Anomalous diffusion in umbrella comb
aa r X i v : . [ c ond - m a t . d i s - nn ] O c t Anomalous diffusion in umbrella comb
A. Iomin
Department of Physics, Technion, Haifa, 32000, Israel
Abstract
Anomalous transport in a circular comb is considered. The circular motiontakes place for a fixed radius, while radii are continuously distributed alongthe circle. Two scenarios of the anomalous transport, related to the reflectingand periodic angular boundary conditions, are studied. The first scenario withthe reflection boundary conditions for the circular diffusion corresponds to theconformal mapping of a 2D comb Fokker-Planck equation on the circular comb.This topologically constraint motion is named umbrella comb model. In thiscase, the reflecting boundary conditions are imposed on the circular (rotator)motion, while the radial motion corresponds to geometric Brownian motionwith vanishing to zero boundary conditions on infinity. The radial diffusion isdescribed by the log-normal distribution, which corresponds to exponentiallyfast motion with the mean squared displacement (MSD) of the order of e t . Thesecond scenario corresponds to the circular diffusion with periodic boundaryconditions and the outward radial diffusion with vanishing to zero boundaryconditions at infinity. In this case the radial motion corresponds to normaldiffusion. The circular motion in both scenarios is a superposition of cosinefunctions that results in the stationary Bernoulli polynomials for the probabilitydistributions. Keywords:
Circular comb model, Conformal mapping, Geometric Brownianmotion, Log-normal distribution, Subdiffusion
Email address: [email protected] (A. Iomin)
Preprint submitted to Journal of L A TEX Templates October 14, 2020 . Introduction
In this paper we consider circular and radial motions in combs of circulargeometry, see Fig. 1, where the radii are continuously distributed over the circle,and the circular motion takes place for the fixed radius r = R , only. Fractionaldiffusion in this geometry has been studied recently, where both outward andinward radial diffusion has been considered analytically [1] and numerically [2].Finite time evolution of both angular and radial probability distribution func-tions as well as the mean squared displacement have been observed analytically[1] and numerically [1, 2] for different realizations of the boundary conditionsfor both angular and radial motions. Further analytical study of the system isimportant to understand asymptotic transport in the system. We consider twopossibilities of boundary conditions for angular diffusions. The first one corre-sponds to the reflecting boundary condition, and the second one correspondsto the periodic boundary condition. Our study of anomalous diffusion in thiscomb geometry is also motivated by consideration of an idealized radial trans-port, which can be also related to the radial transport model for the Tore Supratokamak, considered in Ref. [3]. We however disregard the avalanche dynamics,described by L´evy flights, and concentrate our attention to the geometry impacton the topologically restricted transport in the framework of the circular combmodel, which we call here “umbrella comb model”. It is also related to circularanomalous diffusion in presence of inhomogeneous magnetic fields [4].Anomalous transport in this umbrella comb is described by a probabilitydistribution function P ( r, φ, t ) in polar coordinates to find a particle at theposition ( r, φ ) at time t in the framework of a Fokker-Planck equation as follows ∂ t P = ∆ P , (1.1a)∆ = D φ r ∂ φ + D r r ∂ r r∂ r . (1.1b)Here D φ ( r ) = D δ ( r − R ) and D r = D r ( r ) are diffusion coefficients of theangular and radial directions, respectively. Note that for the singular D φ , the2ransport in the angular direction takes place at r = R only. The radial diffusioncoefficient D r ( r ) is a function of the radius, and this dependence is specified forevery scenario separately. We consider two scenarios of the boundary conditionsfor the angular motion. The first one corresponds to the reflecting boundaries at φ = ± π . That is, there is an infinite wall , at φ = ± π , or a cut along the φ = π axis, where ∂ φ P ( R, φ = ± π, t ) = 0. Note that this scenario results from possiblesymmetry with respect to the x axis for angular diffusion. Such symmetry issupported by symmetrical diffusion obtained numerically in Ref. [2]. In thiscase the diffusive currents at φ = ± π , clockwise and counterclockwise, are equalto each other. These currents can be considered as reflected currents frominfinite walls at φ = ± π , and a cut along the φ = π ray can be performed. Theradial diffusion coefficient is D r ( r ) = D r . The second scenario correspondsto the periodic boundary condition at P ( R, φ = π, t ) = P ( R, φ = − π, t ) andthe radial diffusion coefficient is taken to be a constant value: D r = const. Thezero boundary conditions for the radial directions will be specified separatelyfor each angular scenario.
2. Reflecting boundary conditions
In this section, we concentrate our attention on the geometry impact in theframework of a standard comb model, which can be mapped onto the circleand vice versa by the conformal way. Then the complex plane w = ( r, φ ) ismapped on the complex plane z = ( x, y ). The reflecting boundary conditionsplay important role at this conformal mapping. Indeed, if we take a cut along φ = π that yields the reflecting boundaries at φ = ± π . Then we have x =ln( r/R ) and y = Rφ [7]; the map is shown in Fig. 1. The comb model, whichdescribes anomalous diffusion in the x − y strip, shown in Fig. 1, reads ∂ t P ( x, y, t ) = D δ ( x ) ∂ y P ( x, y, t ) + D ∂ x P ( x, y, t ) . (2.1) Note that this specific choice of the boundary condition can be replaced by a delta poten-tial, which affects the circular diffusion like in the Azbel’-Kaner effect [5], or in chaotic motionof persistent current [6]. π R π R y x (a) (b) Figure 1: Schematic picture of an umbrella comb (a) and its conformal mapping into the x − y strip comb (b). The radii are continuously distributed over the circle of the radius R , andthe angular motion is possible only at r = R . Correspondingly, the x fingers are continuouslydistributed along the y backbone. The boundary and the initial conditions are P ( x = ±∞ , y, t ) = ∂ x P ( x = ±∞ , y, t ) = 0, ∂ y P ( x, y = ± πR, t ) = 0, and P ( x, y, t = 0) = δ ( x ) δ ( y ), re-spectively, and these conditions reflect the boundaries and the initial conditionin the polar coordinates, as well. The radial motion in the umbrella comb inEq. (1.1a) corresponds to a dilation(contraction) operator D ( r∂ r ) , which re-sults from the conformal map D ∂ x → D ( r∂ r ) with the diffusion coefficient D → D r = D r . This inhomogeneous diffusion results from the conformalmap and corresponds to a multiplicative white noise and is known as the so-called geometric, or exponential Brownian motion [12]. For r >
0, it describes e.g. , a stock price behavior as a Wiener process for x = log( r ), whichis known as the Black-Scholes model [13]. In the present consideration, it can be consideredas an exponential instability of plasma in the radial direction in tokamaks [3]. Note thatthe dilation operator in dynamical systems relates to an inverted quartic potential, while indiffusion equation it appear due to inverted harmonic oscillator [8, 9, 10, 11] This leads todilation - contraction operator in the radial diffusion equation in Ref. [3], where it appears
4e solve Eq. (2.1) by standard procedures as follows. Performing theLaplace transformation L [ P ( t )]( s ) = ˜ P ( s ), and substituting it in Eq. (2.1),one has ˜ P ( x, y, s ) = e −| x | √ s/D f ( y, s ) , (2.2a) D ∂ y f ( y, s ) − p sD f ( y, s ) + δ ( y/R ) = 0 . (2.2b)Due to the reflecting boundary conditions f ( y = ± πR, s ) = 0, the solution f ( y, s ) of Eq. (2.2b) is considered as the superposition f ( y, s ) = 1 √ πR ∞ X k =0 f k ( s ) cos( ky/R ) . (2.3)The initial time backbone dynamics is estimated in Appendix A, and the back-bone PDF due to Eq. (A.5) consists of two terms f ( y, t ) = P ( x = 0 , y, t ) = t − / πR √ πD ∞ X n =0 cos( ny/R ) −− t − / D πR √ πD ∞ X n =1 cos( ny/R ) n e − κπn == t − / πR √ πD ( δ ( y/R ) + 1) + t − / D π R √ πD ddκ ϑ ( y/R, κ ) , (2.4)where κ = 2 D t /π / . The first term in Eq. (2.4) relates to the pining initialcondition, while the second term yields the stationary solution in the form of thetheta function ϑ ( y/R, κ ) [14]. A typical behavior of ddκ ϑ ( y/R, κ ) for πκ = 0 . due to a sawtooth field. v d ( v , ) / d Figure 2: An example of derivation of the theta function with respect to κπ for πκ = 0 . d ϑ ( v,κ ) d ( πκ ) , where v = y/R = φ . Here we also use the definition of the theta function ϑ ( v, κ )[14]. of Eq. (2.1) reads P ( x, y, t ) = 1 √ πR ∞ X k =0 cos( ky/R ) L − h f k ( s ) e −| x | √ s/D i == 1 √ πR ∞ X k =0 cos( ky/R ) L − e −| x | √ s/D / √ D (cid:16) √ s + k D (cid:17) == 12 π √ RD t e − x D t + ¯ P ( x, y, t ) , (2.5)where D = D √ D . The term ¯ P ( x, y, t ) is estimated for the large time inAppendix B and reads¯ P ( x, y, t ) = (2 − √ π )4 π √ RD t e − x D t δ ( y/R ) + π / | x | D √ D Rt B (cid:16) y πR (cid:17) e − x D t , (2.6)where B ( z ) is a shifted Bernoulli polynomial [15] defined on z ∈ ( − / , / z -0.1-0.0500.050.10.150.2 B ( z ) (a) -0.5 0 0.5 z -0.1-0.0500.050.10.150.2 B ( z ) (b) Figure 3: Bernoulli polynomial [15] B ( z ) = z − z + 1 / z ∈ [0 ,
1] on panel (a), whichis also the result of summation in Eq. (B.5). Another result of there is of the summation inEq. (B.5) for z ∈ [ − / , /
2] is on panel (b). It consists of two parts for z ∈ [0 , / B ( z ). The second part for z ∈ [ − / ,
0] corresponds to thecurve of the left part of B ( z ) for z ∈ [1 / ,
1] shifted on 1.
In the polar coordinates the obtained result in Eqs. (2.5) and (2.6) reads P ( r, φ, t ) = 12 π √ RD t e − ln2( r/R )4 D t (cid:20) − √ π δ ( φ ) (cid:21) ++ π / | ln( r/R ) | D √ D Rt B (cid:18) φ π (cid:19) e − ln2( r/R )4 D t . (2.7)As obtained, the circular motion is not random due reflections on boundaries.It consists of two stationary distributions: the first one is the initial condition,which relaxes by power law as t − / and the second stationary distribution isaccording to the Bernoulli polynomial B ( φ/ π ), which interacts with the ra-dial motion. The radial motion is random and corresponds to the geometricBrownian motion, which is described by the log-normal distribution [12], andleads to the exponential spreading along the radii with the mean squared dis-7lacement (MSD) h r ( t ) i ∼ e t . This dominant process is also corrected by theL´evy-Smirnov distribution with respect to t ln( r/R ) , see e.g. , [16, 17]. Conclud-ing this section it is worth to stress that the geometric Brownian motion is thegeometry effect of the conformal mapping of the ( x − y ) comb model (2.1) ontothe circular geometry comb by conformal gluing of the backbone ends.
3. Periodic boundary conditions
Figure 4: Schematic picture of the circular/umbrella comb with the periodic boundary con-ditions at φ = ± π and outward diffusion along the radii with zero boundary conditions atinfinity. In the section, we consider the second scenario with periodic circular motionat r = R and the outward radial motion with a constant diffusion coefficient D r = D , see Fig. 4. Equation (1.1) now is ∂ t P = D δ ( r − R ) 1 r ∂ φ P + D r ∂ r r∂ r P . (3.1)The initial condition is P = P ( r, φ, t = 0) = δ ( r − R ) δ ( φ ) The distributionfunction P = P ( r, φ, t ) is a convolution integral P ( r, φ, t ) = Z t G ( r, t − t ′ ) F ( φ, t ′ ) dt ′ , (3.2)which represents two independent motions in the Laplace space:˜ P ( r, φ, s ) = g ( r, s ) f ( φ, s ) . (3.3)8his corresponds to Eq. (3.1) in the Laplace space, s ˜ P − P = D δ ( r − R ) 1 r ∂ φ ˜ P + D r ∂ r r∂ r ˜ P . (3.4)Now, the boundary conditions for the radial motion can be easily specifiedfor ˜ P , g and f . These are ˜ P ( r = R, φ, s ) = f ( φ, s ) and g ( r = R, s ) = 1,and ˜ P ( r = ∞ , φ, s ) = g ( r = ∞ , s ) = ˜ P ′ ( r = ∞ , φ, s ) = g ′ ( r = ∞ , s ) = 0,where prime means differentiation with respect to r . Note also, f ( φ, s ) is amultivalued function, and the conformal map cannot be performed. Therefore,we are treating the problem in the polar coordinates.First, let us consider diffusion in radii - fingers. In the Laplace space, thediffusion equation from Eq. (3.4) leads to the equation rsg = Drg ′′ + Dg ′ (3.5)with the solution g ( r, s ) = A ( s ) K (cid:16) r p s/D (cid:17) θ ( r − R ) , (3.6)where K ( z ) is the modified Bessel function of the second kind, which satisfiesthe boundary conditions at infinity. The second boundary condition g ( R, s ) = 1yields A ( s ) = 2 h K (cid:16) R p s/D (cid:17)i − , where θ (0) = 1 / ∂ φ f + af − bs f + cδ ( φ ) = 0 , (3.7)where a = 2 RD/D and bs / = R K ( λR ) D K ( λR ) √ Ds , and c = R /D . For the initial times, when s → ∞ , and λR = R p s/D ≫
1, one obtains K ( λR ) K ( λR ) ≈ a ≪ bs . Then, Eq. (3.7) is simplified with the solution f ( φ, s ) = ∞ X n = −∞ ce inφ n + b √ s . (3.8)9e also obtain that g ( r, s ) ≈ R e − λ ( r − R ) / √ r [15] and the PDF P ( r, φ, t ) leadsto a chain of estimations in Appendix C. Therefore, the PDF reads P ( r, φ, t ) = L − [ g ( r, s ) f ( φ, s )] ≈ ∞ X n = −∞ cR br e inφ · L − e − √ s ( r − R ) √ D n /b + √ s = (3.9a)= cR b √ rπt e − ( r − R )24 Dt [1 + 2 δ ( φ )] − c ( Dt/r ) b ( r − R ) e − ( r − R )24 Dt + (3.9b)+ cπ p RD p rb ( r − R ) cosh (cid:20) ( π − | φ | ) q D t/b ( r − R ) (cid:21) cosh (cid:20) π q D t/b ( r − R ) (cid:21) e − ( r − R )24 Dt , (3.9c)where φ ∈ [ − π , π ]. The modulus | phi | is due to the symmetry of Eq. (3.7). Wealso stress that the solution (3.9c) is valid for r > R , strictly. It should be notedthat for t →
0, the solution reduces to the transport for r = R , and Eq. (3.9a)reads P ( R, φ, t ) ≈ cb √ πt [1 + 2 δ ( φ )] − c √ t ∞ X n =1 cos( nφ ) n = cb √ πt [1 + 2 δ ( φ )] − cπ √ t B (cid:18) φ π (cid:19) . (3.10)The situation changes dramatically for the long times. For the large times, when λR ≪
1, we have K ( λR ) ≈ ln γλR and K ( λR ) ≈ /λR , where γ is the Euler constant [15]. However, asymptotic behavior of K ( λr ) must correspond to the boundary conditions at r → ∞ . Therefore, wetake the intermediate asymptotic, when λr >
1, which yields K ( λR ) ≈ r π λr e − λr = q π ( D/s ) /re − r ( s/D ) . Correspondingly, in the limits s → rs D − >
1, the radial distribution g ( r, s ) and the coefficient b in Eq. (3.7) are functions of s which are approxi-mated as follows g ( r, s ) ≈ q πD /rs e − r ( s/D ) ln (4 D/γ R s ) , (3.11a) bs = b ( s ) s ≈ DRD ln − (cid:0) D/γ R s (cid:1) ≡ b (cid:2) ln (cid:0) D/γ R s (cid:1)(cid:3) − . (3.11b)10gain neglecting the parameter a in Eq. (3.7), we obtain the solution asfollows f ( φ, s ) = ∞ X n = −∞ ce inφ ln (cid:0) D/γ R s (cid:1) b + n ln ( D/γ R s ) . (3.12)Then the PDF (3.9) for the large time asymptotics reads in the form of theinverse Laplace transformation P ( r, φ, t ) = L − [ g ( r, s ) f ( φ, s )] ≈≈ L − " ∞ X n = −∞ ce inφ n + b ln (4 D/γ R s ) · q π ( D/r s ) e − r ( s/D ) . (3.13)The solution, obtained in Appendix C, reads P ( r, φ, t ) ≈ cπb Γ (cid:0) (cid:1) q π ( D/t r ) ln (4 Dt/γ R ) e − r / Dt × ( ln − (cid:0) Dt/γ R (cid:1) π + cosh h ( π − | φ | ) p b ln (4 Dt/γ R ) i cosh h π p b ln (4 Dt/γ R ) i ) . (3.14)This result is valid for r ≫ R and t → ∞ , and the Tauberian theorem, applied inEq. (C.12), grasps exactly this intermediate asymptotic behavior of K ( λr ) ∝ r − s − due to the power law. The logarithmic evolution in the backboneshown in Fig. 5, relaxes to the radii-fingers. However, in large time asymptoticcalculation this backbone relaxation cannot be separated from the radii one.Note also that the approximate solution (3.14) for the PDF in the specific areadoes not conserve the probability P , namely the latter is P ( t ) = Z ∞ dr Z π − π dφP ( r, φ, t ) ∼ t − . (3.15) This also results to radii subdiffusion with the mean squared displacement(MSD) of the order of h r ( t ) i ∼ Dt . Indeed, the relaxation process in thebackbone contributes to the transport in fingers with the MSD defined from the11 F ( ,t ) Figure 5: Evolution of the braces F ( φ, t ) in PDF of Eq. (3.14). The plots are dependenceof F ( φ, t ) on φ for three times where plot 1 corresponds to t = 10, 2 to t = 100 and 3 to t = 1000. All parameters are taken to be one, while γ = 1 . inverse Laplace transformation in Eq. (3.13) as follows h r ( t ) i = P − ( t ) P − ( t ) 12 π Z ∞ R Z π − π r P ( r, φ, t ) drdφ == L − (cid:20) cb ln (4 D/γ R s ) · q π ( D/s ) Z ∞ R r / e − r ( s/D ) dr (cid:21) ∝∝ D P − ( t ) L − h s − / Γ (cid:18) , R √ s (cid:19) i ( t ) , (3.16)where Γ( α, z ) is the incomplete gamma function [15] and the factor P − ( t ) is dueto the non conserved probability. Performing the Laplace inversion by meansof the Tauberian theorem, see Appendix C, we obtain a subdiffusive growthof the order of Dt / However, the obtained expression should be normalizedby the probability P ( t ). This eventually yields the MSD in the form of normaldiffusion h r ( t ) i ∼ Dt . . Conclusion A circular comb is considered, and two scenarios of the anomalous transportin the circular comb geometry are studied. The first scenario corresponds to theconformal mapping of a comb Fokker-Planck equation on the umbrella comb. Inthis case, the reflecting boundary conditions are imposed on the circular (rota-tor) motion, while the radial motion corresponds to geometric Brownian motionwith vanishing to zero boundary conditions on infinity. The radial diffusion isdescribed by the log-normal distribution, which corresponds to exponentiallyfast motion with the MSD of the order of e t . The second scenario correspondsto circular diffusion with periodic boundary conditions and the outward radialdiffusion with vanishing to zero boundary conditions at infinity. In this casethe radial motion is normal diffusion with the MSD of the order of t . Howeverthe circular motion in both scenarios is a superposition of cosine functions thatresults in a stationary distribution in the form of the Bernoulli polynomials,with the power law relaxation. Appendix A. Backbone dynamics with RBC
Note that “RBC” means reflecting boundary conditions at y = ± πR .Let us consider Eq. (2.2b) D ∂ y f ( y, s ) − p sD f ( y, s ) + δ ( y/R ) = 0 , (A.1)which describes anomalous diffusion, namely subdiffusion, along the y -axis-backbone. Recall that it also corresponds to anomalous diffusion in the ringbackbone of the umbrella comb. Due to the reflection boundary conditions f ′ ( y = ± πR, s ) = 0, the solution can be presented as the superposition of theeven eigenfunctions √ πR cos( ny/R ). It reads f ( y, s ) = 1 √ πR ∞ X n =0 f n ( s ) cos( ny/R ) . (A.2)From Eq. (A.1), this yields f n ( s ) = 12 √ πRD √ s + D n , (A.3)13here D = D / √ D is a subdiffusion coefficient. Performing the inverseLaplace transform, one obtains the solution in the form of the Mittag-Lefflerfunction [14] f n ( t ) = 12 √ πRD πi Z i ∞− i ∞ e st √ s + D n ds == t − / √ πRD E , (cid:16) − n D t (cid:17) . (A.4)Using properties of the Mittag-Leffler functions [14] E α, ( z ) = E α ( z ) and E α,β ( z ) = 1Γ( β ) + zE α,α + β ( z ) ,E α,β ( z ) = ∞ X k =0 z k Γ( kα + β ) , where Γ( ν + 1) = ν Γ( ν ) is a gamma function, we obtain the initial behavior of f n ( t ) in Eq. (A.4): f n ( t ) = 12 √ πRD (cid:20) t − / Γ(1 / − n D E (cid:16) − n D t (cid:17)(cid:21) == 12 √ πRD (cid:26) t − / Γ(1 / − n D · exp h − n D t / Γ(3 / i(cid:27) , (A.5)where Γ(1 /
2) = √ π . Taking into account the expansion (A.2), we obtain thefirst term in Eq. (A.5) in the form of the pining initial condition decaying withtime, while the second term is the theta function ϑ ( y/R, κ ). Then Eq. (A.2)reads f ( y, t ) = P ( x = 0 , y, t ) = t − / πR √ πD ∞ X n =0 cos( ny/R ) −− t − / D πR √ πD ∞ X n =1 cos( ny/R ) n e − κπn == t − / πR √ πD ( δ ( y/R ) + 1) + t − / D π R √ πD ddκ ϑ ( y/R, κ ) , (A.6)where κ = 2 D t /π / . Here we also use the definition of the theta function ϑ ( y/R, κ ) [14]. A typical behavior of ddκ ϑ ( y/R, κ ) for πκ = 0 . ppendix B. Long time asymptotic The long time diffusion can be estimated from Eq. (A.4), as well. To that endwe take into account that the Laplace inversion of the Mittag-Leffler functioncan be presented in the form of the error function Erfc( z ) [18], as follows E , ( at ) = 1 √ πt − ae a t Erfc (cid:16) at (cid:17) . (B.1)However, at the large time asymptotics, diffusion in fingers affects stronglyanomalous diffusion in the backbone, and the former should be taken into ac-count. Therefore, we consider the inverse Laplace transformation in Eq. (2.5),which is the table integral [18] of the form ∞ X n =0 cos( ny/R ) L − " e − a √ s √ s + b = ( πt ) − / e − a (4 t ) − − ∞ X n =1 cos( ny/R ) × h ( πt ) − / e − a (4 t ) − − be ba + b t Erfc (cid:16) − at − / + bt / (cid:17)i . (B.2)Here parameters a and b are determined from Eqs. (2.5) and (A.4). Namely, a = | x | / √ D determines radial diffusion, while b = D n specifies backbonesubdiffusion as obtained above. The error function in Eq. (B.2) for the largeargument readsErfc (cid:16) − at − / + bt / (cid:17) ≈ e − a t − ab − b t at − / + 2 bt / ≈≈ e − a t − ab − b t h (2 b ) − t − / − a (2 b ) − t − / i . (B.3)Taking into account expansion (A.2) and Eqs. (2.5), (B.2), and (B.3), andvalues of a and b , we obtain P ( x, y, t ) = 12 √ πD R ∞ X n =0 cos( ny/R ) L − " e − a √ s √ s + b == 12 π √ D t e x D t + ¯ P ( x, y, t ) == 12 π √ RD t e − x D t [1 + δ ( y/R )] − √ πRD t e − x D t δ ( y/R )+ | x | D √ πD Rt e − x D t ∞ X n =1 cos( ny/R ) n . (B.4)15stimating the last term in Eq. (B.4), which is [19] ∞ X n =1 cos( ny/R ) n = π B (cid:16) y πR (cid:17) , (B.5)we obtain¯ P ( x, y, t ) = (2 − √ π )4 π √ RD t e − x D t δ ( y/R ) + π / | x | D √ D Rt B (cid:16) y πR (cid:17) e − x D t . (B.6)Here B ( z ) is a shifted Bernoulli polynomial presented on z ∈ ( − / , / B ( z ) = z − z + 1 / z ∈ (0 ,
1) [15]
Appendix C. Backbone dynamic with PBC
Note that “PBC” means periodic boundary conditions at φ = ± π .Let us consider the radial motion in Eq. (3.4) according to the radial deriva-tive r ∂ r r∂ r g = g ′′ + r − g ′ , where g = g ( r, s ) is the solution of Eq. (3.5) g ( r, s ) = 2 K ( λr ) K ( λR ) θ ( r − R ) , λ = p s/D (C.1)in the form is the modified Bessel function of the second kind K ( z ), whichsatisfies the boundary conditions at infinity. Using properties of its derivatives[15] as follows K ′ ( z ) = − K ( z ) , (C.2a) K ′′ ( z ) = − K ′ ( z ) = 12 [ K ( z ) + K ( z )] , (C.2b) − K ′ ( z ) = K ( z ) + 1 z K ( z ) , (C.2c)we have g ′ ( r, s ) = 2 K ( λR ) [ − λK ( λr ) θ ( r − R ) + K ( λr ) δ ( r − R )] (C.3)and g ′′ ( r, s ) = 2 K ( λR ) (cid:2) λ K ( λr ) θ ( r − R )++ λr − K ( λr ) θ ( r − R ) − λK ( λr ) δ ( r − R ) (cid:3) , (C.4)16here − K ( λr ) δ ′ ( r − R ) = λK ( λr ) δ ( r − R ) is used. Therefore, the superposi-tion of Eqs. (C.3) and (C.4) yields g ′′ ( r, s ) + 1 r g ′ ( r, s ) = 2 K ( λR ) (cid:2) λ K ( λr ) θ ( r − R )++ 1 R K ( λr ) δ ( r − R ) − λK ( λr ) δ ( r − R ) (cid:21) . (C.5)Inserting the obtained result (C.5) in Eq. (3.4) and taking into account Eq.(3.3), we obtain the equation for f ( φ, s ) as follows D R ∂ φ f + (cid:20) DR − K ( λR ) K ( λR ) √ Ds (cid:21) f + δ ( φ ) = 0 . (C.6) Appendix C.1. Initial time asymptotics
For the initial times, when s → ∞ , and λR = R p s/D ≫
1, it is obtainedthat f ( φ, s ) is determined by Eq. (3.8). Taking into account the solution (3.8)and accounting that g ( r, s ) ≈ R e − λ ( r − R ) / √ r [15], we obtain for the PDF P ( r, φ, t ) the following chain of estimations (see also Appendix B) P ( r, φ, t ) = L − [ g ( r, s ) f ( φ, s )] ≈ L − ∞ X n = −∞ cR br e inφ · e − √ s ( r − R ) √ D n /b + √ s . (C.7)Performing the Laplace inversion, separating term with n = 0, and accountingEqs. (B.2), (B.3), we obtain P ( r, φ, t ) = cR br √ πt e − ( r − R )24 Dt [1 + 2 δ ( φ )] − ∞ X n =1 cRbr cos( nφ ) × e n ( r − R ) b − D − + b − n t Erfc (cid:16) − ( r − R ) D − t − / + n b − t / (cid:17) ≈≈ cR b √ πrt e − ( r − R )24 Dt [1 + 2 δ ( φ )] − cR √ rt ∞ X n =1 cos( nφ ) e − ( r − R )24 Dt − b ( r − R ) D − t − + n == cR b √ rπt e − ( r − R )24 Dt [1 + 2 δ ( φ )] − c ( RDt/r ) b ( r − R ) e − ( r − R )24 Dt ++ cπ p RD p rb ( r − R ) cosh (cid:20) ( π − | φ | ) q D t/b ( r − R ) (cid:21) cosh (cid:20) π q D t/b ( r − R ) (cid:21) e − ( r − R )24 Dt , (C.8)17here φ ∈ [ − π , π ]. Note, that the modulus | phi | is due to the symmetry of Eq.(3.7). We also stress that the solution is valid for r > R , strictly. For r = R ,Eq. (C.8) reads P ( r, φ, t ) ≈ cb √ πt [1 + 2 δ ( φ )] − c √ t ∞ X n =1 cos( nφ ) n == cb √ πt [1 + 2 δ ( φ )] − cπ √ t B (cid:18) φ π (cid:19) . (C.9) Appendix C.2. Large time asymptotics
In this section we estimate Eq. (3.13). Performing the inverse Laplacetransformation in Eq. (3.13), we have P ( r, φ, t ) = L − [ g ( r, s ) f ( φ, s )] ≈≈ L − " ∞ X n = −∞ ce inφ n + b ln (4 D/γ R s ) · q π ( D/r s ) e − r ( s/D ) . (C.10)Performing the Laplace inversion term by term in the summation P n A n ( s ), wehave L − [ A n ( s )] ( t ) = L − (cid:20) d s − / n + S ( s ) e − d √ s (cid:21) , (C.11)where S ( s ) = b ln (cid:0) D/γ R s (cid:1) and e − d √ s ∼ (cid:16) d s (cid:17) − . Making scalingby λ , we obtain from Eq. (C.11) that A n ( λs ) /A n ( s ) ∼ λ − / for s →
0, while[ n + S ( s )] − is slow functions of 1 /s . Therefore, A n ( s ) = s − ρ e − d √ s L (1 /s ),where L (1 /s ) = [ n + S ( s )] − is a slow function of 1 /s and ρ = 1 /
4. Thenapplying the Tauberian theorem, we obtain [20] A n ( t ) = t ρ − e − d / t L ( t ) / Γ( ρ ),see Sec. Appendix C.3 below. Expression (C.10) reads now as follows P ( r, φ, t ) ≈ q π ( D/t r ) e − r / Dt ∞ X n = −∞ ce inφ Γ(1 / n + S ( t )] , (C.12)where S ( t ) = b ln (cid:0) Dt/γ R (cid:1) . The summation yields [19] ∞ X n = −∞ ce inφ n + S ( t ) = 2 cS ( t ) + ∞ X n =1 c cos( nφ ) n + S ( t ) == 2 cS ( t ) + cπ p S ( t ) · cosh h ( π − | φ | ) p S ( t ) i cosh h π p S ( t ) i − cS ( t ) . (C.13)18ventually, one obtains the PDF for the large time asymptotics as follows P ( r, φ, t ) ≈ cπb Γ (cid:0) (cid:1) · q π ( D/t r ) ln (4 Dt/γ R ) e − r / Dt × ( ln − (cid:0) Dt/γ R (cid:1) π + cosh h ( π − | φ | ) p b ln (4 Dt/γ R ) i cosh h π p b ln (4 Dt/γ R ) i ) . (C.14) Appendix C.3. The Tauberian theorem
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