A comparison of escape rates from one and from two holes in an experimental
container (e.g. a laser trap) can be used to obtain information about the
dynamics inside the container. If this dynamics is simple enough one can hope
to obtain exact formulas. Here we obtain exact formulas for escape from a
circular billiard with one and with two holes. The corresponding quantities are
expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate
a direct connection between recent experiments and a major unsolved problem in
mathematics, the Riemann hypothesis.Comment: 5 pages, 4 embedded postscript figures; v2: more explicit on how the
  Reimann Hypothesis arises from a comparison of one and two hole escape rate

C. P. Dettmann

H. Davenport

J. Main

J. B. Conrey

L. A. Bunimovich

English

arXiv

Crossref

Open Circular Billiards and the Riemann Hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis

Bunimovich, LA

Dettmann, CP

Explore Bristol Research

                          Bunimovich, L. A., & Dettmann, C. P. (2005). Open circular billiards and theRiemann hypothesis. Physical Review Letters, 94(10), 1 - 4. [Art. no.100201]. 10.1103/PhysRevLett.94.100201Link to published version (if available):10.1103/PhysRevLett.94.100201Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Open Circular Billiards and the Riemann HypothesisL. A. BunimovichSoutheast Applied Analysis Center, Georgia Institute of Technology, Atlanta, Georgia 30332-1060, USAC. P. DettmannSchool of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom(Received 5 August 2004; published 18 March 2005)A comparison of escape rates from one and from two holes in an experimental container (e.g., a lasertrap) can be used to obtain information about the dynamics inside the container. If this dynamics is simpleenough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circularbilliard with one and with two holes. The corresponding quantities are expressed as sums over zeros of theRiemann zeta function. Thus we demonstrate a direct connection between recent experiments and a majorunsolved problem in mathematics, the Riemann hypothesis.DOI: 10.1103/PhysRevLett.94.100201 PACS numbers: 02.10.De, 02.30.Ik, 05.60.Cd, 45.50.DdBilliard systems, in which a point particle moves freelyexcept for specular reflections from rigid walls, permitclose connections between rigorous mathematics and ex-perimental physics. Very general physical situations, inwhich particles or waves are confined to cavities or otherhomogeneous regions, are related to well understood bil-liard dynamical systems, directly for particles and viasemiclassical (short wavelength) theories for waves.Precise billiard experiments have used microwaves inmetal [1,2] and superconducting [3] cavities and withwave guides [4], visible light reflected from mirrors [5],phonons in quartz blocks [6], electrons in semiconductorsconfined by electric potentials [7,8], and atoms interactingwith laser beams [9–11]. Experiments employ both twoand three [12] dimensional geometries, and both closedand open systems. For instance, escape of cold atoms froma laser trap with a hole was studied in [9,10]. Closedsystems exhibit energy level distributions and scarring ofwave functions as predicted by semiclassical [13] andrandom matrix [14] theories. Open systems exhibit unex-pected and incompletely understood phenomena such asfractal conductance fluctuations [8,15–17]. For both closedand open systems, the behavior depends crucially on theclassical dynamics, which can be tuned to be integrable,chaotic, or mixed [9,10,15,18]. The escape rate is a char-acteristic of open billiards which is both experimentallyaccessible [9,10] and important for transport properties ofmany systems. In this Letter we propose to ask what maybe understood about the intrinsic dynamics of billiardsusing only this experimental escape information, specifi-cally by comparing systems with one and two holes. Thetwo hole escape rate is not measured in the experimentalliterature but presents no fundamental difficulties. Wenaturally begin with the simplest of shapes, the circle,and find remarkable exact expressions based on the mostfamous unsolved mathematical problem, the Riemannhypothesis.At long times, the probability of a particle remaining inan integrable billiard with a hole is well-known to exhibitpower law decay, in contrast to exponential decay fromstrongly chaotic billiards [19]; however the coefficient ofthe power (‘‘escape rate’’) in the integrable case has notbeen computed exactly to our knowledge. Numerical simu-lations can be misleading; for example, a power law decayat long times can be masked by an exponential term at shorttimes.Here we consider the circle billiard, which is integrabledue to angular momentum conservation. Some three di-mensional cases, namely, the cylinder and sphere, can betreated analogously. The circle billiard of unit radius hascollisions defined by the angle around the circumference 2 ; and the angle between the outward trajectoryand the normal  2 =2; =2. The billiard map is then;  !   2 ; where angles are taken modulo2 as usual. The time between collisions is T  2 cos .The dynamical evolution is of two types depending onthe value of  . For    m;n 	 =2m=n with m< ncoprime integers, the trajectory has period n and the values are equally spaced at intervals of 2=n. For  ,FIG. 1. Geometry of the billiard.PRL 94, 100201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 MARCH 20050031-9007=05=94(10)=100201(4)$23.00 100201-1  2005 The American Physical Societywhich are irrational multiples of , the trajectory is uni-formly distributed in .For the escape problem the billiard is filled with auniform density of particles normalized to unity given bycos dd =4 at the boundary. Two (possibly overlap-ping) holes are placed at the boundary at  2 0;  and 2 ;  ; the one hole problem is simply   0. Thenumber of collisions to escape is some function N0;  0(possibly infinite) of the initial conditions and the time toescape is t  NT  2 cos N0;  0. See Fig. 1.Now we compute the density remaining near periodicorbits 0; m; n at long time t. Such a long-lived trajectoryhas    m;n   for  . A prime indicates valuestaken modulo 2=n and lying in 0; 2=n. Thus thedynamics is now 0 ! 0  2 and escape takes placewhen this passes one of the values  or  0 for > 0, or0 or 0 for < 0. The set of initial conditions for whichescape takes at least time t; hence t=2 cos m;n collisionsis thus00 2 tcos m;n; 0[0   tcos m;n;2n(1)for > 0 and similar expressions for < 0. Integratingover this region, including the cos weighting for theinvariant measure and a factor n to account for the equiva-lence modulo 2=n, and summing over all nonescapingperiodic orbits givesPt  14Xm;nng2n  0    g0  tsin2mn;gx  x2 x > 00 x  0; (2)where the sum is restricted to 0  m< n< 2= forcoprime m; n, and faster decaying terms in t have beenneglected. Note that n  1 is included but gives no con-tribution. Now we write the sin functions as sums ofexponentials and apply the Ramanujan identity [20]Xn1m0gcdm;n1e2im=n  n; (3)where  is the Mo¨bius function, defined by 1  1,p  1 for primes p, and mn  mn ifgcdm; n  1; otherwise mn  0. The result isP1	 limt!1tPt18X1n1nnng2n0 g0 : (4)Here n is the Euler totient function, giving the numberof positive integers m  n with gcdm; n  1.Conventionally 1  1. Of course P1 depends on both and , but this is suppressed for brevity.P1 is a finite sum, piecewise smooth when considered asa function of  and/or  with the number of terms un-bounded in the limit of small . For small holes we canextract the asymptotic behavior using Mellin transforms.Writing ~Ps  R10 P1s1d and then P1  12i Rci1ci1 s ~Psds gives the required asymptotic series asa sum of residues. Interchanging the sum and the firstintegral, integrating over  and substituting 0 fn=22=n, where f indicates the fractional part,we findP1  12iZ ci1ci1dss2s12ss 1s 2X1n1nnns11 fn2s2 fn2s2:(5)Now we consider the case of rational angles   2r=q,where gcdr; q  1; the r  0, q  1 case gives a singlehole. In this case the sum over n splits into conjugacyclasses modulo q and the fractional parts are known ra-tional numbers which depend only on the conjugacy class.To evaluate the sum over n we first divide all quantitiesthrough by b  gcda; q, writing n0  n=b, a0  a=b,and q0  q=b. Now we consider Dirichlet characters [20]!n0, defined as follows. The conjugacy classes modulo q0which are coprime to q0 form an Abelian group undermultiplication, of order q0. Since the group is Abelianand finite, there are q0 irreducible representations ! inwhich each n0 coprime to q0 is represented by a complexroot of unity !n0 satisfying !m0!n0  !m0n0;!n0  0 if n0 and q0 have a common factor.Inserting the orthogonality relation for characters [20]into the sum allows n0 to be summed over all integers anddecomposed into prime factors n0  Qpp"p ,-20 -10 0 10 20-20-1001020FIG. 2. Poles of ~Ps [Eq. (9)] for q  1 (one hole) at 2, oddintegers less than or equal to 1, and nontrivial values with realpart 1=2 assuming the Riemann hypothesis. The contour ofEq. (5) is given.PRL 94, 100201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 MARCH 2005100201-2Xn	a modqnnns1 Xn0	a0 modq0bn0bn0bn0s1 1q0X!!a0X1n01!n0bn0bn0bn0s1 ; (6)!n0 Yp!p"p; bn0  b Ypjn0;pj6 b1 p1; bn0 bQp1"pc2j6 bn0 for all c > 10 otherwise:(7)Here, the bar indicates complex conjugation. The condition on  is taken into account by setting "p  0 if pjb andsumming "p  0; 1 otherwise. The sum for  is given by the "  0 term together with a geometric series. The result isXn	a mod qnnns1 1bs1q0X!!a0bLs;!bLs 1;!Qpjb1!pps1 ; Ls;! X1n1!nnsYp1!pps; (8)where characters are taken modulo q0 and Ls; ! is the Dirichlet L function, which in the case q0  1 [i.e., !n  1 for alln] reduces to the Riemann zeta function. Our first main result is the exact expression for the probability Pt of remainingin the unit circular billiard with two holes 0;  and 2r=q; 2r=q limt!1tPt Xjresssj ~Pss;~Ps  2s12ss 1s 2Xqa11 fapq s2  fapq s2bs1q0X!!a0bLs; ! bLs 1; !Qpjb1 !pps1 ;(9)where, as above, b  gcda; q, a0  a=b, q0  q=b, andthe characters are taken modulo q0. In performing thecontour integral (Fig. 2) we recall that the contour lies tothe right of all poles of the integrand (c > 1) and that asemicircular arc to the left may be added that avoids thepoles and for which the integral vanishes in the limit ofinfinite radius.We now consider specific values of q; recall that theangle separating the holes is   2r=q. Thus q  1 is asingle hole and q  2 is two opposite holes.Experimentalists could especially notice a contrast be-tween small values of q, in which the angles are simplerational multiples of  and the escape rate is expressed interms of only a few L functions (in fact, only the Riemannzeta function for q  1; 2; 3; 4; 6), and angles which are notclose to rational numbers with small denominators.Particular values of ~Ps and its residues are given inTable I. Some points to note are that the leading behaviorat s  1, that is, of order 1, is purely given by the totalsize of the holes. The q  1 case is twice as large sincethere is only a single hole in this case. For these values of qthe second to leading order terms, which are not given inthe table, come from the nontrivial zeroes of the %s 1,and for multiplicity m are of orderp lnm1 if all zerosof %s lie in Res  1=2. The latter is a statement of theRiemann hypothesis [21]. An alternative formulation stat-ing that all the nontrivial zeros of %s have Res  1=2 iseasily shown to be equivalent using the functional equation[21] relating %s and %1 s. Thus this celebrated un-solved problem is equivalent to either oflim!0limt!1&1=2tP1t  2=  0; (10)lim!0limt!1&1=2tP1t  2tP2t  0 (11)for every & > 0, and the subscript indicates the one orsymmetric two hole problem. This is our second mainresult: The difference between the escape of the one andtwo hole problems is determined to leading order in thesmall hole limit by the Riemann hypothesis. The general-ized Riemann hypothesis is the equivalent statement for Lfunctions and implies that corrections are of order &1=2 inthe rational two hole case. For irrational  the aboveanalysis breaks down; presumably the poles on the criticalline become dense, blocking analytic continuation.However, the leading order (in ) term can be shown tobe  independent for  > 0, as for the rational values givenabove, as follows. For the leading order behavior, the sumover n can be approximated by an integral, with parts of thesummand replaced by ‘‘mean field’’ averages h i. If  isTABLE I. The function ~Psq=2s1 [see Eq. (9)] and someof the residues of ~Ps for q  1; 2; 3; 4 and r  1.q ~Psq=2s1 1 1 21 %s12ss1s2%s1 2  1312 322 %sss1s2%s1 1  16 03 3sf7%s2s2%s12g%s2s212ss1s23s11%s1 1  14 5 ln29 ln3 344 2sf13%s3s2%s13g%s3s254ss1s22s11%s1 1  13 11 ln316 ln2 3PRL 94, 100201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 MARCH 2005100201-3irrational, the fractional parts are uniformly distributed, sothat we computeg2n 0   g0   n32n 3: (12)We also use hni  6n=2 and hni  0, so thattPt  1242Z 2=0n26n22n 3dn  1(13)as required.We now present some numerical tests. The main resultsof this Letter are exact. However, there are some relatedquestions to do with rates of convergence of various limitswhich are of great importance to numerical or experimen-tal extensions of this work. The rate of convergence oftPt as t!1 is considered in Fig. 3. The convergence ofEq. (9) with the number of residues is tested in Fig. 4.In conclusion, we computed the escape from a circularbilliard with one hole and related it to the Riemann hy-pothesis, computed escape for two holes (rational case),and related it to the generalized Riemann hypotheses andobtained the leading order behavior for the irrational twohole case. Many interesting questions remain concerningone or two hole escape from pseudointegrable, chaotic ormixed billiards, and to the quantum/wave signature ofthese systems [22]. A fuller understanding of open quan-tum billiards should have important practical benefits, forexample, in the design of microlasers [23].We thank E. Croot and Z. Rudnick for helpful discus-sions. L. B. was partially supported by NSF GrantNo. DMS 0140165. C. D. is grateful for the hospitality ofthe Center for Nonlinear Science, Georgia Tech.[1] S. Sridhar, Phys. Rev. Lett. 67, 785 (1991).[2] J. Stein and H. J. Stockmann, Phys. Rev. Lett. 68, 2867(1992).[3] H. D. Graf et al., Phys. Rev. Lett. 69, 1296 (1992).[4] E. Persson et al., Phys. Rev. Lett. 85, 2478 (2000).[5] D. Sweet et al., Physica (Amsterdam) 154D, 207 (2001).[6] C. Ellegard et al., Phys. Rev. Lett. 77, 4918 (1996).[7] T. Sakamoto et al., Jpn. J. Appl. Phys. 30, L1186 (1991).[8] J. P. Bird, J. Phys. Condens. Matter 11, R413 (1999).[9] V. Milner et al., Phys. Rev. Lett. 86, 1514 (2001).[10] N. Friedman et al., Phys. Rev. Lett. 86, 1518 (2001).[11] M. F. Anderson et al., quant-ph/0404118.[12] C. Dembowski et al., Phys. Rev. Lett. 89, 064101 (2002).[13] J. Main, Phys. Rep. 316, 233 (1999).[14] P. J. Forrester, N. C. Snaith, and J. J. M. Verbaarschot, J.Phys. A 36, R1 (2003).[15] R. P. Taylor et al., Phys. Rev. Lett. 78, 1952 (1997).[16] M. Terraneo, J. Mod. Opt. 49, 2027 (2002).[17] R. Crook et al., Phys. Rev. Lett. 91, 246803 (2003).[18] L. A. Bunimovich, Chaos 11, 802 (2001).[19] W. Bauer and G. F. Bertsch, Phys. Rev. Lett. 65, 2213(1990).[20] H. Davenport, Multiplicative Number Theory (Springer-Verlag, New York, 1980).[21] J. B. Conrey, Not. Am. Math. Soc. 50, 341 (2003).[22] R. W. Robinett and S. Heppelmann, Phys. Rev. A 65,062103 (2002).[23] G. D. Chern et al., Appl. Phys. Lett. 83, 1710 (2003). 0.001 0.01 0.1 1 10 0.01  0.1  1  10  100  1000FIG. 3. The scaled survival probability tPt [see Eq. (4)]appears to approach a limiting function as ! 0 with t heldconstant. Here we use 108 random initial conditions,   1 (anirrational multiple of ) and the curves are   10n=2 with n 0    6. At large t the function converges to unity, consistentwith Eq. (13).1 2 3 4 52468101 2 3 4 52468101 2 3 4 52468101 2 3 4 5246810FIG. 4. Numerical computation of P001 for the single holecase q  1, using Eq. (9) and varying the number of polesconsidered on the critical line; real poles are considered for s >10 in all plots. From (4) it is seen that the second derivative ofP1 is a function with uniformly spaced steps and constantaverage gradient in the variables shown on the axes.Convergence is evident except for very small 1 for whichmore real poles would be required; large 1 require morecritical poles for the steps to be visible.PRL 94, 100201 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 MARCH 2005100201-4

Open circular billiards and the Riemann hypothesis

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Open circular billiards and the Riemann hypothesis

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