6 = 1; − (2 /π ) log | p | , α = 1 . (7) Here y is a shift parameter, β is a measure o f asymmetry , with β = 0 yielding a distribution symmetric ab o ut y . In Eq. (6), parameter c is a scale factor, which is a measure of the width of t he distribution a nd α is the expo nen t or index of the distribution. Consider P ( y , t ′ | x, t ) as a symmetric homog eneous L ´ evy a lpha-stable distribution P ( y , t ′ | x, t ) ≡ K ( y − x, t ′ − t ) = 1 2 π Z + ∞ −∞ dp exp  ip ( y − x ) − ( t ′ − t ) C α | p | α  , (0 < α ≤ 2) . (8) F or α = 2, Eq. ( 8 ) gives the Gauss distribution P ( y , t ′ | x, t ) = 1 p 4 π C 2 ( t ′ − t ) exp  − 1 4 C 2 ( t ′ − t ) ( y − x ) 2  . (9) Eq. (8) giv es the function K ( x, t ) = 1 2 π Z + ∞ −∞ dp exp  ipx − tC α | p | α  (10) that can b e presen ted a s a F ourier tr a nsform K ( x, t ) = F − 1  e − tC α | p | α  , (11) where F − 1 ( f ( p )) = 1 2 π Z + ∞ −∞ dp e ipx f ( p ) . (12) F or α = 2, Eq. ( 1 1) g iv es K ( x, t ) = 1 √ 4 π C 2 t exp  − x 2 4 C 2 t  . (13) In the general case , the function K ( x, t ), giv en by Eq. (11), can b e expressed in terms of the F o x H -function [7, 8, 9, 11, 12, 13, 14 ] (see App endix). 3 3 F ractional Kac path in tegral Let us denote by C [ t a , t b ] the set o f tra jectories starting at the p oint x a = x ( t a ) at the time t a and ha ving the endp oint x b = x ( t b ) at the time t b . The Kac functional integral [2, 3, 15] is W ( x b , t b | x a , t a ) = Z C [ t a ,t b ] D W x ( t ) exp  − Z t b t a dτ V ( x ( τ ))  , (14) where V ( x ) is some function, and D W x = lim n →∞ n Y k =1 K (∆ x k , ∆ t k ) dx k . (15) F or ( 1 3), expression (1 5) g ives D W x = lim n →∞ n Y k =1 dx k √ 4 π C 2 ∆ t k exp  − (∆ x k ) 2 4 C 2 ∆ t k  , (16) whic h is the Wiener measure o f functional integration [15]. The in tegral (14) is also called the F eynman-Kac in tegra l. Using (10) for α = 2, the path in tegral (14) can b e written a s W ( x b , t b | x a , t a ) = lim n →∞ 1 (2 π ) n Z R 2 n dx 1 dp 1 ... d x n dp n exp n X k =0  ip k ∆ x k − ∆ t k [ C 2 p 2 k + V ( x k )]  , (17) where the time inte rv al [ t a , t b ] is par t it io ned a s t k = t a + k t b − t a n , t 0 = t a , t n = t b , (18) and ∆ x k = x k +1 − x k , ∆ t k = t k +1 − t k , x k = x ( t k ) , p k = p ( t k ) . (19) The functional integral (17 ) can b e rewritten as W ( x b , t b | x a , t a ) = Z D x D p exp  Z t b t a dt h ip ˙ x − C α p 2 − V ( x ) i . (20) where D x = lim n →∞ n Y k =1 dx k , D p = lim n →∞ n Y k =1 dp k 2 π . (21) 4 The Kac functional in tegral in the for m (20 ) is a classical analog of the F eynman phase-space path in tegral, whic h is also called the path integral in Hamiltonian form. F or the fractional g eneralization of Wiener measure (15) and Kac in tegral (14), w e consider K ( x, t ) given by (10). Substitution of (10) in to W ( x b , t b | x a , t a ) = lim n →∞ Z R n n Y k =1 dx k K (∆ x k , ∆ t k ) exp  − ∆ t k V ( x k )  (22) with K (∆ x k , ∆ t k ) = 1 2 π Z + ∞ −∞ dp k exp  ip k ∆ x k − ∆ t k C α | p k | α  , (0 < α ≤ 2) , (23) giv es W ( x b , t b | x a , t a ) = lim n →∞ Z R 2 n n Y k =1 dx k dp k 2 π exp n X k =0  ip k ∆ x k − ∆ t k [ C α | p k | α + V ( x k )]  . (24) Similarly to (20), (2 1) t his expression can b e written as W ( x b , t b | x a , t a ) = Z D x D p exp  Z t b t a dt h ip ˙ x − C α | p | α − V ( x ) i . (25) This express ion is a fra ctional generalization of (2 0). If w e in tro duce formally imaginary time such that i ˙ x = i dx dt = dx ds , then (25) tra nsforms into the F eynman path in t egral with a generalized action [7, 8] S [ x, p ] = Z t b t a dt h p ˙ x − C α | p | α − V ( x ) i as an action. Hamiltonian- t yp e formal equations o f motion are dx ds = N α | p | α − 1 , dp ds = − ∂ V ( x ) ∂ x , (26) where N α = αC α sig n ( p ). 5 4 F ractional diffusion e quations It is kno wn that the Ka c integral (14) can b e considered as a solutio n of the diffusion equation [2, 15]. Let us deriv e the corresp onding diffusion equation for the fractional generalization of the Kac inte gral (25). In (25) the in tegration is p erformed ov er a set C [ t a , t b ] of tra jectories that star t at p oint x a = x ( t a ) at time t a and end at p oin t x b = x ( t b ) at time t b . F or simplification, t a = 0, x a = 0, and t b = t , x b = x are used. In particular, we can consider t w o follow ing cases of C [ t a , t b ]. (1) The set C f [0 , t ] consists of paths for whic h b o th the initial and final p oints are fixed. The integration ov er this set ob viously giv es the transition probabilit y Z C f [ t a ,t b ] D W x = K ( x b − x a , t b − t a ) = P ( x b , t b | x a , t a ) , or Z C f [0 ,t ] D W x = K ( x, t ) . The conditional fr actional Wiener measure corresp onds to the integration o v er the set C f [0 , t ] of pa ths with fixed endp oints: x a = 0, x b = x . (2) If w e consider a set C a [0 , t ] of tra jectories with arbitrary endp oin t x b = x , the measure is called the unconditional fractional Wiener measure. This measure satisfies the normalization condition Z C a [0 ,t ] D W x = Z + ∞ −∞ dx Z C f [0 ,t ] D W x = Z + ∞ −∞ dx K ( x, t ) = 1 , (27) since it is a probabilit y that the system ends up any where. F or simplification, we intro duce the notation Z [ x, t ] = exp  − Z t 0 dτ V ( x ( τ ))  , (28) and define the field u ( x, t ) = W ( x, t | 0 , 0) . (29) F or t he fractiona l Kac functional in tegra l, w e hav e with resp ect to (27), Z C a [0 ,t ] D W x Z [ x, t ] = Z + ∞ −∞ dx Z C f [0 ,t ] D W x Z [ x, t ] . (30) 6 Using notations (28), (29), express ion (25) for t a = 0, x a = 0, a nd t b = t , x b = x can be presen ted as u ( x, t ) = Z C f [0 ,t ] D W x exp  − Z t 0 dτ V ( x ( τ ))  = Z C [0 ,t ] D W xZ [ x, t ] . (31) T o derive a fractional diffusion equation, w e use the iden t ity [15] exp  − Z t 0 dτ V ( x ( τ ))  = 1 − Z t 0 dτ h V ( x ( τ )) exp  − Z τ 0 dsV ( x ( s ) i . (32) Equation ( 32) can b e prov ed b y using differen tiation b y t , and the v alue of the constan t is found from the condition o f coincidence of b oth sides for t = 0 . F o r the notatio n (29 ), identit y (3 2) has the fo r m Z [ x, t ] = 1 − Z t 0 dτ h V ( x ( τ )) Z [ x, τ ] i . (33) Equation (33) can b e integrated with resp ect to the conditional fra ctional Wiener measure: Z C f [0 ,t ] D W x Z [ x, t ] = Z C f [0 ,t ] D W x 1 − Z C f [0 ,t ] D W x Z t 0 dτ h V ( x ( τ )) Z [ x, τ ] i . (34) Changing the o rder of the integration in the second term in the right hand-side of (34), we get Z C f [0 ,t ] D W x Z t 0 dτ h V ( x ( τ )) Z [ x, τ ] i = Z t 0 dτ Z C f [0 ,t ] D W x h V ( x ( τ )) Z [ x, τ ] i = = Z t 0 dτ Z + ∞ −∞ dx τ Z C f [0 ,τ ] D W x Z C f [ τ , t ] D W x h V ( x ( τ )) Z [ x, τ ] i = = Z t 0 dτ Z + ∞ −∞ dx τ V ( x ( τ )) Z C f [0 ,τ ] D W x Z [ x, τ ] Z C f [ τ , t ] D W x. (35) The first t erm in the righ t hand-side of (34) g iv es Z C f [ t a ,t b ] D W x 1 = lim n →∞ Z R n n Y k =1 dx k K (∆ x k , ∆ t k ) = = lim n →∞ Z R n n Y k =1 dx k K (∆ x k , ∆ t k ) = K ( x b − x a , t b − t a ) . (36) Using (29), (3 6 ), and (35 ) , Eq. (34) giv es the in tegral equation u ( x, t ) = K ( x, t ) − Z t 0 dτ Z + ∞ −∞ dx τ V ( x τ ) u ( x τ , τ ) K ( x − x τ , t − τ ) . (37) 7 F or t his equation there exists the infinitesimal o p erator L α (generator) of time shift suc h that ∂ u ( x, t ) ∂ t = L α u ( x, t ) . (38) Using (31) and (11 ) , we obtain L α u ( x, t ) = C α ∂ α ∂ | x | α u ( x, t ) − lim t → 0 1 t Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) u ( y , τ ) , (39) where ∂ α /∂ | x | α is a fr actional Riesz deriv ative [1 6, 17, 18 , 19] of o r der 0 < α < 2 that is defined b y its F ourier transform ∂ α ∂ | x | α u ( x, t ) = F − 1  | p | α ˜ u ( p, t )  = 1 2 π Z + ∞ −∞ dp | p | α ˜ u ( p, t ) e − ipx , (40) where ˜ u ( p, t ) = Z + ∞ −∞ dx u ( x, t ) e ipx . (41) The initial condition K ( x, 0) = δ ( x ) giv es [15] lim t → 0 1 t Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) u ( y , τ ) = V ( x ) u ( x, t ) . (42) Then (39) giv es L α = C α ∂ α ∂ | x | α − V ( x ) . (43) This generator is an op erato r of fractional differen tiatio n of order α . As a result, w e obtain ∂ u ( x, t ) ∂ t = C α ∂ α u ( x, t ) ∂ | x | α − V ( x ) u ( x, t ) , (44) whic h is a diffusion equation with fractional co ordinate deriv ativ es. F o r α = 2, Eq. (44 ) is the usual diffusion equation. It is worth while to men tion that the wa y of obtaining fractional equation (44) is based on the exploiting the prop erties of integral eq uation (37), while the expansion of exp onents in (24) o v er small ∆ t k has b een used in [7, 8] fo r F eynman path in tegral. 8 5 F ractional diffusion e quations b y Kac approac h It is useful also to deriv e the fractional diffusion equation f rom (14) using Kac approac h de- scrib ed in Sec. 4. of [2]. The mathematical expectation v alue of Z [ x, t ] is defined as E  exp  − Z t 0 dτ V ( x ( τ ))   = Z C a [0 ,t ] D w x exp  Z t 0 dτ V ( x ( τ )  . (45) Using the expansion exp  − Z t 0 dτ V ( x ( τ ))  = ∞ X m =0 ( − 1) m m !  Z t 0 dτ V ( x ( τ ))  m , (46) w e get E  exp  − Z t 0 dτ V ( x ( τ ))   = ∞ X m =0 ( − 1) m m ! Z C a [0 ,t ] D W x  Z t 0 dτ V ( x ( τ )  m . (47) The expression (47 ) can b e presen ted as E  exp  − Z t 0 dτ V ( x ( τ ))   = ∞ X m =0 ( − 1) m Z + ∞ −∞ dx Q m ( x, t ) , (48) where Q m ( x, t ) = 1 m ! Z C f [0 ,t ] D W x  Z t 0 dτ V ( x ( τ ))  m . (49) These f unctions ( 4 9) satisfy the recurrence equations [2] Q m +1 ( x, t ) = Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) Q m ( y , τ ) , (50) and Q 0 ( x, t ) = K ( x, t ) . (51) Let us introduce Q ( x, t ) = ∞ X m =0 ( − 1) m Q m ( x, t ) . (52) Then Q ( x, t ) = ∞ X m =1 ( − 1) m m ! Z C f [0 ,t ] D W x exp  Z t 0 dτ V ( x ( τ ))  m = 9 = Z C f [0 ,t ] D W x exp  Z t 0 dτ V ( x ( τ )  , (53) and E  exp  − Z t 0 dτ V ( x ( τ ))   = Z + ∞ −∞ dx Q ( x, t ) . (54) It follo ws from (50) and (51) that the field Q ( x, t ) satisfies the integral equation Q ( x, t ) = Q 0 ( x, t ) − Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) Q ( y , τ ) . (55) There exists an infinitesimal op erator L α of t ime shift suc h that ∂ Q ( x, t ) ∂ t = L α Q ( x, t ) . (56) Using (50) (4 9), and (11), this g enerator can b e expressed through a fractional differen t ial op erator L α Q ( x, t ) = C α ∂ α ∂ | x | α Q ( x, t ) − lim t → 0 1 t Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) Q ( y , τ ) . (57) The initial condition K ( x, 0) = δ ( x ) giv es similar to (42) lim t → 0 1 t Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) Q ( y , τ ) = V ( x ) Q ( x, t ) . (58 ) As a result, w e obtain ∂ Q ( x, t ) ∂ t = C α ∂ α Q ( x, t ) ∂ | x | α − V ( x ) Q ( x , t ) , (59) whic h is fractiona l diffusion equation that coincides with ( 44). Then Q ( x, t ) = W ( x, t | 0 , 0) = Z C f [0 ,t ] D W x exp  − Z t 0 dτ V ( x ( τ ))  . (60) Using (52), the approximate solution of (44) can b e presen ted as u ( x, t ) ≈ Q 0 ( x, t ) − Q 1 ( x, t ) + Q 2 ( x, t ) = = K ( x, t ) − Z t 0 dτ Z + ∞ −∞ dy K ( x − y , t − τ ) V ( y ) K ( y , τ )+ + Z t 0 dτ Z τ 0 dt ′ Z + ∞ −∞ dy Z + ∞ −∞ dy ′ K ( x − y , t − τ ) V ( y ) K ( y − y ′ , τ − t ′ ) V ( y ′ ) K ( y ′ , t ′ ) . (61) for small enough V ( x ). 10 6 Nonlinear fractional equations Equations ( 4 4) and (59) are linear equations with respect to the fields u ( x, t ) and Q ( x, t ). In general, nonlinear equations can b e deriv ed from the functional in tegral ov er the space of branc hing paths (see [21] and Sec. VI.4. of [2 0]). Note that F eynman path integral ov er the branc hing paths has b een suggested in [22] (see also [23, 2 4]). The m ultiplicative represen tations of nonlinear diffusion equations a re also considered in [25, 26, 27]. As a n example of no nlinear diffusion equation, whic h can b e deriv ed from in tegrals o v er the branching paths, is an equation with the p olynomial nonlinearit y [20, 21]: U ( u ) = m X k =2 a k [ u ( x, t )] k . (62) Using fractional Kac in tegral ov er the branc hing L´ evy paths [28, 29], a nonlinear generalization of f ractional equation (44) can b e deriv ed in the f orm ∂ u ( x, t ) ∂ t = C α ∂ α u ( x, t ) ∂ | x | α − V ( x ) u ( x, t ) + m X k =2 a k [ u ( x, t )] k . (63) F or example, fra ctional equations with cubical nonlinearity can b e obtained ∂ u ( x, t ) ∂ t = C α ∂ α u ( x, t ) ∂ | x | α − V ( x ) u ( x, t ) + a 3 [ u ( x, t )] 3 . (64) Equation (64) is the fractional g eneralization of the Gross-Pitaevskii equation [30, 31]. F or V ( x ) = const , Eq. (64) is fractional Ginzburg-Landau equation that is suggested in [32] (see also [33, 3 4]) to describ e complex media with fractional disp ersion la w. 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A 337 (2005) 75-80 ( cond- mat/030957 7). 14 App endix: F o x function r e presentation for K ( x, t ) In this sec tion, w e use t he results of the pap er [7] (see also [8]) to demonstrate ho w the function K ( x, t ) defined b y Eq. (10) can b e expressed in the terms of the F ox H -function [9, 11, 1 2 , 13, 14]. The F o x function represen tation of K ( x, t ) can b e considered as a fractional a na log of expression (13). T o presen t K ( x, t ) in terms of the F o x H -function, w e consider the Mellin transform of (10). C omparing of the inv erse Mellin transform with the definition of the F ox function , w e obtain an express ion in terms o f F ox H - function. Using the relation K ( x, t ) = K ( − x, t ), it is sufficien t to consider K ( x, t ) for x ≥ 0 only . The Mellin tra nsfor ma t io n of (10) is ∧ K ( s, t ) = Z ∞ 0 dx x s − 1 K ( x, t ) = 1 2 π Z ∞ 0 dx x s − 1 Z + ∞ −∞ dp exp  ipx − C α | p | α t  . (65) Changing the v a riables p → ( C α t ) − 1 /α η , x → ( C α t ) 1 /α ξ , w e presen t ∧ K ( s, t ) as ∧ K ( s, t ) = 1 2 π  ( C α t ) 1 /α  s − 1 Z ∞ 0 dξ ξ s − 1 Z + ∞ −∞ dη e iηξ −| η | α . (66) The integrals o ver d ξ and dη can b e ev a lua ted b y using the equation [13]: Z ∞ 0 dξ ξ s − 1 Z ∞ 0 dη e iηξ − η α = 4 s − 1 sin π ( s − 1) 2 Γ( s )Γ  1 − s − 1 α  , (67) where s − 1 < α ≤ 2 and Γ( s ) is the Gamma function. Inserting of (67) into (66) and using the relations Γ(1 − z ) = − z Γ( − z ) , Γ( z )Γ(1 − z ) = π / sin π z , (68) w e find ∧ K ( s, t ) = 1 α  ( C α t ) 1 /α  s − 1 Γ( s )Γ( 1 − s α ) Γ( 1 − s 2 )Γ( 1+ s 2 ) . (69) Then the inv erse Mellin transform of (69) is K ( x, t ) = 1 2 π i c + i ∞ Z c − i ∞ dsx − s ∧ K ( s, t ) = 1 2 π i 1 α c + i ∞ Z c − i ∞ ds  ( C α t ) 1 /α  s − 1 x − s Γ( s )Γ( 1 − s α ) Γ( 1 − s 2 )Γ( 1+ s 2 ) , (70) 15 where t he integration con tour is t he straight line from c − i ∞ to c + i ∞ with 0 < c < 1 . Replacing s b y − s , w e get K ( x, t ) = 1 α ( C α t ) − 1 /α 1 2 π i − c + i ∞ Z − c − i ∞ ds  ( C α t ) − 1 /α x  s Γ( − s )Γ( 1+ s α ) Γ( 1+ s 2 )Γ( 1 − s 2 ) . (71) The integration contour may b e deformed in to one running clo ck wise around [ − c, ∞ ). Com- parison with the definition of the F o x H -function [9, 11, 12] giv es K ( x, t ) = 1 α ( C α t ) − 1 /α H 1 , 1 2 , 2  ( C α t ) − 1 /α x    (1 − 1 /α, 1 / α ) , (1 / 2 , 1 / 2) (0 , 1) , (1 / 2 , 1 / 2)  . (72) Using the prop erties of the F ox H - f unction [9, 11, 12], w e o bta in K ( x, t ) = 1 α | x | H 1 , 1 2 , 2  ( C α t ) − 1 /α | x |    (1 , 1 /α ) , (1 , 1 / 2) (1 , 1) , (1 , 1 / 2)  . (73) Let us sho w by analogy with [7] (see also [8]) that Eq. (73) includes as a particular case at α = 2 the w ell kno wn Gauss distribution (13). Assuming α = 2 in Eq. (73), K ( x, t ) | α =2 = H 1 , 1 2 , 2  ( C 2 t ) − 1 / 2 | x |    (1 , 1 / 2) , ( 1 , 1 / 2) (1 , 1) , (1 , 1 / 2)  . (74) The series expansion of the function (74) giv es K ( x, t ) | α =2 = 1 2 ( C 2 t ) − 1 / 2 ∞ X k =0  − ( C 2 t ) − 1 / 2  k | x | k k ! 1 Γ( 1 − k 2 ) . (75) Substituting of k → 2 l in to (75), and using Γ  1 2 − l  = √ π ( − 1) l (2 l )! (2) 2 l l ! , (76) the function K ( x, t ) can b e rewritten as K ( x, t ) | α =2 = ( C 2 t ) − 1 / 2 2 √ π ∞ X l =0  − ( C 2 t ) − 1 / 2  2 l ( − 1) l x 2 l 2 2 l l ! = 1 √ 4 π C 2 t exp  − x 2 4 C 2 t  . (77) Th us, it is sho wn that (13) can b e deriv ed fro m equation (73) with α = 2. 16