PHYSICAL REVIEW E, VOLUME 63, 041112

Stochastic resonance in a suspension of magnetic dipoles under shear flow Toma´s Alarco´n and Agustı´n Pe´rez-Madrid Departament de Fı´sica Fonamental and CER on Physics of Complex Systems, Facultat de Fı´sica, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain 共Received 25 July 2000; published 27 March 2001兲 We show that a magnetic dipole in a shear flow under the action of an oscillating magnetic field displays stochastic resonance in the linear response regime. To this end, we compute the classical quantifiers of stochastic resonance, i.e., the signal to noise ratio, the escape time distribution, and the mean first passage time. We also discuss the limitations and role of the linear response theory in its applications to the theory of stochastic resonance. DOI: 10.1103/PhysRevE.63.041112

PACS number共s兲: 05.40.⫺a

I. INTRODUCTION

The dynamics of periodically driven stochastic systems has been an active field of research in recent years 关1兴. This kind of system arises frequently in the fields of physics, chemistry, and biology. Examples are found in problems involving transport at the cellular level 关2,3兴, optical and electronic devices 关4兴, and signal transduction in neuronal tissue 关5,6兴, to cite just a few. A particularly interesting phenomenon, occurring in periodically driven nonlinear noisy systems, is stochastic resonance 共SR兲 关7兴. This term refers to the enhancement of the response of the system to a coherent signal when the intensity of the noise grows, instead of the degradation that one naively expects. The mechanism leading to this phenomenon is quite simple. Imagine a system that exhibits an energetic activation barrier. In the presence of noise, the system can be assumed to surmount this barrier with a rate proportional to e ⫺⌬E/D , where ⌬E is the height of the barrier and D is the intensity of the noise acting on the system. The inverse of this rate defines the average waiting time T(D) between two noise-induced transitions. In the presence of a periodic forcing, the height of the barrier is periodically raised and lowered. When the period of the external force synchronizes with 2T(D), the barrier surmounting will be enhanced by the cooperative effect of the noise and the periodic forcing. Although originally proposed for systems in a doublewell potential, this original scheme has been extended. In fact, it is known that SR is exhibited by several classes of monostable system, among which one might mention excitable and threshold systems 关8–11兴 or systems that do not follow an activated dynamics but a relaxational dynamics 关12,13兴. In this paper we will show that a magnetic dipole immersed in a shear flow exhibits stochastic resonance when a weak oscillating magnetic field is acting on it. The presence of this flow takes the system out of equilibrium causing certain peculiarities in the behavior of the system. In order to treat this problem we will analyze the response of the system in the linear regime. A previous study of the dynamics of a dipole under an oscillating magnetic field has revealed that linear response theory 共LRT兲 predicts a monotonically decreasing behavior for the ratio between the output signal and the output noise or signal-to-noise ratio 共SNR兲, i.e., for very 1063-651X/2001/63共4兲/041112共12兲/$20.00

weak applied fields the dipole does not exhibit SR 关13兴. In the present case there is a new ingredient, absent in 关13兴: the presence of the shear flow, which is the determinant for many interesting aspects of the dynamics of this system. Additionally, although we show the existence of SR in the linear regime, we discuss the limitations and role of LRT in its application to the theory of SR, mainly related to questions concerning the fluctuation-dissipation theorem. The paper is organized as follows. In Sec. II we analyze the dynamics of a dipole in a shear flow and find the fixed points. Section III is devoted to studying the response of the system to an oscillating magnetic field by computing the susceptibility. In Sec. IV we calculate the power spectrum and the signal-to-noise ratio. In Sec. V we compute the escape time distribution and from it the mean first passage time. Finally, in Sec. VI we discuss our results. II. DYNAMICS OF A DIPOLE IN A SHEAR FLOW: FIXED POINTS AND THEIR STABILITY

We consider a dilute colloidal suspension of ferromagជ netic dipolar spherical particles, with magnetic moment m ជˆ , where Rជˆ is a unit vector accounting for the orienta⫽m s R tion of the dipole; the magnetic moment is therefore rigidly attached to the particles. Each dipole is under the influence ជ ⫽2 ␻ zជˆ , with zជˆ being the unit of a shear flow with vorticity ⍀ 0

ជ vector along the z axis, and of an oscillating field H ⫽He ⫺i ␻ t xជˆ , with xជˆ being the unit vector along the x axis. The dynamics of these dipoles is governed by the following equation of motion:

I





ជp 1 d⍀ ជ ⫻H ជ ⫹␰r ⍀ ជ ⫺⍀ ជp , ⫽m dt 2

共1兲

where I is the moment of inertia of the particles, ␰ r ⫽8 ␲ ␩ 0 a 3 is the rotational friction coefficient, ␩ 0 the solvent viscosity, and a the radius of the particle. For tⰇ ␶ r , with ␶ r ⫽I/ ␰ r being the inertial time scale, the motion of the particle enters the overdamped regime. This time scale defines a cutoff frequency ␻ r ⫽ ␶ r⫺1 , such that the condition for overdamped motion is equivalent to ␻ Ⰶ ␻ r . In this case Eq.

63 041112-1

©2001 The American Physical Society

´ S ALARCO ´ N AND AGUSTI´N PE´REZ-MADRID TOMA

PHYSICAL REVIEW E 63 041112

共1兲 yields the balance condition between the magnetic and hydrodynamic torques acting on each particle,

ជ ⫻H ជ ⫹␰r m





1 ជ ⫺⍀ ជ p ⫽0, ⍀ 2

ជ B (t) is a Gaussian white noise of zero mean and where F correlation function

具 Fជ B 共 t 兲 Fជ B 共 t ⬘ 兲 典 ⫽2 ␰ r k B T ␦ 共 t⫺t ⬘ 兲 .

共2兲

The Fokker-Planck equation corresponding to Eq. 共8兲 is given by

which, together with the rigid rotor evolution equation

ជˆ dR ជ p ⫻Rជˆ , ⫽⍀ dt

⳵ t ⌿ 共 Rជˆ ,t 兲 ⫽ 关 L0 ⫹␭ 共 t 兲 L1 兴 ⌿ 共 Rជˆ ,t 兲 ,

共3兲

dt

ជˆ ⫻xជˆ 兲 其 ⫻Rជˆ . ⫽ ␻ 0 兵 zជˆ ⫹␭ 共 t 兲共 R

共4兲

ជ p being the angular Here ␭(t)⬅(m s H/ ␰ r ␻ 0 )e ⫺i ␻ t , with ⍀ velocity of the particle. The computation of the fixed points of Eq. 共4兲 when the magnetic field is held constant, i.e., ␭(t)⫽␭ 0 ⫽const, and their linear stability analysis are given in detail in Appendix A. After some algebra Eq. 共4兲 becomes ˆ dRជ ជˆ ⫹␭xជˆ ⫺␭Rជˆ 共 Rជˆ •xជˆ 兲兴 . ⫽ ␻ 0 关 zជˆ ⫻R dt

共5兲

For ␭ 0 ⭓1, this equation has only a linearly stable stationary state. The orientation of the suspended particles is fixed to

ជˆ s ⫽ 冑1⫺␭ ⫺2 xជˆ ⫹␭ ⫺1 yជˆ . R

共6兲

This means that in this regime the hydrodynamic torque, which tends to cause the rotation of the particles, is insufficient to overcome the magnetic torque, which maintains their constant orientation. For ␭ 0 ⬍1, which is the case we are interested in, the particles undergo a rotation around a fixed axis lying in the y-z plane, the director of this axis being given by ˆ ˆ Rជ s ⫽⫾ 冑1⫺␭ 2 yជ ⫹␭zជˆ .

共7兲

In this case the hydrodynamic torque is strong enough to ˆ make the dipole precess around the orientation Rជ s , Eq. 共7兲 共see Appendix A兲.

The analysis of Sec. II was carried out for the deterministic dynamics of a magnetic dipole in a shear flow. Fluctuations are introduced by means of a Brownian torque. The corresponding Langevin equation is



ជˆ 1 ˆ dR ជ ⫻Fជ B 共 t 兲兲 ⫻Rជˆ , 共8兲 ⫽ ␻ 0 ␭ 共 t 兲共 Rជˆ ⫻xជˆ 兲 ⫹zជˆ ⫹ 共R dt ␰ r␻ 0

ជ ⫹D r R ជ 2, L0 ⫽⫺ ␻ 0 zជˆ •R

共11a兲

ជˆ •xជˆ ⫺ ␻ 0 共 Rជˆ ⫻xជˆ 兲 •R ជ, L1 ⫽2 ␻ 0 R

共11b兲

with D r ⫽k B T/ ␰ r being the rotational diffusion coefficient ជ ⫽Rជˆ ⫻ ⳵ / ⳵ Rជˆ the rotational operator. Notice that the first and R and second terms on the right hand side of Eq. 共11a兲 correspond to convective and diffusive terms, respectively. Moreover, Eq. 共10兲 which, according to Eq. 共8兲, rules the Brownian dynamics in the case of overdamped motion, is valid in the diffusion regime. This regime is also characterized by the condition tⰇ ␶ r , or equivalently ␻ Ⰶ ␻ r , which implicitly involves the white noise assumption. To solve the Fokker-Planck equation 共10兲 we will assume that ␭ 0 ⬅ 兩 ␭(t) 兩 constitutes a small parameter such that this equation can be solved perturbatively. Thus up to first order in ␭ the solution of the Fokker-Planck equation 共10兲 is

ជˆ ,t 兲 ⫽e (t⫺t 0 )L 0 ⌿ 0 共 t 0 兲 ⫹ ⌿共 R



t

t0

dt ⬘ ␭ 共 t ⬘ 兲 e (t⫺t ⬘ )L0 L1 ⌿ 0 共 t ⬘ 兲 . 共12兲

Here ⌿ 0 (t ⬘ )⫽e (t ⬘ ⫺t 0 )L 0 ⌿ 0 (t⫽t 0 ) is the zero order solution at time t ⬘ , and ˆ ជˆ ⫺Rជˆ 0 兲 , ⌿ 0 共 Rជ ,t⫽t 0 兲 ⫽ ␦ 共 R

共13兲

with Rជˆ 0 being an arbitrary initial orientation. As follows from Eq. 共11a兲, the unperturbed operator L0 is composed of the operators Rz and R 2 , which are proportional to the orbital angular momentum operators of quantum mechanics L z and L 2 , respectively, and, therefore, their eigenfunctions are the spherical harmonics 关14兴

III. RESPONSE TO AN OSCILLATING MAGNETIC FIELD



共10兲

where L0 and L1 are operators defined by

ជˆ , leads to the dynamic equation for R ជˆ dR

共9兲

ជˆ 兲 ⫽imY l m 共 Rជˆ 兲 , Rz Y l m 共 R

共14a兲

ជˆ 兲 ⫽⫺l 共 l⫹1 兲 Y l m 共 Rជˆ 兲 . R 2Y l m共 R

共14b兲

ជ acts on the spherical harmonGiven that we know how R ics, it is convenient to expand the initial condition in series of these functions, since the spherical harmonics constitute a complete set of functions that are a basis in the Hilbert space of the integrable functions over the unit sphere 关15兴:

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ˆ ជˆ ⫺Rជˆ 0 兲 ⫽ ⌿ 共 Rជ ,t 0 兲 ⫽ ␦ 共 R 0



PHYSICAL REVIEW E 63 041112

l

兺 兺 Y l*m共 Rជˆ 0 兲 Y l m共 Rជˆ 兲 . l⫽0 m⫽⫺l

␹ i共 ␻ 兲 ⫽ 共15兲

Using this expansion in Eq. 共12兲, for the first order correction to the probability density ⌬⌿⬅⌿⫺⌿ 0 , we obtain

ជˆ ,t 兲 ⫽ ⌬⌿ 共 R

l

␹ x共 ␻ 兲 ⫽

t

1 3

0

ˆ ⫻e (t⫺t ⬘ )L0 L 1 e (t ⬘ ⫺t 0 )L0 Y l m 共 Rជ 兲 .

再 冕

t

⫺⬁

dt ⬘ ␭ 共 t ⬘ 兲 e

(t⫺t ⬘ )L0

再冋 冋



共16兲

Notice that the integral of ⌬⌿(Rជˆ ,t) over the entire solid angle is zero, in agreement with the fact that the unperturbed ជˆ ,t) is normalized. solution ⌿ 0 (R Since we are interested in the asymptotic behavior we will set t 0 →⫺⬁. In this limit, Eq. 共12兲 becomes 1 ជˆ ,t 兲 ⫽ 1⫹ ⌿共 R 4␲





⫺⬁





ជˆ Rˆ i e ␶ L0 2Rជˆ •xជˆ . dR

␹ y共 ␻ 兲⫽



1 3

ជˆ •xជˆ , 共17兲 2R

2D r ⫹i 2 4D r ⫹ 共 ␻ ⫹ ␻ 0 兲 2

再冋 冋



2D r ⫺i 2 4D r ⫹ 共 ␻ ⫺ ␻ 0 兲 2

共24兲

共 ␻ 0⫹ ␻ 兲 ⫺i 2 4D r ⫹ 共 ␻ ⫹ ␻ 0 兲 2

册 册冎

共 ␻ 0⫺ ␻ 兲 2 4D r ⫹ 共 ␻ ⫺ ␻ 0 兲 2

共 ␻ 0⫹ ␻ 兲 2 4D r ⫹ 共 ␻ ⫹ ␻ 0 兲 2

共 ␻ 0⫺ ␻ 兲 ⫹i 2 4D r ⫹ 共 ␻ ⫺ ␻ 0 兲 2

册 册冎

⫺⬁

ជˆ •xជˆ , dt ⬘ ␭ 共 t ⬘ 兲 e (t⫺t ⬘ )L0 2R

共18兲

共25兲

2D r 2 4D r ⫹ 共 ␻ ⫺ ␻ 0 兲 2

2D r 2 4D r ⫹ 共 ␻ ⫹ ␻ 0 兲 2

␹ z 共 ␻ 兲 ⫽0. t

,

, 共26兲

where now 1 ជˆ ,t 兲 ⫽ ⌬⌿ 共 R 4␲

d ␶ e i␻␶

From this equation we obtain the components of the susceptibility:

兺 兺 冕t dt ⬘ ␭ 共 t ⬘ 兲 Y l*m共 Rជˆ 0 兲 l⫽0 m⫽⫺l ⬁

1 4␲

共27兲

The quantities ␹ x and ␹ y have poles at ␻ ⫽⫾ ␻ 0 ⫾2D r i. The inverse of the imaginary part of these poles (2D r ) ⫺1 defines the Brownian relaxation time.

and IV. POWER SPECTRUM

1 ជˆ ,t 兲 ⫽ ⌿ 0共 R 4␲

共19兲

is the uniform distribution function on the unit sphere. From Eq. 共18兲 the contribution of the ac field to the mean ˆ value of the orientation vector Rជ can be obtained as

ជˆ 共 t 兲 ⫽ R



1 ជˆ Rជˆ ⌬⌿⫽ dR 4␲



t

⫺⬁

dt ⬘ ␭ 共 t ⬘ 兲



ជˆ e (t⫺t ⬘ )L0 2Rជˆ •xជˆ . dRជˆ R 共20兲

In order to discern whether or not SR is present in the relaxation process of the system under consideration we compute the power spectrum, which, following the WienerKhinchine theorem, is given by the Fourier transform of the correlation function 关17,1兴. Since we will take as output sigជˆ parallel to the magnetic field, i.e., Rˆ x , nal the projection of R we compute only the correlation function of this quantity. The correlation function of Rˆ x is defined by

具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 兩 Rជ 0 共 t 0 兲 典 ˆ

This equation can be written in the more compact form Rˆ i 共 t 兲 ⫽



t

⫺⬁

dt ⬘ ␭ 共 t ⬘ 兲 ␹ i 共 t⫺t ⬘ 兲 ,

共21兲

where the response function 关16兴 has been defined as

␹ i共 ␶ 兲 ⫽

1 4␲



ជˆ Rˆ i e ␶ L0 2Rជˆ •xជˆ dR



共22兲

ជˆ 0 ,t 0 兲 , 共28兲 duជˆ uˆ x vˆ x ⌿ 共 vជˆ ,t;uជˆ ,t⫹ ␶ 兩 R

⌿ 共 vជˆ 1 ,t 1 ; . . . ; vជˆ n ,t n 兲 ⫽⌿ 共 vជˆ 1 ,t 1 兲 ⌿ 共 vជˆ 1 ,t 1 兩 vជˆ 2 ,t 2 ; . . . ; vជˆ n ,t n 兲 , ⌿ 共 vជˆ 1 ,t 1 兩 vជˆ 2 ,t 2 ; . . . ; vជˆ n ,t n 兲 ⫽⌿ 共 vជˆ 1 ,t 1 兩 vជˆ 2 ,t 2 兲 .

共23兲

where ␹ i ( ␻ ) is the generalized susceptibility, which is the Fourier transform of ␹ i ( ␶ ),

d vˆ

ជˆ ,t 0 )⫽ ␦ (Rជˆ ⫺Rជˆ 0 ). where the initial condition is taken as ⌿(R The above quantity can be calculated from the solution of the Fokker-Planck equation simply by recalling the following properties of a Markov process 关17兴:

for ␶ ⬎0. By causality, we can write t→⬁ in the upper limit of the integral in Eq. 共21兲; hence, this equation becomes Rˆ i 共 t 兲 ⫽ ␹ i 共 ␻ 兲 ␭ 共 t 兲 ,

冕 ជ冕

共29兲

where t 1 ⬎t 2 ⬎•••⬎t n . By combination of these two properties Eq. 共28兲 becomes

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具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 兩 Rជ 0 共 t 0 兲 典

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ˆ







ជˆ 0 ,t 0 兲 d vជˆ vˆ x ⌿ 共 vជˆ ,t 兩 R

duជˆ uˆ x ⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 . 共30兲

To proceed further, we compute first the integral over uជˆ in Eq. 共30兲. From Eq. 共12兲, if ␶ ⬎0, ⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⫽e ␶ L0 ␦ 共 uជˆ ⫺ vជˆ 兲 ⫹



t

t⫹ ␶

first order term vanishes, and consequently we do not worry about it and compute only those whose average gives a nonzero contribution, i.e., the zeroth and second order terms. Taking this into account, and by applying Eq. 共12兲 to ជˆ 0 ,t 0 ), ⌿( vជˆ ,t 兩 R

具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 兩 Rជ 0 共 t 0 兲 典 ˆ

⬃ ds ␭ 共 s 兲

共31兲 thus we have

冕 ⫹

duជˆ uˆ x ⌿ 0 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲



d vˆ

ជˆ 0 ,t 0 兲 ⌿ 0 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 duˆ uˆ x vˆ x ⌿ 0 共 vជˆ ,t 兩 R ˆ duˆ uˆ x vˆ x ⌬⌿ 共 vជˆ ,t 兩 Rជ 0 ,t 0 兲 ⌬⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 , 共34兲

⫽⌿ 0 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⫹⌬⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ;

duជˆ uˆ x ⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⫽

d vˆ



⫻e (t⫹ ␶ ⫺s)L0 L 1 e sL0 ␦ 共 uជˆ ⫺ vជˆ 兲



冕 ជ冕 ជ 冕 ជ冕 ជ

where the sign ⬃ indicates that the terms which vanish after averaging over the period of the driving have been neglected 共although the average has not been performed yet兲. After ˆ introducing the corresponding expressions for ⌿( vជˆ ,t 兩 Rជ 0 ,t 0 ) and by using Eq. 共33兲 we obtain

具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典 ⬃

duជˆ uˆ x ⌬⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 . 共32兲

冉 冊 4␲ 3

duជˆ uˆ x ⌿ 0 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⫽⫺



具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典 ⬅ 共33a兲

⫽⫺

兺 兺 Y lm* 共 vជˆ 兲 冕t l⫽0 m⫽⫺l





⫹e

l

t⫹ ␶

ds ␭ 共 s 兲



duជˆ

具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典 ⫽

2 ␲ ⫺(2D ⫺i ␻ )(t⫹ ␶ ⫺s) r 0 Y 11共 uជˆ 兲 兵关 e 3

⫺(2D r ⫹i ␻ 0 )(t⫹ ␶ ⫺s)

⫻L1 Y lm 共 uជˆ 兲 其 .



0

dt ⬘ e i ␻ t ⬘ ␹ x 共 t ⬘ 兲



2␲ 3

3

␭ 2共 t 兲

2

共35兲

1 4␲



ជˆ 0 共 t 0 兲 典 . dRជˆ 0 具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 兩 R 共36兲

At this stage, and before applying the Fourier transform to the correlation function to obtain the power spectrum of the process Rˆ x (t), we average Eq. 共35兲 to obtain

duជˆ uˆ x ⌬⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⬁

冏冕

冉 冊

共the details of this computation are given in Appendix B兲, where we have defined

2 ␲ ⫺(2D ⫺i ␻ ) ␶ ˆ r 0 Y 共v 关e 11 ជ 兲 3

⫹e ⫺(2D r ⫹i ␻ 0 ) ␶ Y 1⫺1 共 vជˆ 兲兴 ,



2e ⫺2D r ␶ cos共 ␻ 0 ␶ 兲 ⫹

⫻e i ␻ ␶

The results of these integrals are



2



Y 1⫺1 共 uជˆ 兲兴 e ⫺[l(l⫹1)D r ⫹im ␻ 0 ]s



2␲/␻

0

冉 冊 冉 冊 4␲ 3



共33b兲

共for the detailed derivation, see Appendix B兲. After introducing these expressions into Eq. 共30兲 we obtain three terms corresponding to an expansion of the correlation function in powers of ␭(t), of zeroth, first, and second order, respectively. The presence of this driving yields an explicit dependence of the correlation function on the time t, instead of its depending only on the time difference, as occurs in the stationary case. The method for removing this dependence on the initial time is to average the correlation function over a period of the driving 关1兴. After doing this the

␻ 2␲

2

dt 具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典

2e ⫺2D r ␶ cos共 ␻ 0 ␶ 兲

2␲ 3

3

冏冕

␭ 20 e i ␻ ␶



0



2

dt ⬘ e i ␻ t ⬘ ␹ x 共 t ⬘ 兲 . 共37兲

This computation has been carried out with the assumption that ␶ is a positive quantity. To extend our computation to ␶ ⬍0 we have to use the backward Fokker-Planck equation. The operator that generates the backward evolution of the probability distribution is ⫺L † 关18兴, L being the FokkerPlanck operator and L † its adjoint operator. Consequently the formal solution of the backward Fokker-Planck, equivalent to Eq. 共12兲, is given by (t⬍t 0 )

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STOCHASTIC RESONANCE IN A SUSPENSION OF . . .

PHYSICAL REVIEW E 63 041112

具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典 ⫽

冉 冊 冉 冊 4␲ 3

2

2␲ 3



2e 2D r ␶ cos共 ␻ 0 ␶ 兲 3

冏冕

␭ 20 e i ␻ ␶



0

dt ⬘ e i ␻ t ⬘ ␹ x 共 t ⬘ 兲



2

共40兲 for ␶ ⬍0. We now apply to this averaged correlation function 共now defined for ⫺⬁⬍ ␶ ⬍⬁) the Wiener-Khinchine theorem, which states that the power spectrum and the correlation function are related through a Fourier transform. Thus, Q共 ⍀ 兲⫽

FIG. 1. Signal-to-noise ratio as a function of the inverse of the Pe´clet number Pe⫺1 ⫽D r / ␻ 0 . We have represented nondimensional quantities. †





t0

dt ⬘ ␭ 共 t ⬘ 兲 e

N共 ⍀ 兲⫽

⫺(t⫺t ⬘ )L 0 †

L 1⌿ 共 t ⬘ 兲. 0

共38兲

As in the case of the operator L0 , the spherical harmonics are eigenfunctions of ⫺L †0 with eigenvalues given by

ជˆ 兲 ⫽ 关 l 共 l⫹1 兲 D r ⫺im ␻ 0 兴 Y lm 共 Rជˆ 兲 . ⫺L †0 Y lm 共 R

共39兲

Thus the process to follow in the calculation of the correlation function for ␶ ⬍0 is identical to the corresponding computation for ␶ ⬎0 but changing the eigenvalues of the operator L0 to those of ⫺L †0 , which yields

R⫽



⫺⬁

d ␶ 具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典 e i⍀ ␶

⫽N 共 ⍀ 兲 ⫹S 共 ␻ 兲 ␦ 共 ⍀⫺ ␻ 兲 ,

⌿ 共 Rជˆ ,t 兲 ⫽e ⫺(t⫺t 0 )L 0 ⌿ 0 共 t 0 兲 t



冉 冊冋 4␲ 3

2

2D r ⫹ 2 4D r ⫹ 共 ⍀⫹ ␻ 0 兲 2

S共 ␻ 兲⫽

冉 冊 2␲ 3

共41a兲

2D r 2 4D r ⫹ 共 ⍀⫺ ␻ 0 兲 2



,

共41b兲

3

␭ 20 兩 ␹ x 共 ␻ 兲 兩 2 .

共41c兲

Since our purpose is to discern whether or not SR is present in the relaxation process of the quantity Rˆ x (t), we proceed to compute the signal-to-noise ratio R, i.e., the ratio between the weight of the ␦ function in Eq. 共41a兲 and the noisy part of Q(⍀) computed at the frequency of the driving. From Eqs. 共25兲 and 共41兲 we achieve

6 S共 ␻ 兲 ⫽␭ 20 N共 ␻ 兲 ␲ ⫻

兵 2D r / 关 4D r2 ⫹ 共 ␻ ⫹ ␻ 0 兲 2 兴 ⫹2D r / 关 4D r2 ⫹ 共 ␻ ⫺ ␻ 0 兲 2 兴 其 2 ⫹ 兵 共 ␻ ⫹ ␻ 0 兲 / 关 4D r2 ⫹ 共 ␻ ⫹ ␻ 0 兲 2 兴 ⫹ 共 ␻ ⫺ ␻ 0 兲 / 关 4D r2 ⫹ 共 ␻ ⫺ ␻ 0 兲 2 兴 其 2 2D r / 关 4D r2 ⫹ 共 ␻ ⫹ ␻ 0 兲 2 兴 ⫹2D r / 关 4D r2 ⫹ 共 ␻ ⫺ ␻ 0 兲 2 兴

.

共42兲

This quantity has been plotted in Fig. 1 as a function of the inverse of the Pe´clet number Pe⫺1 ⫽D r / ␻ 0 , which measures the ratio between the time scales associated with diffusion 共thermal noise兲 and flow. The presence of a maximum in R for a nonzero value of this parameter shows the existence of stochastic resonance in the relaxation process of a dipole in a shear flow. In addition to the slow relaxation to the single attractor of the dynamics, our model includes another effect, which hides, to some extent, the SR profile. To understand this, note that even though the signal is too weak, it nevertheless causes the position of the attractor of the dynamics to vary, and so the output will always have a nonzero compo-

nent at the signal frequency. This fact causes the SNR to go to infinity in the zero noise limit 关19兴. V. MEAN FIRST PASSAGE TIME

In this section we study the behavior of the escape time distribution and the mean first passage time of the magnetic dipole immersed in a shear flow. To this end, we have to account for the fixed point orientations of Eq. 共4兲 in the case ␭ 0 ⬍1. In this situation there is a single fixed point corresponding to an orientation contained in the plane x⫽0 or, equivalently, ␾ ⫽ ␲ /2. However, when ␭(t)⬎0 this station-

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PHYSICAL REVIEW E 63 041112

ary orientation is in the subspace z⬎0 (cos ␪⬎0) and in the subspace z⬍0 (cos ␪⬍0) if ␭(t)⬍0. Therefore, we are going to study the escape from the region z⬎0 (cos ␪⬎0) assuming that the initial orientation of the dipole is contained in this region. Consequently, we have to solve the FokkerPlanck equation 共10兲 with absorbing boundary conditions in the plane cos ␪⫽0 关20,21兴, i.e., ⌿ 共 cos ␪ ⫽0,␾ ,t 兲 ⫽0.

共43兲

Since this escape problem will be treated perturbatively, the first step is to analyze the eigenvalue problem of the operator L0 关Eq. 共11兲兴 under the boundary condition 共43兲. It is easy to check that the eigenfunctions and the eigenvalues are the same with the restriction that only those spherical harmonics that vanish at cos ␪⫽0 are solutions of this eigenvalue problem. From the parity properties of the associated Legendre functions 关15兴, one can see that Eq. 共43兲 selects only the spherical harmonics such that l⫹m⫽2n⫹1 with n⫽0,1, . . . . Thus, we have

ជˆ ,t 兲 ⫽ ⌿共 R



具 l,m 典

ជˆ 兲 , a lm 共 t 兲 Y lm 共 R

␳ 共 Rជˆ 0 ,t 兲 ⫽⫺

dt

,

(0) ជˆ 0 兲 . a 10 共 t 兲 ⫽e ⫺2D r t Y 10共 R



冕 冕 ␾冕 R

⫺␻0 ⫺i

1

d

0

0

4␲ ជˆ 兲 Ry Y 共R 3 10



2␲ ជˆ 兲 ⫺Y 1⫺1 共 Rជˆ 兲兴 Rz Y lm 共 Rជˆ 兲 , 关 Y 11共 R 3

ជˆ ) is given by Eq. 共14a兲 and where the action of Rz on Y lm (R

共45兲 1 ˆ ជˆ 兲 Ry Y lm 共 Rជ 兲 ⫽⫺ 兵 冑共 l⫺m 兲共 l⫹m⫹1 兲 Y lm⫹1 共 R 2

ជˆ 兲 其 . ⫺ 冑共 l⫹m 兲共 l⫺m⫹1 兲 Y lm⫺1 共 R

ជˆ 0 兲 , 共46兲 d 共 cos ␪ 兲 ⌿ 共 cos ␪ , ␾ ,t 兩 R

with R the region from which we are studying the escape ជˆ 0 苸R the initial problem 共in the present case cos ␪⬎0), and R orientation of the dipole. The probability distribution ˆ ˆ ⌿(Rជ ,t 兩 Rជ 0 ) is obtained from Eq. 共12兲 with the boundary conditions 共43兲 and the initial condition

ជˆ ,t⫽0 兲 ⫽ ␦ 共 Rជˆ ⫺Rជˆ 0 兲 ⫽ ⌿共 R

2␲ ជˆ 兲 ⫹Y 1⫺1 共 Rជˆ 兲兴 Y lm 共 Rជˆ 兲 关 Y 11共 R 3

共49兲

ជˆ ⌿ 共 Rជˆ ,t 兩 Rជˆ 0 兲 dR

2␲

冑 冋冑 冑

ជˆ 兲 ⫽⫺2 ␻ 0 L1 Y lm 共 R

ជˆ 0 ,t) is defined by where S(R ជˆ 0 ,t 兲 ⫽ S共 R

共48兲

(1) Obtaining the first order contribution a 10 (t) requires somewhat more elaborate calculation. To proceed further ជˆ ) with this computation, the operator L1 acting on Y lm (R yields

共44兲

where 具 l,m 典 denotes that the sum is carried out over 0⭐l ⬍⬁ and ⫺l⭐m⭐l restricted by l⫹m⫽2n⫹1. In order to evaluate the mean first passage time 共MFPT兲, ជˆ 0 ,t) we have to compute first the survival probability S(R and the escape time distribution 共ETD兲, which are related through

ជˆ 0 ,t 兲 d S共 R

⫹m⫽2n⫹1 reduces to keeping only the odd values of l. In addition, we are interested only in the modes with greater relaxation times. Therefore, from the whole series Eq. 共44兲 we are interested only in the term l⫽1, m⫽0. Thus, our purpose is to obtain the coefficient a 10(t) up to first order in ␭(t) from Eqs. 共10兲 and 共43兲. Up to zeroth order, we have



具 l,m 典

* 共 Rជˆ 0 兲 Y lm 共 Rជˆ 兲 . Y lm

共47兲

Before proceeding to obtain the survival probability, there are some facts to consider that will facilitate further computation. Looking at Eq. 共46兲, one can see that, due to the integration over the azimuthal angle, only terms with m⫽0 ˆ contribute to S(Rជ 0 ,t). Consequently, the selection rule l

共50兲

From Eqs. 共49兲 and 共50兲 together with the rules for the addition of angular momenta familiar from quantum mechanics 关14兴 and the selection rule l⫹m⫽2n⫹1 imposed by the boundary condition 共43兲, one can deduce that only the term ˆ (1) (t). The rules of addition of anY 2⫾1 (Rជ ) contributes to a 10 gular momenta imply that the product of two spherical harmonics Y lm Y pq has a projection onto a third spherical harmonic Y rs only when m⫹q⫽s. On the other hand, these same rules impose the restriction that the product Y lm Y pq projects only onto subspaces such that 兩 l⫺p 兩 ⭐r⭐l⫹p. By using these restrictions one can see that when one takes l ⫽1 and m⫽0 in Eq. 共49兲 one obtains a vanishing contribution and only when l⫽2 and m⫽⫾1 is the contribution to (1) (t) different from zero. All other contribution of higher a 10 values l are explicitly excluded by the rule 兩 l⫺p 兩 ⭐r⭐l⫹p. Taking these considerations into account and by using the results

041112-6

STOCHASTIC RESONANCE IN A SUSPENSION OF . . .

3 4␲

冕 ␾冕 冑␲ ␲ 2␲

1

d

0

0

ជˆ 兲 Y 2⫺1 共 Rជˆ 兲 Y 10共 Rជˆ 兲 d 共 cos ␪ 兲 Y 11共 R

S 共 Rជˆ 0 ,t 兲 ⫽ ␲

5 , 3

9 80



3 4␲

PHYSICAL REVIEW E 63 041112

0

9 80



3 4␲

d␾

1

0

ជˆ 兲 Y 21共 Rជˆ 兲 Y 10共 Rជˆ 兲 d 共 cos ␪ 兲 Y 1⫺1 共 R

1

0



0

ជˆ 0 兲 ⫽ T共 R

共51兲







d ␭ 共 ␶ 兲 e ⫺(4D r ⫹i ␻ 0 ) ␶

t

0





d ␭ 共 ␶ 兲 e ⫺(4D r ⫺i ␻ 0 ) ␶ .

0

ជˆ 0 ,t 兲 ⫽ dt t ␳ 共 R

冑 冉

15 冑2 ⫹2 冑3 4␲ 2



ជˆ 0 兲 ⫹Y 2⫺1 共 R

␭ 20 ˆ ជ ⌬T 共 R 0 兲 ⫽T 0 40



4D r ⫹i 共 ␻ ⫺ ␻ 0 兲 ជˆ 0 兲 ⫻ Y 21共 R 16D r2 ⫹ 共 ␻ ⫺ ␻ 0 兲 2 4D r ⫹i 共 ␻ ⫹ ␻ 0 兲 16D r2 ⫹ 共 ␻ ⫹ ␻ 0 兲 2

⫻3 ␻ 0



⫹ 共52兲

.

共53兲







0

ជˆ 0 ,t 兲 , dt S 共 R

共54兲

3 1⫺␭ 2 , 4 ␲ 2D r

15 2 ⫹2 冑3 4␲ 2



5 共␻⫺␻0兲 24␲ 16D r2 ⫹ 共 ␻ ⫺ ␻ 0 兲 2

共␻⫹␻0兲

16D r2 ⫹ 共 ␻ ⫹ ␻ 0 兲 2

冎册

共55兲

,

ជˆ s . In Fig. 2 we have plotted the where we have taken Rជˆ 0 ⫽R quantity ⌬T/T 0 . The figure shows that this quantity exhibits a minimum, as required for the appearance of SR. The knowledge of the survival probability allows us to ជˆ 0 ,t). From Eqs. 共45兲 and 共53兲 the ETD is obtain the ETD ␳ (R given by

Equations 共48兲 and 共52兲 together with Eqs. 共44兲 and 共46兲 ˆ allows us to obtain the survival probability S(Rជ 0 ,t), which is given by

␳ 共 Rជˆ 0 ,t 兲 ⫽e ⫺2D r t 2D r ␲



冑 冑 冑 冉冑 冑 再

ជˆ 0 兲 ⫽ ␲ T 0共 R

we obtain the first order correction to the coefficient a 10(t): ␭0 (1) a 10 共 t 兲 ⫽⫺ ␻ 0 40

0

15 2 ⫹2 冑3 4␲ 2

where we have used Eq. 共45兲. Consequently, the MFPT is given by ជˆ 0 兲 ⬅T 0 共 Rជˆ 0 兲 ⫹⌬T 共 Rជˆ 0 兲 , T共 R

ជˆ 兲 Y 20共 Rជˆ 兲 Y 10共 Rជˆ 兲 d 共 cos ␪ 兲 Y 10共 R 5 . 4

3 20

t

冑 冉冑

This quantity is directly related to the MFPT, since

冕 ␾冕 冑␲ ␲ d

冕␶ 冕␶

ជˆ 0 兲 e ⫺2D r t ⫹Y 2⫺1 共 R

5 , 3

2␲



3 ␻0 Y 共 Rជˆ 兲 e ⫺2D r t ⫺ 4 ␲ 10 0 40

ជˆ 0 兲 e ⫺2D r t ⫻ Y 21共 R

冕 冕 冑␲ ␲ 2␲





3 2D r ជˆ 0 兲 ⫺␭ 0 Y 10共 R 4␲ 40

冑 冉冑

15 2 ⫹2 冑3 4␲ 2





冑 冉冑

2D r ⫹i 共 ␻ ⫺ ␻ 0 兲 2D r ⫹i 共 ␻ ⫹ ␻ 0 兲 1 ជˆ 0 兲 ជˆ 0 兲 ⫻ Y 21共 R Y ⫹␭ 0 共R 2 2 2 2⫺1 2 40 16D r ⫹ 共 ␻ ⫺ ␻ 0 兲 16D r ⫹ 共 ␻ ⫹ ␻ 0 兲

ជˆ 0 兲 e ⫺[4D r ⫹i( ␻ ⫹ ␻ 0 )]t 其 ⫻兵 Y 21共 Rជˆ 0 兲 e ⫺[4D r ⫹i( ␻ ⫺ ␻ 0 )]t Y 2⫺1 共 R



15 2 ⫹2 冑3 4␲ 2

冊 共56兲

ˆ ជˆ s , we finally obtain By taking Rជ 0 ⫽R e 2D r t 2D r 冑1⫺␭ 20

⌬␳⫽

␻ 0 ␭ 20 40␲ ⫺



冑75



冑2 2

⫹2 冑3

共␻⫺␻0兲

16D r2 ⫹ 共 ␻ ⫺ ␻ 0 兲 2

冊再



e ⫺4D r t 2

兵 sin关共 ␻ ⫺ ␻ 0 兲 t 兴 ⫹sin关共 ␻ ⫹ ␻ 0 兲 t 兴 其

共␻⫹␻0兲

16D r2 ⫹ 共 ␻ ⫹ ␻ 0 兲 2

041112-7

册冎

,

共57兲

´ S ALARCO ´ N AND AGUSTI´N PE´REZ-MADRID TOMA

PHYSICAL REVIEW E 63 041112

FIG. 2. Mean first passage time as a function of the parameter ␣ ⫽ ␻ / ␻ 0 . The presence of a minimum reveals the existence of stochastic resonance. Pe⫺1 ⫽D r / ␻ 0 ⫽0.08. We have represented nondimensional quantities.

where ⌬ ␳ ⬅ ␳ ⫺ ␳ 0 , ␳ 0 being the corresponding ETD when the amplitude of the oscillating field is set to zero. The succession of maxima in the ETD 共see Fig. 3兲 indicates that the dynamics of a magnetic dipole suspended in a shear flow under a periodic field exhibits SR. VI. DISCUSSION

We have shown that the relaxation process of a dipole immersed in a shear flow exhibits SR upon application of a weak periodic field. To this end we have computed three quantities typically used to characterize SR, namely, the signal-to-noise ratio, the escape time distribution, and the mean first passage time. All of them behave as expected for a process in which SR occurs. Previous work devoted to analyzing whether or not SR is present in the relaxation process of an overdamped dipole in a fluid at rest has shown that this phenomenon does not occur in the linear regime 关13兴. Effectively, linear response theory predicts a maximum in the signal, i.e., in the susceptibility, as a function of the noise level. However, the SNR decreases monotonically with the noise level. This behavior can be easily understood. In the limit of zero noise the output of the system has a small component 共proportional to the applied field兲 at the frequency of the signal whereas the background noise vanishes at zero noise level, this behavior being responsible for the monotonic dependence of the SNR on the noise intensity. In our case, the situation is completely different. When the fluid in which the dipole is suspended is submitted to a pure rotation 共vortex flow兲, both output signal and output background noise exhibit a peak at the same value of D r 共see Fig. 4兲. Consequently, although the background noise vanishes when D r goes to zero, the characteristic SR profile of the SNR cannot be completely hidden, as shown in Fig. 1. This feature arises as a consequence of the presence of shear acting on the suspension; thus, the appearance of SR in the system studied in this paper is a nonequilibrium feature. In a sense, the mechanism yielding SR in this system is similar to the one operating in SR in threshold devices

FIG. 3. Escape time distribution for a dipole in a shear flow under an oscillating magnetic field. The succession of maxima is a signature of the presence of stochastic resonance in this system. Pe⫺1 ⫽D r / ␻ 0 ⫽0.08 and ␣ ⫽ ␻ / ␻ 0 ⫽0.7. We have represented nondimensional quantities.

关9,10兴. Due to the presence of noise, the dipole can eventually acquire enough energy to get out from its stable orientation by crossing the absorbing barrier 共the threshold兲 cos ␪⫽0. After this, the system is driven to its stable position. This process produces a short spike in the magnetization. Of course, the time that the system takes to return to the fixed point has to be smaller than the semiperiod of the oscillating magnetic field. Thus, SR in this system can be understood in the same way as, for example, the SR in level crossing detectors 关9兴. LRT has been one of the most widely used tools in the study of stochastic resonance 关22兴. When the system is in thermal equilibrium in the absence of the external periodic force, a very adequate way of describing stochastic resonance is in terms of the susceptibility. This is because the noisy part of the power spectrum is given directly by the susceptibility through the fluctuation-dissipation theorem, Im ␹ 共 ⍀ 兲 ⫽

⍀ N共 ⍀ 兲. 2D r

共58兲

FIG. 4. Output signal and output background signal as functions of the inverse of the Pe´clet number. Solid line represents the quantity S(Pe⫺1 ) ␻ 0 /␭ 20 whereas dashed line represents N(Pe⫺1 ). We have taken ␣ ⫽0.1. We have represented nondimensional quantities.

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STOCHASTIC RESONANCE IN A SUSPENSION OF . . .

PHYSICAL REVIEW E 63 041112

ជˆ s 兩 ⫽1, we From this equation, and taking into account that 兩 R obtain that the stationary orientation is ˆ ជˆ ជˆ ⫹␭ ⫺1 Rជ s ⫽⫾ 冑1⫺␭ ⫺2 0 x 0 y.

共A2兲

This solution exists only when ␭ 0 ⭓1 and corresponds to a fixed orientation of the dipoles, given that the intensity of the magnetic field is high enough to maintain this fixed direction. The second possibility leads to

ជˆ s ⫽⫺␭ 0 共 Rជˆ s ⫻xជˆ 兲 ⫻Rជˆ s ⫽⫺␭ 0 关 xជˆ ⫺ 共 xជˆ •Rជˆ s 兲 Rជˆ s 兴 . 共A3兲 zជˆ ⫻R

FIG. 5. Comparison between the imaginary part of the susceptibility of the signal Rˆ x 共solid line兲 and the noisy part of the spectrum computed from the Fokker-Planck equation 共dashed line兲. Pe⫺1 ⫽D r / ␻ 0 ⫽0.08. We have represented nondimensional quantities.

This result is correct when the fluctuations whose spectral density is given by N(⍀) have the thermal equilibrium state as reference state 关16兴. However, in the present case we are dealing with a system that is maintained in an out-of-equilibrium steady state due to the presence of a shear flow. It is evident from Fig. 5, where we have plotted the imaginary part of the susceptibility corresponding to Rˆ x and the noisy part of the power spectrum, that these two quantities are clearly different. Note that, if ␻ 0 ⫽0, i.e., the system in the absence of the periodic field is in equilibrium, the relation 共58兲 is fulfilled. Thus we have shown that, although we can define a susceptibility that describes the response of our system to a small perturbation, we cannot describe SR by means of LRT. The reason can be found in the fact that due to the nonequilibrium nature of the attractor of the dynamics the fluctuation-dissipation theorem fails to be valid.

Equation 共A3兲 provides two equations for three unknowns. If one sets Rˆ z ⫽0 one recovers Eq. 共A2兲. If, by contrast one makes Rˆ x ⫽0 then a different stationary orientation is obtained,

ជˆ s ⫽␭ 0 yជˆ ⫾ 冑1⫺␭ 20 zជˆ , R

which exists only when ␭ 0 ⭐1. This orientation gives rise to a rotation of the dipoles with angular velocity

ជ s ⫽ ␻ 0 冑1⫺␭ 20 兵 冑1⫺␭ 20 zជˆ ⫾␭ 0 yជˆ 其 , ⍀

ជˆ ⫻xជˆ 兲 ⫻Rជˆ ⫽xជˆ ⫺Rជˆ 共 Rជˆ •xជˆ 兲 , 共R

1 dRˆ x ⫽␭ 0 共 1⫺Rˆ 2x 兲 ⫺Rˆ y , ␻ 0 dt 1 dRˆ y ⫽⫺␭ 0 Rˆ x Rˆ y ⫹Rˆ x , ␻ 0 dt

共A7兲

1 dRˆ z ⫽⫺␭ 0 Rˆ x Rˆ z . ␻ 0 dt After expressing the components of Rជˆ in spherical coordinates, we obtain the following bidimensional dynamical system: 1 d␪ ⫽␭ 0 cos ␪ cos ␾ , ␻ 0 dt

From Eqs. 共3兲 and 共4兲 one can see that the time derivative ជˆ vanishes either when ⍀ ជ p ⫽0ជ or when ⍀ ជ p ⫻Rជˆ ⫽0ជ . In the of R first case we have

sin ␾ 1 d␾ ⫽⫺␭ 0 ⫹1, ␻ 0 dt sin ␪

共A1兲

共A6兲

we obtain

APPENDIX A: LINEAR STABILITY ANALYSIS OF THE FIXED POINTS OF EQ. „4…

ជ p ⫽ ␻ 0 兵 zជˆ ⫹␭ 0 共 Rជˆ s ⫻xជˆ 兲 其 ⫽0⇒zជˆ ⫽⫺␭ 0 Rជˆ s ⫻xជˆ . ⍀

共A5兲

since, in this case, the field is not strong enough to inhibit the rotation caused by the shear flow. The linear stability of these fixed points is better analyzed in spherical coordinates. Taking into account that

ACKNOWLEDGMENTS

The authors thank Miguel Rubı´ for valuable discussions. This work was supported by DGICYT of the Spanish Government under Grant No. PB98-1258. One of us 共T.A.兲 wishes to thank DGICYT of the Spanish Government for financial support.

共A4兲

共A8兲

where ␪ and ␾ are the polar and azimuthal angles, respectively. By linearization of Eqs. 共A8兲 around the ␭ 0 ⭓1 fixed points we obtain the matrix

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´ S ALARCO ´ N AND AGUSTI´N PE´REZ-MADRID TOMA

A共 ␭ 0 ⭓1 兲 ⫽



⫿␭ 0

冑 冉 冊 1 ␭0

1⫺

2

冑 冉 冊冊

PHYSICAL REVIEW E 63 041112

and therefore Eq. 共B2兲 reads

0

⫿␭ 0

0

1 1⫺ ␭0

2

,

A共 ␭ 0 ⬍1 兲 ⫽



0 ⫺␭ 20





1⫺␭ 20

1⫺␭ 20 0



.

2␲ 3

⫽⫺



共A11兲

duជˆ 关 e ⫺(2D r ⫺i ␻ 0 ) ␶ Y 11共 uជˆ 兲

2 ␲ ⫺(2D ⫺i ␻ ) ␶ ⫺(2D r ⫹i ␻ 0 ) ␶ ˆ r 0 Y 共v Y 1⫺1 共 vជˆ 兲兴 , 关e 11 ជ 兲 ⫹e 3 共B5兲

leading to Eq. 共33a兲. In Eq. 共B5兲 we have used the relation

共A10兲

The eigenvalues of this matrix are given by

␣ ⫽⫾i␭ 0 2 冑1⫺␭ 20 .

冑 冕

⫹e ⫺(2D r ⫹i ␻ 0 ) ␶ Y 1⫺1 共 uជˆ 兲兴 ␦ 共 uជˆ ⫺ vជˆ 兲

共A9兲

which implies that, if ␭ 0 is positive, the orientation corresponding to chosing the sign ⫹ in Eq. 共A2兲 is stable, while the other one is unstable. The same linearization procedure carried out around the ␭⬍1 fixed points leads to ␭ 20

I 1 ⫽⫺

uˆ x ⫽⫺



2␲ 关 Y 11共 uជˆ 兲 ⫹Y 1⫺1 共 uជˆ 兲兴 . 3

To compute the integral I 2 , we have to use the following representation of the ␦ function: ⬁

冕 I 1⫽

I 2⫽ ⫽

冕 冕 冕

duជˆ uˆ x ⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⫽I 1 ⫹I 2 ,

duជˆ uˆ x ⌿ 0 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲 ⫽



兺 Y lm* 共 vជˆ 兲 Y lm共 uជˆ 兲 . l⫽0 m⫽⫺l

duជˆ uˆ x



t⫹ ␶

t

兺 兺 Y lm* 共 vជˆ 兲 冕t l⫽0 m⫽⫺l ⬁

I 2⫽

l

兺 兺 Y lm* 共 vជˆ 兲 冕t l⫽0 m⫽⫺l ⬁



duជˆ uˆ x e ␶ L0 ␦ 共 uជˆ ⫺ vជˆ 兲 , 共B1b兲



l



† duជˆ 共 e ␶ L 0 uˆ x 兲 ␦ 共 uជˆ ⫺ vជˆ 兲 ,





† duជˆ 共 e (t⫹ ␶ ⫺s)L 0 uˆ x 兲

t⫹ ␶

ds ␭ 共 s 兲 e ⫺[l(l⫹1)D r ⫹im ␻ 0 ]s 共B8兲

duជˆ 共 L †0 A兲 B

兺 m⫽⫺l 兺 Y lm* 共 vជˆ 兲 冕t l

t⫹ ␶

ds ␭ 共 s 兲

duជˆ 兵 关 e ⫺(2D r ⫺i ␻ 0 )(t⫹ ␶ ⫺s) Y 11共 uជˆ 兲

⫻e ⫺[l(l⫹1)D r ⫹im ␻ 0 ]s L1 Y lm 共 uជˆ 兲 其 . 共B3兲

共B9兲

Once these expressions have been obtained we can compute the correlation function given by Eq. 共34兲,

with A and B two arbitrary observables. Explicitly, L †0 is given by

ជˆ 0 •R ជ ⫹D r R 2 , L †0 ⫽ ␻ 0 R ជˆ 兲 ⫽ 关 ⫺l 共 l⫹1 兲 D r ⫹i ␻ 0 m 兴 Y lm 共 Rជˆ 兲 , L †0 Y lm 共 R

冑 冕



2␲ 3 l⫽0

⫹e ⫺(2D r ⫹i ␻ 0 )(t⫹ ␶ ⫺s) Y 1⫺1 共 uជˆ 兲兴

共B2兲

where L †0 is the adjoint operator of L0 defined by duជˆ A共 L0 B兲 ⫽



† duជˆ 共 e (t⫹ ␶ ⫺s)L 0 uˆ x 兲 L1 Y lm 共 uជˆ 兲 ,

I 2 ⫽⫺

To begin with we focus on the integral I 1 , which can be rewritten as



ds ␭ 共 s 兲

and, by using Eq. 共B6兲, Eq. 共B8兲 yields Eq. 共33兲, i.e.,

ds ␭ 共 s 兲 e (t⫹ ␶ ⫺s)L0 L1 e sL0 ␦ 共 uជˆ ⫺ vជˆ 兲 .



t⫹ ␶

⫻L 1 e sL0 Y lm 共 uជˆ 兲

共B1c兲

I 1⫽

共B7兲

After introducing this expression into Eq. 共B1c兲 we obtain

共B1a兲

duជˆ uˆ x ⌬⌿ 共 uជˆ ,t⫹ ␶ 兩 vជˆ ,t 兲

l

␦ 共 uជˆ ⫺ vជˆ 兲 ⫽ 兺

APPENDIX B: COMPUTATION OF EQS. „33… AND „35…

In this Appendix we work out in detail some steps of the computation of the power spectrum corresponding to the relaxation process of a dipole under an oscillating magnetic field in a shear flow; in particular, we calculate the integrals that yield Eqs. 共33兲 and 共35兲. From Eqs. 共31兲 and 共32兲,

共B6兲

共B4兲 041112-10

具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 兩 Rជ 0 共 t 0 兲 典 ⬃ ˆ



ជˆ 0 ,t 0 兲 I 1 d vជˆ vˆ x ⌿ 0 共 vជˆ ,t 兩 R





ជˆ 0 ,t 0 兲 I 2 , d vជˆ vˆ x ⌬⌿ 共 vជˆ ,t 兩 R 共B10兲

STOCHASTIC RESONANCE IN A SUSPENSION OF . . .

PHYSICAL REVIEW E 63 041112

the average of the correlation function over initial conditions is given by

ជˆ 0 ,t 0 ) and ⌬⌿( vជˆ ,t 兩 Rជˆ 0 ,t 0 ) being given by ⌿ 0 ( vជˆ ,t 兩 R ជˆ 0 兲 ⫽e tL0 ␦ 共 vជˆ ⫺Rជˆ 0 兲 ⌿ 0 共 vជˆ ,t 兩 R ⬁



具 Rˆ x 共 t 兲 Rˆ x 共 t⫹ ␶ 兲 典 ⬃I 3 ⫹I 4 ,

共B13a兲



共B13b兲

l

兺 兺

* 共 Rជˆ 0 兲 e ⫺[l(l⫹1)D r ⫹im ␻ 0 ]t Y lm 共 vជˆ 兲 , Y lm

l⫽0 m⫽⫺l



ជˆ 0 兲 ⫽ ⌬⌿ 共 vជˆ ,t 兩 R

t

0

ជˆ 0 兲 dr ␭ 共 r 兲 e (t⫺r)L0 L 1 e rL0 ␦ 共 vជˆ ⫺R





I 3⬅

l

兺 兺

l⫽0 m⫽⫺l

* 共 Rជˆ 0 兲 Y lm



t

0

I 4⬅

dr ␭ 共 r 兲

I 3⫽

共B11兲

1 I 4 ⫽⫺ 3

* 共 Rជˆ 0 兲 ⫽ ␦ l,0␦ m,0 , dRជˆ 0 Y lm



l



t

t⫹ ␶

* ( vជˆ ) ⫹e ⫺(2D r ⫹i ␻ 0 )(t⫺r) Y 1⫺1 ( vជˆ )]Y lm



t

0

dr ␭ 共 r 兲 e (t⫺r)L0 L1 Y 00共 vជˆ 兲 I 2 . 共B13c兲



ds ␭ 共 s 兲

1 4␲



d vជˆ

2␲ 关 Y 11共 vជˆ 兲 ⫹Y 1⫺1 共 vជˆ 兲兴 3

4 ␲ ⫺2D ␶ r cos共 ␻ ␶ 兲 , e 0 9

共B14兲

where the orthogonality relation for the spherical harmonics,



4 ␲ 共 l⫹m 兲 ! d vជˆ Y * ជˆ 兲 Y lm 共 vជˆ 兲 ⫽ ␦ ␦ , pq 共 v 2l⫹1 共 l⫺m 兲 ! l,p m,q

共B15兲

has been used. On the other hand, from Eqs. 共B9兲 and 共B13c兲,

共B12兲

兺 m⫽⫺l 兺 冕0 dr ␭ 共 r 兲 冕t ⬁

2␲ 3 l⫽0



⫻ 关 e ⫺(2D r ⫺i ␻ 0 ) ␶ Y 11共 vជˆ 兲 ⫹e ⫺(2D r ⫹i ␻ 0 ) ␶ Y 1⫺1 共 vជˆ 兲兴

where the initial time t 0 has been fixed to zero and Eq. 共B7兲 has been used. Equations 共B11兲 provide the evolution of the probability distribution under the condition of the system being initially ˆ in the state Rជ 0 . Since a priori nothing is known about this ជˆ 0 is a random variable initial condition, we assume that R uniformly distributed over the orientation space; consequently we average the correlation function over the distribution of initial states 关Eq. 共36兲兴. Taking into account that



4

d vជˆ vˆ x

d vជˆ vˆ x I 1 ,

After introducing Eqs. 共B5兲 and 共B6兲 into 共B13b兲, we obtain

⫻e ⫺[l(l⫹1)D r ⫹im ␻ 0 ]r e (t⫺r)L0 L1 Y lm 共 vជˆ 兲 ,

1 4␲

冕 冑 ␲ 1

1 4␲

冕 ជ再

d vˆ [Y 11( vជˆ )⫹Y 1⫺1 ( vជˆ )][e ⫺(2D r ⫺i ␻ 0 )(t⫺r) Y 11( vជˆ )

duជˆ 兵 [e ⫺(2D r ⫺i ␻ 0 )(t⫹ ␶ ⫺s) Y 11(uជˆ )⫹e ⫺(2D r ⫹i ␻ 0 )(t⫹ ␶ ⫺s) Y 1⫺1 (uជˆ )]

⫻e ⫺[l(l⫹1)D r ⫹im ␻ 0 ]s L1 Y lm (uជˆ ) .

共B16兲

Let us focus our attention on the integral over vជˆ :



* 共 vជˆ 兲 d vជˆ 关 Y 11共 vជˆ 兲 ⫹Y 1⫺1 共 vជˆ 兲兴关 e ⫺(2D r ⫺i ␻ 0 )(t⫺r) Y 11共 vជˆ 兲 ⫹e ⫺(2D r ⫹i ␻ 0 )(t⫺r) Y 1⫺1 共 vជˆ 兲兴 Y lm ⫽

冕 ⫹

* 共 vជˆ 兲 ⫹ d vជˆ Y 11共 vជˆ 兲 e ⫺(2D r ⫺i ␻ 0 )(t⫺r) Y 11Y lm





* 共 vជˆ 兲 d vជˆ Y 1⫺1 共 vជˆ 兲 e ⫺(2D r ⫹i ␻ 0 )(t⫺r) Y 1⫺1 共 vជˆ 兲 Y lm

* 共 vជˆ 兲 ⫹ d vជˆ Y 1⫺1 共 vជˆ 兲 e ⫺(2D r ⫺i ␻ 0 )(t⫺r) Y 11共 vជˆ 兲 Y lm 041112-11



* 共 vជˆ 兲 . d vជˆ Y 11共 vជˆ 兲 e ⫺(2D r ⫹i ␻ 0 )(t⫺r) Y 1⫺1 共 vជˆ 兲 Y lm

共B17兲

´ S ALARCO ´ N AND AGUSTI´N PE´REZ-MADRID TOMA

PHYSICAL REVIEW E 63 041112

From the rules of addition of angular momenta familiar from quantum mechanics, which imply that the product of two spherical harmonics Y pq Y rs has a nonvanishing projection over a third spherical harmonic Y lm only when the relations 兩 p⫺r 兩 ⭐l⭐r⫹ p and q⫹s⫽m are fulfilled, it is easy to see that these integrals will give a nonzero result only when l⫽0,1,2 关14兴. In addition, for the integrals containing ( vជˆ )Y ( vជˆ ) the parameter m has to be the products Y 1⫾1

tributions finally yield, by the orthogonality property of the spherical harmonics, a vanishing result. Thus, only l⫽0 and m⫽0 contributes to I 3 . Taking this into account and using Eq. 共B15兲,

关5兴 关6兴 关7兴 关8兴 关9兴 关10兴 关11兴 关12兴

冉 冊冕 2␲ 3





3

t

0

t⫹ ␶

t

dr ␭ 共 r 兲 兵 e ⫺(2D r ⫺i ␻ 0 )(t⫺r) ⫹e ⫺(2D r ⫹i ␻ 0 )(t⫺r) 其

ds ␭ 共 s 兲 兵 e ⫺(2D r ⫺i ␻ 0 )(t⫹ ␶ ⫺s)

⫹e ⫺(2D r ⫹i ␻ 0 )(t⫹ ␶ ⫺s) 其 .

共B18兲

1⫾1

m⫽⫾2 whereas it must be m⫽0 for the integrals with Y 1⫾1 ( vជˆ )Y 1⫿1 ( vជˆ ) to yield a nonzero contribution. However, although these integrals give a nonvanishing contribution in principle, note that when we perform the integral over the variable uជˆ in Eq. 共B16兲 the terms introduced by these con-

关1兴 关2兴 关3兴 关4兴

I 4⫽

P. Jung, Phys. Rep. 234, 175 共1993兲. M.O. Magnasco, Phys. Rev. Lett. 71, 1477 共1993兲. R. Dean Astumian, Science 276, 917 共1997兲. B. McNamara, K. Wiesenfeld, and R. Roy, Phys. Rev. Lett. 60, 2626 共1988兲. A. Longtin, J. Stat. Phys. 70, 309 共1993兲. J.M.G. Vilar, R.V. Sole´, and J.M. Rubı´, Phys. Rev. E 59, 5920 共1999兲. L. Gammaitoni, P. Ha¨nggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 共1998兲. K. Wiesenfeld, D. Pierson, E. Pantazelou, C. Dames, and F. Moss, Phys. Rev. Lett. 72, 2125 共1994兲. Z. Gingl, L.B. Kiss, and F. Moss, Europhys. Lett. 29, 191 共1995兲. F. Chapeau-Blondeau and X. Godivier, Phys. Rev. E 55, 1478 共1997兲. J.M.G. Vilar, G. Gomila, and J.M. Rubı´, Phys. Rev. Lett. 81, 14 共1998兲. M.I. Dykman, D.G. Luchinsky, R. Manella, P.V.E. McClintock, N.D. Stein, and N.G. Stocks, J. Stat. Phys. 70, 479

Finally, performing the changes of variables t ⬘ ⫽t⫺r and t ⬙ ⫽t⫹ ␶ ⫺s and using Eq. 共25兲 we obtain

I 4⫽

冉 冊 2␲ 3

3

␭ 2 共 t 兲 e ⫺i ␻ ␶

冉冕



0

dt e ⫺i ␻ t ␹ x 共 t 兲



2

.

共B19兲

In this integral the upper limit goes to infinity by causality.

共1993兲. 关13兴 J.M.G. Vilar, A. Pe´rez-Madrid, and J.M. Rubı´, Phys. Rev. E 54, 6929 共1996兲. 关14兴 J.J. Sakurai, Modern Quantum Mechanics 共Addison-Wesley, Reading, MA, 1985兲. 关15兴 R. Courant and D. Hilbert, Methods of Mathematical Physics 共Interscience Publishers, New York, 1966兲, Vol. I. 关16兴 P.M.V. Re´sibois and M. de Leener, Classical Kinetic Theory of Fluids 共Wiley, New York, 1977兲. 关17兴 N. Van Kampen, Stochastic Processes in Physics and Chemistry 共North-Holland, Amsterdam, 1992兲. 关18兴 H. Risken, The Fokker-Planck Equation 共Springer-Verlag, Berlin, 1984兲. 关19兴 K. Wiesenfeld and F. Jaramillo, Chaos 8, 539 共1998兲. 关20兴 G.H. Weiss, Adv. Chem. Phys. 13, 1 共1967兲. 关21兴 J.E. Fletcher, S. Havlin, and G.H. Weiss, J. Stat. Phys. 51, 215 共1988兲. 关22兴 M.I. Dykman, D.G. Luchinsky, R. Manella, P.V.E. McClintock, N.D. Stein, and N.G. Stocks, Nuovo Cimento D 17, 661 共1995兲.

041112-12

Stochastic resonance in a suspension of magnetic dipoles under ...

Mar 27, 2001 - Tomás Alarcón and Agustın Pérez-Madrid. Departament de Fısica Fonamental and CER on Physics of Complex Systems, Facultat de Fısica, ...

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