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JOURNAL OF MATHEMATICAL PHYSICS 48, 052901 共2007兲 Dynamics of a classical Hall system driven by a time-dependent Aharonov-Bohm flux J. Ascha兲 CPT-CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France P. Šťovíček Department of Mathematics, Faculty of Nuclear Science, Czech Technical University, Trojanova 13, 120 00 Prague, Czech Republic 共Received 4 September 2006; accepted 16 March 2007; published online 10 May 2007兲 We study the dynamics of a classical particle moving in a punctured plane under the influence of a homogeneous magnetic field, an electric background, and driven by a time-dependent singular flux tube through the hole. We exhibit a striking 共de兲localization effect: when the electric background is absent we prove that a linearly time-dependent flux tube opposite to the homogeneous flux eventually leads to the usual classical Hall motion: the particle follows a cycloid whose center is drifting orthogonal to the electric field; if the fluxes are additive, the drifting center eventually gets pinned by the flux tube whereas the kinetic energy is growing with the additional flux. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2723550兴 I. INTRODUCTION The motivation to study the dynamics of this classical system is to sharpen our intuition on its quantum counterpart which is, following Laughlin’s13 and Halperin’s11 proposals, widely used for an explanation of the integer quantum Hall effect. Of special interest is how the topology influences on the dynamics. In the mathematical physics literature Bellissard et al.5 and Avron et al.3,4 used an adiabatic limit of the model to introduce indices. The indices explain the quantization of charge transport observed in the experiments.12 See Refs. 6, 9, 7, 8, and 10 for recent developments. We discussed aspects of the adiabatics of the quantum system in Ref. 2, its quantum and semiclassical dynamics will be treated elsewhere. The dynamics of the classical system without magnetic field were discussed in Ref. 1. We state the model and our main results. Consider a classical point particle of mass m ⬎ 0 and charge e ⬎ 0 moving in the punctured plane R2 \ 共0兲. Suppose that a magnetic flux line with time varying strength ⌽ pierces the origin and further the presence of a homogeneous magnetic field of strength B orthogonal to the plane and an interior electric field with smooth bounded potential V. Let ⌽:R → R and V:R ⫻ R2 → R be smooth functions. Denote q⬜ ª 共−q2 , q1兲. The force on the particle at q 苸 R2 \ 共0兲 with velocity q̇ is 冉 − e Bq̇⬜ − 冊 t⌽ q ⬜ + qV = e共q̇ ∧ rot共A兲 − tA兲, 2 兩q兩2 with a兲 Electronic mail: [email protected] 0022-2488/2007/48共5兲/052901/14/$23.00 48, 052901-1 © 2007 American Institute of Physics Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-2 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček A共⌽共t兲,q兲 ª 冉 冊 B ⌽共t兲 − q⬜ + tqV共q兲. 2 2兩q兩2 Remark that the electromotive force induced by the flux line has circulation et⌽, constant torque et⌽ / 2, vanishing rotation, and is long range with a 1 / r singularity at the origin, we call it the circular part. V is smooth on the entire plane so the circulation of the corresponding field is zero. The total magnetic flux through a circle of radius R is BR2 − ⌽ if it encircles the flux line, else BR2. So the two fluxes are “opposite” if B and ⌽ have the same sign. The equations of motions are Hamiltonian. For a point 共q , p兲 = 共共q1 , q2兲 , 共p1 , p2兲兲 in phase space P = R2 \ 共0兲 ⫻ R2 , the time-dependent Hamiltonian is H共t,B;q,p兲 = 1 共p − eA共⌽共t兲,q兲兲2 . 2m Suppose ⌽共− t兲 = − ⌽共t兲, then there is the time reversal symmetry H共− t,− B;q,− p兲 = H共t,B;q,p兲. So in order to fix the ideas we convene that growing time means growing flux opposite to the homogeneous flux and suppose furthermore B ⬎ 0, t⌽共t兲 艌 ⌽0 ⬎ 0. Recall that when only the constant magnetic field is present, the particle follows the Landau orbits: circles around a fixed center with the cyclotron frequency = eB m and radius R such that the magnetic flux through the Landau circle satisfies e / 2共BR2兲 = H. With the above convention this motion is clockwise. If V = 0 then intuitively the physics is the following: the additional field makes the center q − 共q̇⬜ / 兲 drift orthogonal to the electric field. The constant torque exerced by the circular part accelerates orbits which encircle the origin, the orbit shrinks with growing flux and grows with decreasing flux; orbits not encircling the flux line should have constant radius. We have the following results for the case ⌽ = ⌽0t, V = 0 共see Corollary 7.2兲. a. The above intuition is asymptotically correct. Furthermore H共t,B兲⬃⌽共t兲→−⬁ e 兩⌽共t兲兩 2 H共t,B兲→⌽共t兲→+⬁const. b. In the accelerating regime the center c is eventually trapped by the flux line, the particle is spiraling outward c共t兲→⌽共t兲→−⬁const, Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-3 J. Math. Phys. 48, 052901 共2007兲 Hall effect driven by an Aharonov-Bohm flux FIG. 1. Typical trajectory of the Hamiltonian 1 / 2共p − 共共1 / 2兲q⬜ − s共q⬜ / q2兲兲兲2. 冑 c. B q共t兲⬃⌽共t兲→−⬁共cos共− t兲,sin共− t兲兲. 兩⌽共t兲兩 In the decelerating regime the orbit ends up drifting diffusively orthogonal to the field 冑 B q共t兲→⌽共t兲→⬁const. ⌽共t兲 In addition we expand the solution up to an error O共1 / t3/2兲 关see Theorem 共5.1兲兴. We further compute the adiabatic limit 共i.e., the solution of the equations of motions averaged over the Landau orbits兲 in the perspective to obtain information on the transition between the two dynamics. We find 关see Theorem 共6.2兲兴 that in the adiabatic limit the transition between the two dynamics is sharp and that the center gets stuck after a finite time if there is no electric background; it is a challenging problem to study if the adiabatic limit provides an approximation of the true dynamics. For a general increasing flux and a background field whose torque is controlled by the constant torque of the circular part we show 关see Corrollary 共7.1兲兴 2 H共t,B兲 ⬎ 0, ⌽共t兲→−⬁ e 兩⌽共t兲兩 lim inf Bc2共t兲 ⬎ 0. ⌽共t兲→⬁ ⌽共t兲 lim inf We may state our observation as follows: States if submitted to an accelerating flux line will eventually become energy conducting; if no electric background is present they get trapped by the flux tube. We give two numerical illustrations in Figs. 1 and 2. II. DYNAMICS OF THE FROZEN SYSTEM Upon scaling q, p, t to dimensionless coordinates, which we call q, p, s, we work with the Hamilitonian, H共s兲 where 1 H共s;q,p兲 = 共p − a共s,q兲兲2 , 2 with Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-4 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček FIG. 2. Typical trajectory of the Hamiltonian 1 / 2共p − 共共1 / 2兲q⬜ − s共q⬜ / q2兲 + sqV兲兲2 with the background potential V chosen to be V共x , y兲 = 1 / 10共sin x + sin y兲 on a region 关−10, 10兴2. The background shows the potential lines of V共x , y兲 − arg共x , y兲. and N, Ṽ smooth functions, N共− s兲 = − N共s兲, sN 艌 1, and a dimensionless parameter. We discuss the scaling in Sec. VII. The function aE is smooth on R2 \ 共0兲 with rot共aE兲 = 0. Define E共s兲 : R2 \ 共0兲 → R2 by 冉 E共s兲 ª − saE共s兲 = 共sN兲共s兲 冊 q⬜ − qṼ共q兲 . q2 共1兲 We discuss first the solution of the equation of motions for a frozen time 苸 R. As saE共 ; q兲 = 0, the solution of the frozen equations generated by the Hamiltonian H共兲 goes along the lines of the classical Landau problem 关which means the case ⌽ = 0, V共q兲 = 0兴. For 苸 R define 1. 2. 3. the velocity field: v共兲 : P → R2, v共 ; q , p兲 ª p − a共 ; q兲; the center: c共兲 : P → R2, c共 ; q , p兲 ª q − v⬜共 ; q , p兲; the angular momentum: L : P → R, L共q , p兲 ª q ∧ p. Denote the Poisson bracket: 兵f , g其 = q f pg − p f qg. We list some useful formulas. Proposition 2.1: The following identities hold as functions on phase space P for all 苸 R: 1. 2. 3. 兵v1 , v2其 = 1, 兵c1 , c2其 = −1, 兵c , c2 / 2其 = c⬜, 兵ci , v j其 = 0; H = 21 v2, 兵v , H其 = −v⬜, 兵c , H其 = 0; 1 2 c 共兲 = H共兲 + L − q ∧ aE共兲 = H + L + 共N共兲 − q ∧ qṼ兲; 2 4. 共2兲 the frozen flow 共q共 ; s兲 , p共 ; s兲兲 defined by sq共 ; s兲 = pH共兲, s p共 ; s兲 = −qH共兲, Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-5 J. Math. Phys. 48, 052901 共2007兲 Hall effect driven by an Aharonov-Bohm flux 共q共 ; 0兲 , p共 ; 0兲兲 = 共q , p兲 is q共 ;s兲 = c共兲 + cos共s兲v⬜共兲 + sin共s兲v共兲 1 p共 ;s兲 = 共c⬜共兲 + cos共s兲v共兲 − sin共s兲v⬜共兲兲 + aE共 ;q共 ;s兲兲. 2 Proof: a. b. 1, 2, 3: 兵v1 , v2其 = 兵p1 − a1共 , q兲 , p2 − a2共 , q兲其 = rot共a共兲兲 = 1, 兵qi , v j其 = ␦ij. H = 21 v2 so 兵q , H其 = v , 兵v , H其 = −v⬜. c2 = q2 + v2 + 2q ∧ v; on the other hand, L = q ∧ v + 21 q2 + q ∧ aE共 ; q兲. 4: The force is −q̇⬜ independently of ; Newton’s equation q̈ = −q̇⬜ is readily verified. On the other hand, p = v + a = a + c⬜ − q⬜ = c⬜ − 21 q⬜ + aE共 ; q兲. So p共s兲 follows from q共s兲. 䊐 Remark 2.1: 1. Since the energy H共兲 = 21 v共兲2 is conserved under the frozen flow, the projections of the trajectories to q space are circles around c共兲 with radius 冑2H共兲. An orbit encircles the origin (has nontrivial homotopy) in R2 \ 共0兲 if and only if c2 ⬍ 2H ⇔ L − q ∧ aE共 ;q兲 ⬍ 0; 2. 1 2 1 ⬜ 2 2 c = 2 共c 兲 is the Hamiltonian for the reversed magnetic field. III. GENERAL FEATURES Set = 1 and denote by abuse of notation O共s兲 ª O共s ; q共s兲 , p共s兲兲 for an observable O. We have the following general qualitative behavior. Proposition 3.1: Suppose that there exists a 苸 关0 , 1兲 such that for all s , q: 兩q ∧ qṼ共q兲兩 艋 sN共s兲a, then for any initial condition there exists a unique “hitting” time s0 such that ±c2共s兲 艌 ± 2H共s兲, ± s ⬎ ± s0 . Furthermore, 2H共s兲 艌 共1 − a兲 ⬎ 0, N共s兲→−⬁ 兩N共s兲兩 lim inf c2共s兲 艌 共1 − a兲 ⬎ 0. N共s兲→⬁ 2N共s兲 lim inf For radially symmetric potentials it holds 兩q ∧ qṼ共q兲兩 = 0 ⇒ c2共s兲 − H共s兲 = s − s0 . 2 共3兲 Proof: By Eq. 共2兲 and as 兵c2 , H其 = 0, 冉 冊 冉 冊 c2 d c2 − H = s − H = 共sN − q ∧ qṼ兲 艌 sN共1 − a兲 艌 共1 − a兲, 2 ds 2 this gives the first claim. The second follows from positivity, Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-6 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček 冉 冊 冉 冊 c2共s兲 c2 c2 艌 H共s兲 + − H 共0兲 + 共1 − a兲N共s兲 艌 − H 共0兲 + 共1 − a兲N共s兲, 2 2 2 which implies lim inf s→⬁ c2共s兲 c2共s兲 = lim inf 艌 共1 − a兲. 2N共s兲 N共s兲→⬁ 2N共s兲 Analogously, lim inf s→−⬁ H共s兲 H共s兲 = lim inf 艌 共1 − a兲. − N共s兲 N共s兲→−⬁ 兩N共s兲兩 IV. ACTION ANGLE COORDINATES In order to discuss the dynamics we introduce action angle coordinates. The frozen dynamics as discussed in Proposition 2.1 suggests to take as coordinates the angles and absolute values of c and v⬜, i.e., with the notation, e共兲 ª 共cos ,sin 兲, q = c + v⬜ = 兩c兩 c v⬜ + 兩v兩 ¬ 兩c兩e共1兲 + 兩v兩e共− 2兲, 兩c兩 兩v兩 1 1 p = 共c⬜ + v兲 + aE共 ;q兲 = 共兩c兩e⬜共1兲 − 兩v兩e⬜共− 2兲兲 + aE共 ;q兲. 2 2 Motivated by this we define for 苸 R, q共 ; ,I兲 ª 冑2I1e共1兲 + 冑2I2e共− 2兲 ¬ q共,I兲, 1 p共 ; ,I兲 ª 共冑2I1e⬜共1兲 − 冑2I2e⬜共− 2兲兲 + aE共 ;q共 ; ,I兲兲, 2 and denote by C the nullset 兵共 , I兲 ; 1 + 2 = , I1 = I2其 where q共 ; , I兲 = 0, by D the nullset 兵共q , p兲 ; v2 = 0 or c2 = 0其. Thus for each frozen time 苸 R the transformation to action angle coordinates T共兲 is defined by T共兲:S1 ⫻ S1 ⫻ 兵共I1,I2兲;I1 艌 0,I2 艌 0其 \ C → P \ D, T共 ; ,I兲 = T共 ; 1, 2,I1,I2兲 ª 共q共 ; ,I兲,p共 ; ,I兲兲. We have the following. Lemma 4.1: 1. 2. T共兲 is a canonical diffeomorphism. T−1共兲 is determined by I 1共 兲 = 冉 冉 冊冊 冉 冉 冊冊 1 c 2共 兲 1 = p − − q⬜ + aE共 ;q兲 2 2 2 I2共兲 = H共兲 = 1 1 p − q⬜ + aE共 ;q兲 2 2 2 , 2 , Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-7 J. Math. Phys. 48, 052901 共2007兲 Hall effect driven by an Aharonov-Bohm flux e共1共兲兲 = 共1/2兲q − p⬜ + aE⬜共 ;q兲 c 共兲 = 冑2共H共兲 + L − q ∧ aE共 ;q兲兲 , 兩c兩 e共− 2共兲兲 = 共1/2兲q + p⬜ − aE⬜共 ;q兲 v⬜ . 共兲 = 冑2H共兲 兩v兩 Proof: These identities follow immediately from Proposition 2.1: 兵I1,I2其 = 0, 兵e共1兲,e共2兲其 = 0, 兵e共1兲,I1其 = 兵I1,e共2兲其 = 0 = 兵I2,e共1兲其, 再 冎 1 c2 c⬜ c, = = e⬜共1兲. 2 兩c兩 兩c兩 On the other hand, 兵e共1兲 , I1其 = e⬜共1兲兵1 , I1其, so 兵1 , I1其 = 1. Similarly, 兵2 , I2其 = 1. 䊐 We now write the equations of motion for time-dependent flux in these action angle coordinates. As rot共E兲 = 0 there exists a 共possibly multivalued兲 function which we denote by m = m共s ; q兲 such that qm共s兲 = E共s兲 = − saE共s兲. Then T共s兲 is generated by m: sT共s; ,I兲 = 共0, saE共s,q共,I兲兲兲 = 共 pm,− qm兲 ⴰ T共s; ,I兲. Denote by U共s兲 : P → P the Hamiltonian flow of H共⑀s兲 defined by U共s兲 ª 共q共s兲 , p共s兲兲, q̇共s兲 = pH,ṗ共s兲 = − qH, 共q共0兲,p共0兲兲 = 共q,p兲, then for the flow Û共s兲 = 共共s兲 , I共s兲兲 in action angle coordinates defined by T共s兲 ⴰ Û共s兲 = U共s兲 ⴰ T共s = 0兲, it holds ˙ 共s兲 = IK ⴰ Û共s兲, İ共s兲 = − K ⴰ Û共s兲, 共共0兲,I共0兲兲 = 共,I兲, where the Hamiltonian in action angle coordinates, K = H ⴰ T − m ⴰ T, is K共s; ,I兲 = I2 − m共s;q共,I兲兲 and the equations of motion are 共with the notation 具·,·典 for the scalar product兲 ˙ 共s兲 = IK = 冉冊 0 1 − 具E共s,q共,I兲兲, Iq典, İ共s兲 = − K = 具E共s;q共,I兲兲, q典. 共4兲 共5兲 Remark 4.1: Another way to derive these equations is to start from Newton’s equation, q̈ = − q̇⬜ + E共s;q兲. From the very definition of c and v one gets ċ = − E⬜共c + v⬜兲, v̇ = − v⬜ + E共c + v⬜兲, which in action angle coordinates gives Eqs. (4) and (5). Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-8 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček V. LARGE TIME ASYMPTOTICS, POTENTIAL FREE CASE For the case N共s兲 = s, V = 0 we can precise the large time asymptotics and develop the solution up to order O共1 / s3/2兲 We have E共s兲 = q⬜ . q2 So m共q兲 = arg共q兲. Observe that K = K共,I兲 = I2 − arg共冑2I1e共1兲 + 冑2I2e共− 2兲兲 is an integral of motion. Theorem 5.1: Let V = 0, N共s兲 = s. Denote by I = 共I1 , I2兲, = 共1 , 2兲 the solution of the equations of motion (4) and (5), ˙ 共s兲 = IK共共s兲,I共s兲兲, I共0兲 = 共I01,I02兲, İ共s兲 = − K共共s兲,I共s兲兲, 共0兲 = 共01, 02兲, then the following asymptotic behaviors hold. a. In the future, s → ⬁. The following limits exist and define the constants a0 ⬎ 0, b0: lim I2共s兲 ¬ s→⬁ a20 , 4 lim 共1共s兲 + 2共s兲 − s兲 ¬ b0, s→⬁ lim 共I2共s兲 − 1共s兲兲 = K. s→⬁ The asymptotics are I2共s兲 = 冉 冊 冉 冊 a20 a0 a2 1 1 1 1 − sin共s + b0兲 + 1 + 0 sin共2共s + b0兲兲 + O 3/2 , 冑s 4 4 2 2 s s I1共s兲 = I2共s兲 + 共s − s0兲, 1共s兲 = 2共s兲 = s + b0 − +O b. 冉 冊 a20 1 1 4 1 1 cos共s + b0兲 + − 1 + 2 cos共2共s + b0兲兲 − 2 sin共2共s + b0兲兲 +K− 冑s 8 4 a0 s a0 冉 冊 1 s 冉 冊 a20 1 1 −K− + O 3/2 , 4 4s s 3/2 , with s0 defined as in Eq. (3). In the past, s → −⬁. The following limits exist and define the constants ã0 ⬎ 0, b̃0: lim I1共s兲 ¬ s→−⬁ ã20 , 4 lim 共1共s兲 + 2共s兲 − s兲 ¬ b̃0, s→−⬁ lim 共I2共s兲 + 2共s兲兲 = K. s→−⬁ The asymptotics are I1共s兲 = 冉 冊 冉 冊 ã2 ã20 ã0 1 1 1 1 − 1 − 0 sin共2共s + b̃0兲兲 + O , + sin共s + b̃0兲 冑 4 2 2 s 兩s兩3/2 兩s兩 4 Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-9 J. Math. Phys. 48, 052901 共2007兲 Hall effect driven by an Aharonov-Bohm flux I2共s兲 = I1共s兲 − 共s − s0兲, 1共s兲 = s0 + b̃0 + − ã20 1 1 cos共s + b̃0兲 −K+ 冑兩s兩 , 4 ã0 冉 冊 冉 冊 1 4 1 1 1 − 2 cos共2共s + b̃0兲兲 − 2 sin共2共s + b̃0兲兲 + O , 8 s 兩s兩3/2 ã0 2共s兲 = s − s0 − 冉 冊 ã20 1 1 . +K− +O 4 4s 兩s兩3/2 Proof: We give an outline of the main steps of the proof for the case t → ⬁. Some particular computations in the proof turned out to be quite tedious and thus computer algebra systems were employed to facilitate them. Suppose t ⬎ 0. Step 1. From Eq. 共2兲 we know I1共s兲 − I2共s兲 = 共s − s0兲. So the equations of motion only involve J ª I1 + I2 and ª 1 + 2 and transform to ˙ = 1 + s sin 冑J2 − s2共J + 冑J2 − s2 cos 兲 , J̇ = s , J + 冑J2 − s2 cos Step 2. Do a second transformation, x1 ª 冑J2 − s2 cos , x2 ª 1 + 冑J2 − s2 sin , the J, equations transform to ẋ1 − x1 + x2 = F共s,x1,x2兲, s ẋ2 − x1 = 0, with F共s,x1,x2兲 ª 1 − x1 s . − s 冑x21 + 共x2 − 1兲2 + s2 + x1 The corresponding homogeneous system is equivalent to ẍ1 − 冉 冊 1 ẋ1 + 1 + 2 x1 = 0 s s or sÿ + ẏ + sy = 0, with y defined by x1 = sy. The latter is Bessel’s equation of order 0 so one has two independent solutions of the homogeneous system: 冉 冊冉 冊 冉 冊冉 冊 x1共s兲 sJ0共s兲 = x2共s兲 sJ1共s兲 and x1共s兲 sY 0共s兲 = x2共s兲 sY 1共s兲 with the Bessel functions Jm 共Y m兲 of the first 共second兲 kind. Step 3. Transform the x-differential equation to the integral equation, x1共s兲 = c1sJ0共s兲 + c2sY 0共s兲 − s 2 冕 ⬁ 共Y 0共s兲J1共兲 − J0共s兲Y 1共兲兲F共,x1共兲,x2共兲兲d , s Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-10 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček x2共s兲 = c1sJ1共s兲 + c2sY 1共s兲 − s 2 冕 ⬁ 共Y 1共s兲J1共兲 − J1共s兲Y 1共兲兲F共,x1共兲,x2共兲兲d , s where the numbers c1, c2 involve the initial conditions. The equation is of the form x = K共x兲; the solution is constructed as the limit of the sequence xn+1 = K共xn兲 starting from x0 = 0. To verify the convergence one can apply yet another substitution x共s兲 = y共s兲 / 冑s, G共s , y兲 = s−1/2F共s , s−1/2y兲. Consequently, the integral equation takes the form y共s兲 = y 0共s兲 − 冕 ⬁ F共s, 兲G共,y 1共兲,y 2共兲兲d , s where y 0j共s兲 = c1冑sJ j−1共s兲 + c2冑sY j−1共s兲, F j共s, 兲 = j = 1,2, 冑 s共Y j−1共s兲J1共兲 − J j−1共s兲Y 1共兲兲, 2 j = 1,2. Considering the new integral equation in the Banach space L⬁共关s* , ⬁关兲 丢 R2, one can show that the iteration process is indeed contracting provided s* 艌 1 is sufficiently large. It is then straightforward to derive from the integral equation the asymptotic expansion of the solution x共s兲. One finds that x共s兲 = a0e共s + b0兲冑s + 冉 冊 冉冊 a30 5 1 1 +O . e共s + b0兲 − a0e⬜共t + b0兲 冑 8 8 s s Step 4. Transforming back first to the J, then to I1, I2, 1, 2 variables gives the claimed asymptotic expansion. 䊐 VI. AVERAGED DYNAMICS In the perspective to investigate the behavior at the transition point between the two dynamics in the case of small epsilon we analyze the equation averaged over the fast angle 2. This might provide a good approximation for the actions for small for times of order 1 / .14 We consider again the case N共s兲 = s. We set up the averaged equations and show that in the adiabatic limit the energy grows if and only I1 ⬍ I2. We then solve the equations explicitly for the case V = 0 and show that in the adiabatic limit the transition between the pinned and the “Hall dynamics” happens for a unique value of the driving flux and that the center moves if and only if the Landau orbit does not encircle the origin. We apply averaging with respect to the fast angle 2 to the system 共4兲 and 共5兲. E共s兲 = 冉 冊 q⬜ − qṼ共q兲 . q2 Further, choose m and thus K, m共q兲 = 共arg共q兲 − Ṽ兲, K共,I兲 = I2 − m共冑2I1e共1兲 + 冑2I2e共− 2兲兲. Denote the average of a function f on the phase space by Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-11 J. Math. Phys. 48, 052901 共2007兲 Hall effect driven by an Aharonov-Bohm flux f av共1,I兲 ª 1 2 冕 2 f共1, 2,I兲d2 . 0 In particular, for a function f defined on the plane thus depending only on the variable q we denote f av共1,I兲 = 1 2 Making use of the identities 冓 冔 冕 2 f共冑2I1e共1兲 + 冑2I2e共− 2兲兲d2 . 0 冊 冓 冔冉 冉 sin共1 + 2兲 冑I2/I1 q⬜ , 2 , Iq = q q2 − 冑I1/I2 the system 共4兲 and 共5兲 reads 冉冊 0 ˙ 共s兲 = 1 İ共s兲 = − 冊 关共I1 − I2兲/q2兴 + 1/2 q⬜ , q = , 关共I1 − I2兲/q2兴 − 1/2 q2 sin共1 + 2兲 2共I1 + I2 + 2冑I1I2 cos共1 + 2兲兲 I1 − I2 2共I1 + I2 + 2冑I1I2 cos共1 + 2兲兲 冉 冑冑 冊 I2/I1 − I1/I2 + IṼ共q共I, 兲兲, 冉冊 冉 冊 1 1 + 1 − Ṽ共q共I, 兲兲. 2 −1 The averaged quantities are readily calculated: using 冉冊 1 q2 = av 1 , 2兩I1 − I2兩 冉 sin共1 + 2兲 q2 冊 = 0, av one finds for the averaged vector field 共IK兲av共1,I兲 = 冉冊 0 1 + IṼav共1,I兲 − 共K兲av共1,I兲 = 冉 冊 冉 冊 共I1 ⬎ I2兲 1Ṽav共1,I兲 − , − 共I1 ⬍ I2兲 0 共6兲 where we used the binary function : 共True兲 ª 1, 共False兲 ª 0. Remark 6.1: Remark that the averaged vector field is the Hamiltonian vector field derived from the from the “averaged” Hamiltonian Kav. Indeed, using the splitting of arg共q兲, which is a multivalued function defined on the covering space of R2 \ 共0兲, into a linear and oscillating part, arg共q共,I兲兲 = 冦 冉 冑 冉 冑 1 + arg 共1,0兲 + − 2 + arg 共1,0兲 + I2 e共− 1 − 2兲 I1 I1 e共1 + 2兲 I2 冊 冊 if I1 ⬎ I2 if I2 ⬎ I1 , 冧 and the equality 冕 2 arg共共1,0兲 + ae共s兲兲ds = 0 for 0 艋 a ⬍ 1, 0 one finds that for Kav共,I兲 ª I2 − 共共1共I1 ⬎ I2兲 − 2共I1 ⬍ I2兲兲 − Ṽav共1,I兲兲, it holds Kav = 共K兲av, IKav = 共IK兲av. The result on the averaged dynamics now is as follows. Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-12 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček Theorem 6.1: Let N共s兲 = s. Denote by J = 共J1 , J2兲, = 共1 , 2兲 the solution of the averaged equations (6) ˙ 共s兲 = IKav共共s兲,J共s兲兲, J̇共s兲 = − Kav共共s兲,J共s兲兲, J共0兲 = 共J01,J02兲, 共0兲 = 共01, 02兲, then it holds the following. 1. Let V = 0, denote ⌬J = J02 − J01 then 冉 冉 冊 J共s兲 = min兵J01,J02其 + 共s − ⌬J兲 共s兲 = 2. For any V and any s1 , s2 苸 R, 兩J2共s2兲 − J2共s1兲兩 = 01 02 + s 冏冕 s2 冊 共s ⬎ ⌬J兲 , − 共s ⬍ ⌬J兲 . 冏 共J1共u兲 ⬍ J2共u兲兲du . s1 Proof: Using that for V = 0 it holds J1共s兲 − J2共s兲 − s = ⌬J the first assertion follows by inspection. The second assertion follows from integration of Eq. 共6兲. 䊐 Remark 6.1: 1. Loosely speaking the second assertion of the theorem means that, on the average, one has 兩energy change兩 = 兩flux change through the orbit during stay time兩, where the stay time means the time where the “orbit surrounds the origin.” This should be like this as the change in energy equals the work of the electric field along the orbit: H共s;q共s兲兲 − H共s0 ;q共s0兲兲 = 冕 s 具aE共s兲,ds典. s0 2. 3. 4. In situations where the averaging approximation is valid one gets estimates of the type 兩I共s兲 − J共s兲兩 = O共兲 for 兩s兩 艋 O共1 / 兲. Because of the singular behavior of the averaged equation it is a challenging problem to investigate if such an estimate is true or not and how the error would depend on the initial conditions. In the case when a smooth potential is present in view of the second assertion of the above theorem, it would be interesting to investigate if the kinetic energy only fluctuates by small amounts as soon as the particle is in the region c2 / 2 ⬎ H. Remark that a finite time adiabatic analysis would apply also to the case where ⌽ is a switching function which is linear for some time. VII. SCALING Let T, L, B, 关⌽兴 be units of time, length, magnetic field, and flux. Define dimensionless parameters T¬−1; 关⌽兴 / 共2BL2兲¬, denote N共s兲 ª ⌽共sT兲 / 关⌽兴 and choose L such that = 1 then H共t;q,p兲 = e 关⌽兴H共t/T,q/L,p/共eLB兲兲, 2 where Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-13 J. Math. Phys. 48, 052901 共2007兲 Hall effect driven by an Aharonov-Bohm flux 1 H共s;q,p兲 = 共p − a共s,q兲兲2 , 2 with and 2T V共Lq兲. e关⌽兴 Ṽ共q兲 ª The scaled function 共qsc共s兲 , psc共s兲兲 ª 共q / L共s / 兲 , p / 共eBL兲共s / 兲兲 then solves the Hamilton equations for the Hamiltonian H共s兲. Corollary 7.1: (To Proposition 7.1). Suppose that the torque of the background field is smaller than the circular one, i.e., that there exists a 苸 关0 , 1兲 such that for all t, q: e兩q ∧ qV共q兲兩 艋 e t⌽共t兲 a, 2 then for any initial condition there exists a unique hitting time t0 such that ± Bc2共t兲 艌 ± 2 H共t,B兲, e ± t ⬎ ± t0 . Furthermore for any initial condition we have 2 H共t,B兲 艌 共1 − a兲 ⬎ 0, e 兩⌽共t兲兩 lim inf ⌽共t兲→−⬁ lim inf ⌽共t兲→⬁ Bc2共t兲 艌 共1 − a兲 ⬎ 0. ⌽共t兲 䊐 We have H共t,B兲 = e 关⌽兴I2共t兲, 2 q共t兲 = Lqsc共t兲, H共t,− B兲 = e 关⌽兴I1共t兲, 2 qsc = 冑2I1e共1兲 + 冑2I2e共− 2兲. Corollary 7.2: (To Theorem 5.1). Let 共t兲 = 关⌽兴t / T, V = 0. The following limits are valid for any fixed initial condition: Bq2共t兲 →兩⌽共t兲兩→⬁1, 兩⌽共t兲兩 H共t,B兲 兩⌽共t兲兩 →⌽共t兲→−⬁ H共t,− B兲 e , ←⌽共t兲→⬁ 2 ⌽共t兲 H共t,B兲→⌽共t兲→⬁ a20 , 4 Downloaded 11 May 2007 to 139.124.7.126. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp 052901-14 J. Math. Phys. 48, 052901 共2007兲 J. Asch and P. Šťovíček H共t,− B兲→⌽共t兲→−⬁ B 冑兩⌽共t兲兩 冑 q共t兲→t→⬁e ã20 , 4 冉 冊 a20 −K , 4 B q共t兲⬃t→−⬁e共− t兲. 兩⌽共t兲兩 Remark also that for the rescaled center it holds H共t,− B兲 = 1 2 c 2m2 so Bc2共t兲⬃t→⬁⌽共t兲. ACKNOWLEDGMENTS One of the authors 共P. Š.兲 wishes to acknowledge gratefully partial support from the Grant Nos. 201/05/0857 of the Grant Agency of the Czech Republic and LC06002 of the Ministry of Education of the Czech Republic. Asch, J., Benguria, R. 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