Applied Mathematics, 06, 7, 709-70 Published Online Apil 06 in SciRes. http://www.scip.og/jounal/am http://dx.doi.og/0.46/am.06.77065 Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in a Lage Sphee Hongyun Wang, Hong Zhou Depatment of Applied Mathematics and Statistics, Baskin School of Engineeing, Univesity of Califonia, Santa Cuz, CA, USA Depatment of Applied Mathematics, Naval Postgaduate School, Monteey, CA, USA Received 0 Febuay 06; accepted 5 Apil 06; published 8 Apil 06 Copyight 06 by authos and Scientific Reseach Publishing Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY). http://ceativecommons.og/licenses/by/4.0/ Abstact We study the poblem of a diffusing paticle confined in a lage sphee in the n-dimensional space being absobed into a small sphee at the cente. We fist non-dimensionalize the poblem using the adius of lage confining sphee as the spatial scale and the squae of the spatial scale divided by the diffusion coefficient as the time scale. The non-dimensional nomalized absoption ate is the poduct of the physical absoption ate and the time scale. We deive asymptotic expansions fo the nomalized absoption ate using the invese iteation method. The small paamete in the asymptotic expansions is the atio of the small sphee adius to the lage sphee adius. In paticula, we obseve that, to the leading ode, the nomalized absoption ate is popotional to the (n )-th powe of the small paamete fo n. Keywods Diffusion Equation, Bownian Diffusion, Asymptotic Solutions, Absoption Rate. Intoduction Seach theoy epesents the bith of opeations analysis []-[4]. One of the classical seach poblems involves a seache equipped with a cookie-cutte senso looking fo a single moving taget. A cookie-cutte senso can detect a taget instantly when the taget gets within distance R to the seache and thee is no deteciton when the taget ange is lage than R. One inteesting mathematical challenge is to find the pobability of a diffusing taget avoiding detection by a stationay cookie-cutte senso. This poblem has been adessed by Eagle [5] whee the seach egion is a two-dimensional disk. Recently we have evisited this poblem and have deived a unified asymptotic expession fo the decay-ate of the non-detection poblability which is valid fo the cases How to cite this pape: Wang, H.Y. and Zhou, H. (06) Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in a Lage Sphee. Applied Mathematics, 7, 709-70. http://dx.doi.og/0.46/am.06.77065
whee the seach egion is eithe a disk o a squae [6]. In this pape, we would like to extend ou ealie wok [6] to high dimensions. Moe specifically, we investigate the absoption ate into a small sphee such as a cookie-cutte senso fo a difusing paticle (i.e. taget) confined in a lage sphee (i.e. seach egion). Fom the next section, the pape is outlined as follows. We fist pesent the mathematical fomulation of the poblem in Section. Then we conside the special case of the thee dimensions in Section and deive the exact solution fo this case in Section 4. Section 5 and Section 6 descibe the solutions fo dimension fou and dimension five, espectively. These asymptotic solutions ae validated against the accuate numeical solutions of a Stum-Liouville poblem in Section 7. Finally, Section 8 summaizes the pape.. Mathematical Fomulation We conside a paticle in the n-dimensional space n, undegoing a Bownian diffusion with diffusion n B 0, R denote the ball in, of adius R and centeed at the oigin coefficient D. Let ( 0, ) = { n and } B R x x x R R is the bounday of R. We conside the situation whee the diffusing paticle is confined fom outside by a lage sphee S R whee. Figue shows the geomety of the poblem setpup in the thee dimensional space ( n = ). Let p( xt, ) be the pobability of the paticle being at position x at time t. pxt (,) is govened by the diffusion equation with bounday and initial conditions: p = D p t p = 0, p = 0 () S( 0, R n ) n Let S( 0, R ) denote the sphee in, of adius R and centeed at the oigin. S( 0, ) B( 0, ) S( 0, R ) and is absobed nea the oigin by a a small sphee ( 0, ) R R S( 0, R ) (,0) = p x p x 0 Figue. A diffusing paticle confined fom outside by S 0, R and absobed nea the oigin a lage sphee ( ) by a small sphee S( 0, R ) in an outside eflecting bounday is placed at S( 0, R ) and an inside absobing bounday at S( 0, R ).. Mathematically, 70
p whee denotes the Laplace opeato and n p epesents the diectional deivative of p along the n S R. We fist pefom non-dimensionalization to make the poblem dimensionless. Let nomal vecto n of ( 0, ) The function (, ) x t new = x = t old R D new old R (, ) = (, ) p x t R p x t new new new old old old pnew xnew t new has the meaning of pobability density with espect to x new initial bounday value poblem below (we op the subscipt new fo simplicity): p = p t p = 0, p = 0 S ( 0, n ) S ( 0,) (,0) = p x p x 0. It satisfies the whee R R. Afte non-dimensionalization, the outside confining sphee has adius and the inside absobing sphee has adius. The solution of initial bounday value poblem () can be expessed in tems of exponentially decays of eigenfunctions. Hee { 0 < λ < λ < < λj < } tions of the Stum-Liouville poblem = ( λ ) α p xt, exp jt juj x () j= ae the eigenvalues and u n S( 0,) { j,,, } () u x j = ae the associated eigenfunc- u = λu (4) = 0, u = 0 S( 0, ) In (), the slowest decaying tem is exp ( λt ) decay and pobability density (, ) p( xt, ) α exp ( λt) u( x). Ove long time, the dominant tem is the one with the slowest p xt has the appoximate expession = (5) We conside the suvival pobability: S t p xt, d x. Ove long time, the decay of suvival B 0, \ B 0, pobability S( t ) is descibed by the smallest eigenvalue λ exp ( λ ) S t = c t (6) Quantity S 0,. Fo small, time scale λ is lage. In contast, λ is appoximately detemined by the time scale of pobability density elaxing to equilibium within the egion B( 0, ) \ B( 0, ). As a esult, λ = O, appoximately independent of. This sepaation of time scales makes it possible to deive asymptotic expessions fo the smallest eigenvalue λ. The nomalized decay ate of suvival pobability is λ, which is dimensionless. The physical decay ate (befoe non-dimensionalization) of suvival pobability is elated to the nomalized decay ate λ as Decay ate = D R λ (7) λ coesponds to the time scale of the paticle being absobed by sphee 7
In the two-dimensional case ( n = ), we showed that fo small the smallest eigenvalue λ has the expansion (Wang and Zhou, 06) λ = 7 η + + O 8 96 η η = + + O fo n = η 4η η η whee η log In this study, we deive asymptotic expansions fo the smallest eigenvalue λ in the cases of n =, n = 4 and n = 5. Fo simplicity, we op the subscipt, and use λ to denote the smallest eigenvalue and u( x ) to denote a coesponding eigenfunction. Since an eigenfunction fo the smallest eigenvalue is axisymmetic, function u( x ) depends only on = x. We wite u( x ) as u( ). The axisymmetic Stum-Liouville u and λ ) has the fom poblem fo the eigenpai ( d n du λu n = u = 0, = 0 u( ) We use the invese iteation method to deive an asymptotic expansion fo λ, stating with an initial guess fo eigenfunction: ( 0 ) ( = δ ) u Specifically, we solve the linea diffeential equation with bounday conditions below to update the k k+ appoximation fom u to u. ( k+ ) d n du ( k ) u n = ( k+ ) k+ u = 0, u = 0 In the fist iteation ( k = 0 ), the delta function on the ight hand side can be conveniently incopoated into the bounday condition at =. Fo k = 0, Equation (0) becomes d n du 0 n = () u =, u = 0 An appoximation to the smallest eigenvalue λ is calculated as λ ( k ) n n ( k ) ( k+ ) = u u In the subsequent sections, we show that 9 459 λ = + + + ) 5 75 fo n = () 8 λ = 8 + + fo n = 4 (4) (8) (9) (0) () 5 λ = 5 + + fo n = 5 7 (5) 7
. The Thee Dimensional Case: n = Fo the thee dimensional case ( n = ), the diffeential equation in (0) has the fom d dw = We fist solve fo two independent solutions of (6) in the case of g 0 without any bounday condition Next we solve Fo j = 0, the solution is Fo j =, the solution is Fo j =, the solution is Fo j =, the solution is, w w g = = d dw j = w = 0, = 0 w( ) w0 = + + w ( 0 = + + O ) 9 5 w = + + w O 6 8 6 d = + + ( ) w = + + 4 4 5 w O 6 d = + + ( ) 4 4 w = + + 5 4 5 4 () w ( = + + O ) 5 8 With these esults, we stat the invese iteation. Fo the fist iteation ( k = 0 ), the solution of () is a linea combination of two independent solutions and. ( u ) = The coesponding appoxomation fo λ using () is ( 0 ) δ u = = u O ( ) d = d = + ( ) (6) (7) (8) (9) (0) 7
0 u ( 0) λ = = = + ( u ) ( + O ( )) In the second iteation ( k = ), the ight hand side of (0) is fomed using w0 ( ) and w descibed above. ( O ) ( ) u = w w0 = + + + + + The coesponding appoxomation fo λ using () is ( ) u ( ) d = 9 0 8 O + + ( u ) = and the solution of (0) is u + O ( ) 9 λ = = = O ( ) ( u ) + + 5 + O 9 0 ( ) In the thid iteation ( k = ), the ight hand side of (0) is The solution of (0) is constucted using 0 u ( ) O = + + 6 w, w, w ( ) and w. u 0 w O w w 6 w ( ) = + ( ) + The coesponding appoxomation fo λ using () is 7 0 u d = 7 90 80 O + + λ 7 u + + O ( ) 9 0 8 = = ( u ) 5 78 O 7 + + 0 80 ( ) 9 459 = + + + O 5 75 ( ) Theefoe, in the thee dimensional case, λ has the expansion 9 459 λ = + + + fo n = 5 75 () Fo the thee dimensional case, the smallest eigenvalue λ can be witten as the exact solution of a tanscendental equation, which povides an altenative way of deiving the asymptotic expansion. This is caied out in the next section. 4. Exact Solution fo the Special Case of n = Fo the special case of n =, we wite u( ) as u = φ () 74
Substituting it into (9) fo The bounday condition fo φ at = is The bounday condition fo φ at = is n =, we deive the diffeential equation fo dφ = λ Thus, φ satisfies the Stum-Liouville poblem φ( ) = 0 φ φ φ = 0 = d φ = λφ φ φ = 0, φ = 0 A geneal solution of the diffeential equation has the expession sin φ : cos ( ) φ = c λ + c λ Enfocing the bounday condition φ( ) = 0, we have c = 0 and ( ) sin H. Y. Wang, H. Zhou (4) φ = λ (5) Hee we have set c = because an eigenfunction must be non-tivial and can be multiplied by any non-zeo φ φ = 0 yields constant. Enfocing the bounday condition λ tan ( λ ( ) ) = (6) The smallest eigenvalue λ is the smallest (positive) solution of Equation (6). The coesponding eigenfunction is given by (5). This exact solution specified by Equation (6) povides an altenative deivation fo the asymptotic expansion µ λ. Substituting it into Equation (6) gives us an equation fo µ. of λ. Let Using the Taylo expansion of tan ( x ) and subtacting fom both sides of (7), we get tan ( µ ) = µ 5 7 tan ( x) = + x + x + 7 x + 5 5 7 5 5 Based on (8), we constuct an iteative fomula fo expanding µ The iteative fomula gives us (7) = µ + µ + µ + (8) ( k+ ) ( k) 7 k µ = µ µ + (9) 5 05 ( 0) µ = µ = 5 75
Going fom µ back to λ, we obtain ( ) 4 µ = + 5 75 4 9 459 λ = + + = + + + (0) 5 75 5 75 which is the same as the asymptotic expnsion deived using invese iteation method. 5. The Fou Dimensional Case: n = 4 Fo the fou dimensional case ( n = 4 ), the diffeential equation in (0) has the fom d dw = We fist solve fo two independent solutions of () in the case of g 0 without any bounday condition: Next we solve Fo j = 0, the solution is Fo j =, the solution is w w g =, w = w d dw j = w = 0, = 0 w( ) 4 4 0 = + + d = + + ( ) w 0 O = log + 6 8 4 d = + log + + ( ) w O With these esults, we stat the invese iteation. Fo the fist iteation ( k = 0 ), the solution of () is a linea combination of two independent solutions and. ( ) u = The coesponding appoxomation fo λ using () is ( 0 ) δ u = = u 8 4 O ( ) d = d = + ( ) 0 ( 0) u = = = 8 + O u + O 8 ( ) λ () () () (4) 76
( In the second iteation ( k = ), the ight hand side of (0) is ) u = and the solution of (0) is descibed above. ( ) u = w w 0 The coesponding appoxomation fo λ using () is fomed using w0 ( ) and w λ u ( ) d 7 log 4 64 96 O = + ( + O ( 4 )) u = = 8 ( u ) 4 4 O log 4 + 64 8 4 = 8 + + O log Theefoe, in the fou dimensional case, λ has the expansion 6. The Five Dimensional Case: n = 5 8 λ = 8 + + fo n = 4 (5) Fo the five dimensional case ( n = 5 ), the diffeential equation in (0) has the fom d dw 4 4 = We fist solve fo two independent solutions of (6) in the case of g 0 without any bounday condition Next we solve Fo j = 0, the solution is Fo j =, the solution is w g =, w = w d 4 dw j 4 = w = 0, = 0 w( ) 5 5 75 0 = + + d = + + ( ) w 4 0 O w = + + (9) 4 w ( O ) = + + 0 0 4 With these esults, we stat the invese iteation. Fo the fist iteation ( k = 0 ), the solution of () is a linea combination of two independent solutions and. (6) (7) (8) 77
u ( ) The coesponding appoxomation fo λ using () is 4 4 = ( 0 ) δ u = = u 5 6 O ( ) d = d = + ( ) 4 4 4 0 ( 0) u = = = 5 + O 4 u + O 5 ( ) λ In the second iteation ( k = ), the ight hand side of (0) is fomed using w0 ( ) and w descibed above. u w w 0 ( ) = The coesponding appoxomation fo λ using () is λ u ( 7 ) d 6 5 60 O = + 4 5 4 u + O 5 = = 4 ( u ) 85 O 5 6 + 4 5 5 = 5 + + O ( ) 7 Theefoe, in the fou dimensional case, λ has the expansion 7. Accuacy of Asymptotic Solutions ( ) u = and the solution of (0) is ( ) 5 ( ) 5 5 λ = 5 + + fo n = 5 7 (40) To demonstate the accuacy of asymptotic expansions we obtained above, we solve numeically Stum- Liouville poblem (9). Instead of using a unifom gid in vaiable, we use a unifom gid in vaiable s = log, which povides a moe unifom numeical esolution fo the whole egion even when is small. Let In vaiable s log ( exp ) v s u s =, Stum-Liouville poblem (9) becomes d v dv + = ds ds v 0 = 0, log = 0 ( n ) λ exp ( ) v( ) 4 We use the cental diffeence with N = points to discetize Stum-Liouville poblem (4). The discete vesion of (4) is an eigenvalue poblem of a tidiagonal matix. The smallest eigenvalue of this spase matix povides a vey accuate appiximation to λ, the smallest eigenvalue of Stum-Liouville poblem (9). Below we teat this vey accuate numeical solution as the tue solution and use it to judge the pefomance of asymptotic solutions. s v (4) 78
Figue compaes asymptotic solutions and a vey accuate numeical solution in the thee dimensional case ( n = ). Solutions ae compaed fo in the inteval of [ 0.,0.4 ]. The one-tem asymptotic solution is not vey good in this ange of. Nevetheless, as is educed, the one-tem asymptotic solution conveges slowly to the tue solution, which is epesented by the accuate numeical solution in Figue. The two-tem asymptotic solution is bette than the one-tem solution. The thee-tem asymptotic solution is even bette. Fo 0., the thee-tem asymptotic solution is indistinguishable fom the tue solution. Figue compaes asymptotic solutions and a vey accuate numeical solution in the fou dimensional case ( n = 4 ). The one-tem asymptotic solution in the fou dimensional case (Figue ) is much moe accuate than that in the thee dimensional case (Figue ). In Figue, the one-tem solution is vey close to the tue solution fo 0.. This indicates that as n is inceased, the leading ode asymptotic solution becomes moe accuate. The two-tem asymptotic solution in Figue coincides with the tue solution fo 0.. Figue 4 compaes asymptotic solutions and a vey accuate numeical solution in the five dimensional case ( n = 5 ). The one-tem asymptotic solution in Figue 4 is aleady vey close to the tue solution fo 0., confiming the tend that in highe dimensional space (lage n), the leading ode asymptotic solution is moe Figue. Compaison of asymptotic solutions and a vey accuate numeical solution in the thee dimensional case (n = ). Figue. Compaison of asymptotic solutions and a vey accuate numeical solution in the fou dimensional case (n = 4). 79
Figue 4. Compaison of asymptotic solutions and a vey accuate numeical solution in the five dimensional case (n = 5). accuate than that in lowe dimensional space (smalle n). The two-tem asymptotic solution in Figue 4 is indistinguishable fom the tue solution even at = 0.4. In each case ( n =, n = 4, o n = 5 ), the most accuate asymptotic solution coincides with the tue solution, at least, fo 0.. 8. Concluding Remaks The focus of this pape was to calculate the absoption ate into a small sphee fo a diffusing paticle which was confined in a lage sphee. Unde the assumption that the atio of the small sphee adius to the lage sphee adius was small, we deived asymptotic expansions fo the nomalized absoption ate with the invese iteation method. Acknowledgements Hong Zhou would like to thank Naval Postgaduate School Cente fo Multi-INT Studies fo suppoting this wok. The views expessed in this document ae those of the authos and do not eflect the official policy o position of the Depatment of Defense o the U.S. Govenment. Refeences [] Dobbie, J.M. (968) A Suvey of Seach Theoy. Opeations Reseach, 6, 55-57. [] Koopman, B.O. (999) Seach and Sceening: Geneal Pinciples with Histoical Applications. The Militay Opeations Reseach Society, Inc., Alexania. [] Stone, L.D. (989) Theoy of Optimal Seach. nd Edition, Academic Pess, San Diego. [4] Washbun, A.R. (00) Seach and Detection, Topics in Opeations Reseach Seies. 4th Edition, INFORMS. [5] Eagle, J.N. (987) Estimating the Pobability of a Diffusing Taget Encounteing a Stationay Senso. Naval Reseach Logistics, 4, 4-5. [6] Wang, H. and Zhou, H. (06) Non-Detection Pobability of a Diffusing Taget by a Stationay Seache in a Lage Region. Applied Mathematics, 7, 50-66. 70