Physica A 283 (2000) 140–145www.elsevier.com/locate/physa

The rigid rotator with L�evy noise

Manuel O. C�aceres∗; 1Centro At�omico Bariloche, Instituto Balseiro CNEA, Universidad Nacional de Cuyo, Av. Ezequiel

Bustillo Km 9.5, 8400 San Carlos de Bariloche, R ��o Negro, Argentina

Abstract

We have used an exact functional approach to solve a plane rotator in presence of L�evy noise.The cosine relaxation has been calculated. Its non-autonomous generalization can also be solvedin the context of our functional analysis. c© 2000 Elsevier Science B.V. All rights reserved.

PACS: 05.40.+j; 02.50.Ey; 47.27.Qb

Keywords: Linear response; Plane stochastic rotator; L�evy noise

1. Stochastic N th-order di�erential equations

Consider the non-autonomous stochastic di�erential equation (SDE), with naturalboundary conditions in its associated probability distribution

dN

dtNX (t) = f(t)�(t); X (0) ≡ X (0)0 ; : : : ;

dN−1

dtN−1X (0) ≡ X (N−1)0 : (1)

Here f(t) is a non-random force, and �(t) is an arbitrary noise (correlated or not)characterized by some characteristic functional [1], for t ∈ [0;∞),

G�([k(·)]) =⟨exp(i∫ ∞

0�(t)k(t) dt

)⟩: (2)

The sp X(t) is completely characterized if we know its functional GX ([k(·)]). Thisnotation emphasizes that GX depends on the whole test function k(t), not just on thevalue it takes at one particular time tj.

∗ Fax: +54-2944-445-299.E-mail address: caceres@cab.cnea.gov.ar (M.O. C�aceres)1 Senior Independent Research associated at CONICET.

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(00)00141 -2

M.O. C�aceres / Physica A 283 (2000) 140–145 141

Following Ref. [2], it is possible to prove that the characteristic functional of thesp X(t) is, for any noise �(t), and for t ∈ [0;∞) given by

GX ([Z(·)]) = exp i(q0X0 + q1X (1)0 + · · ·+ qN−1X (N−1)0 )

×G�([f(t)

∫ ∞

tds∫ ∞

sds1 · · ·

∫ ∞

sN−2

dsN−1Z(sN−1)]

); (3)

where qj; j = 0; 1; 2; : : : ; N − 1 are functionals of Z(t) given by

q0 =∫ ∞

0Z(s) ds; q1 =

∫ ∞

0ds∫ ∞

sds1Z(s1); : : :

qj =∫ ∞

0ds · · ·

∫ ∞

sj−1

dsjZ(sj) : (4)

The functionals of the “auxiliaries” processes X( j)(t) and X( j+1)(t) = (d=dt)X( j)(t);06j¡N − 1, are characterized by the functional relationship

GX (j) ([Z(·)]) = exp i(X ( j)0

∫ ∞

0Z(s) ds

)GX (j+1)

([∫ ∞

tZ(s)ds

]): (5)

Note that the process X(0)(t) ≡ X(t) is a marginal sp. On the other hand, the functionalof the sp X(N−1)(t), ful�lling (d=dt)X(N−1)(t) = f(t)�(t), is

GX (N−1) ([Z(·)]) = exp i(X (N−1)0

∫ ∞

0Z(s) ds

)G�

([f(t)

∫ ∞

tZ(s) ds

]):

(6)

Therefore, from (5) and (6) any m-time joint probability distribution can be obtainedby quadrature. This issue is simple to realize by introducing the Fourier representationof the �-function, in the following expression:

P(x( j)1 ; t1; x( j)2 ; t2; : : : ; x

( j)m ; tm)

=〈�(X( j)(t1)− x( j)1 )�(X( j)(t2)− x( j)2 ) · · · �(X( j)(tm)− x( j)m )〉 :

We can invert the characteristic functional by introducing the m-dimensional Fouriertransform

P(x( j)1 ; t1; x( j)2 ; t2; : : : ; x

( j)m ; tm) =

1(2�)m

∫· · ·∫dk1 · · · dkm

× exp(−i

m∑i=1

kix( j)i

)

× [GX (j) ([Z(t)])]Z(t)=k1�(t−t1)+···+km�(t−tm) :(7)

142 M.O. C�aceres / Physica A 283 (2000) 140–145

Alternatively all m-time moments 〈X (j)(t1)X (j)(t2) · · ·X (j)(tm)〉, and cumulants 〈〈X (j)(t1)X (j)(t2) · · ·X (j)(tm)〉〉 can be obtained by functional di�erentiation of GX ( j) ([Z(·)]):Solution (3) can be used for any stochastic N th-order di�erential problem. Con-

sider for example the instantaneous displacement X (t) of a free particle of mass 1subjected to a purely random force �(t). In this case N = 2; f(t) = 1, and the statis-tics is characterized by the noise �(t). The Gaussian case: if noise’s functional isG�([k(·)]) = exp(−12

∫∞0 ds1

∫∞0 ds2 k(s2)k(s1)�(s1 − s2)) can immediately be worked

out in agreement with previous reports [3]. The analysis of the phase di�usion is an-other problem that can also be studied in a similar way, for example, by introducingthe rigid rotator model. At this point we should emphasize that because we do notmake use of any partial di�erential equation approach, the results that we present areexact solutions for any noise, in particular long-range correlated noises can also besolved [4].

2. The rigid rotator with L�evy noise

A plane Brownian rotor is a useful model to represent a Spherical molecule [1]when there is only one relevant variable, the angle �(t). Then the angular velocity(t) = (d=dt)�(t) is assumed to follow a stochastic motion where the random torqueis represented by an additive noise and the dissipation by a given coe�cient , i.e.,the evolution equation of the stochastic angle is: (d2=dt2)� + (d=dt)� = �(t). Manymagnetic systems instead of having numerous weak collisions can have strong or verystrong collisions (random torque), then the model of a Gaussian rigid rotator breaksdown to describe the transient and also the long-time cosine relaxation of such a “planemolecule”.Using our functional approach, it is simple to see that the cosine relaxation 〈cos(�(t))〉

is, at long times, exponential provided that the random torque is represented by ashort-range Gaussian noise. Hence, it is worthwhile to study the case when the randomtorque is characterized by an external L�evy noise. Because a rigid rotator is equivalentto the translation of a particle on a circular track, this situation corresponds to assumethat the random torque can produce short and very long angle excursions (withoutany characteristic length) on the circular track. Then the evolution equation (Langevindynamics) of a L�evy plane rotator is de�ned, here, by the SDEs

ddt(t) + (t) = �(t);

ddt�(t) = (t); {�;} ∈ (−∞;∞) ; (8)

where �(t) is the L�evy noise [5]. In general these SDEs can be solved, for anynoise �(t), by using our functional approach. The non-autonomous case can also besolved [5].First, note that the functional G([k(·)]) of the angular velocity is given in terms

of G�([k(·)]). Hence, the functional of the angle � is equivalent to a generalized

M.O. C�aceres / Physica A 283 (2000) 140–145 143

Wiener process [2] where, now, the noise is characterized by the functional G([k(·)]).Therefore, the general solution of the stochastic angle �(t) is given by the functional

G�([Z(·)]) = eik0�0G([∫ ∞

tZ(s) ds

]); (9)

where k0=∫∞0 Z(s) ds and �0 is the angle initial condition; so using the explicit expres-

sion G ([M (·)])=e+iq00G�([∫∞t e (t−t

′)M (t′) dt′]) we get, with M (t)=∫∞t Z(t′′) dt′′,

the functional

G�([Z(·)]) = eik0�0+iq00G�([∫ ∞

te (t−t

′)∫ ∞

t′Z(t′′) dt′′ dt′

]); (10)

where q0 =∫∞0 M (s)e− s ds =

∫∞0 e− s

∫∞s Z(s′) ds′ ds and 0 ≡ �̇0 is the angular

velocity initial condition.Second, if the noise �(t) appearing in (8) is a L�evy (white) noise its functional is

given [5] by G�([k(·)]) = exp(−b∫∞0 |k(s)|� ds); 0¡�62; b¿ 0, then from (10) we

obtain the solution

G�([Z(·)]) = eik0�0+iq00 exp(−b∫ ∞

0

∣∣∣∣∫ ∞

se (s−t

′)∫ ∞

t′Z(t′′) dt′′ dt′

∣∣∣∣�

ds):

(11)

The cosine relaxation function is obtained from the cosine functional 2⟨cos∫ ∞

0�(t)Z(t) dt

⟩= Re[G�([Z(·)])] : (12)

For the plane L�evy rotator, from (12), we see that the 1-time characteristic functionis G�(k1; t1) = exp(ik1[�0 + (1= )(1 − e− t1 )�̇0] − b|k1= |��(t1)); hence P(�; t) is amomentless distribution as expected. Nevertheless, the cosine relaxation is �nite and isgiven by

〈cos�(t1)〉= cos[�0 + 1 (1− e− t1 )�̇0]exp(−b= ��(t1)) ; (13)

where the function �(t) has the expression

�(t) =∫ t

0|e (s−t) − 1|� ds= 1

B(�+ 1; 0; (1− e− t)) ; (14)

here B(a; b; z) is the incomplete Beta function [6]. From (14) we see that at long-time,t/ −1, the behavior is �(t) ∼ t. This can be seen by de�ning �(t) ˙ t�e� (t), wherethe e�ective exponent is given by

�e� (t) =d ln

∑(t)

d ln t=

(1− e− t)� tB(�+ 1; 0; (1− e− t)) ; (15)

2 Note that the Gaussian (white) noise case is immediately reobtained from (11) by taking � = 2.

144 M.O. C�aceres / Physica A 283 (2000) 140–145

so in the long-time limit t/ −1; �e� (t) → 1. Then the shape of the transient regimeof the cosine relaxation can be anomalous (depending on L�evy’s exponent � ∈ (0; 2]),but for any �, in the long-time limit the relaxation is exponential.

2.1. The rigid rotor with Gaussian long-range correlated torques

The long-range Gaussian case can also be worked out in a similar way [4]. Considera Gaussian functional for the noise �(t), when 〈〈�(s1)�(s2)〉〉 is a power-law temporalcorrelation; for example characterized by

〈〈�(t1)�(t2)〉〉= �2�−1

(1 + |t1 − t2|=�)� ; � ∈ (0;∞); �¿0 : (16)

The case � = 0 corresponds to a ballistic-like situation [7]. Then, if � ∈ [0; 1) it ispossible to see that the cosine relaxation will not be exponential at long-times. In factfor times t/� the behavior looks like

〈cos�(t)〉 ∼ exp(− 2�2�(�−1) −2

(1− �)(2− �) t2−� +O(t) + O(t2−�exp(− t)

); (17)

hence, the long-time regime of the cosine relaxation depends on the time-scale of thepower-law correlation, �, rather than on the dissipative parameter . The faster relax-ation occurs when �→ 0, i.e., in the ballistic-like limit. If �=1, there are logarithmiccorrections; only if �¿ 1 the cosine relaxation, at long time, will be exponential.

3. Conclusions

Herein we summarize the main results of the paper. The functional of a stochasticN th-order di�erential equation has been found. The plane rotator (8) in presence ofL�evy random torques �(t) (non-hard-collision model) has been solved. We have showedthat the cosine relaxation has a non-exponential transient regime, and we have provedthat its shape depends on L�evy’s parameter � (but in the long-time limit the relaxationis exponential).On the other hand, the long-range correlated Gaussian model has also been solved,

showing in this case that at long-times the cosine relaxation is non-exponential: e−C t2−�

with C=constant and � ∈ [0; 1). As a matter of fact, using our functional technique, itis also possible to prove that at long-times the cosine–cosine correlation behaves like

〈cos[�(t1)− �(t2)]〉˙ cos

(�̇0 (e− t2 − e− t1 )

)exp(−C|t1 − t2|2−�) :

Acknowledgements

M.O.C thanks CONICET (grant PIP No. 4948).

M.O. C�aceres / Physica A 283 (2000) 140–145 145

References

[1] N.G. van Kampen, in: Stochastic Processes in Physics and Chemistry, 2nd Edition, North-Holland,Amsterdam, 1992; and references therein

[2] M.O. C�aceres, A.A Budini, J. Phys. A 30 (1997) 8427.[3] J. Heinrichs, Phys. Rev. E 47 (1993) 3007.[4] M.O. Caceres, Phys. Rev. E 60 (1999) 5208.[5] M.O. C�aceres, J. Phys. A 32 (1999) 6009.[6] J. Spanier, K.B. Oldham, in: An Atlas of Functions, Springer, Berlin, 1987.[7] A. Compte, M.O. C�aceres, Phys. Rev. Lett. 81 (1998) 3140.