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Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise

The diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied. Using Laplace analysis we derive exact expressions for the relaxation functions of the particle in terms of generalized Mittag-Leffler functions and its derivatives from a generalized Langevin equation. Our results show that the oscillator displays an anomalous diffusive behavior. In the strictly asymptotic limit, the dynamics of the harmonic oscillator corresponds to an oscillator driven by a noise with a pure power-law autocorrelation function. However, at short and intermediate times the dynamics has qualitative difference due to the presence of the characteristic time of the noise.


INTRODUCTION
The study of anomalous diffusion in complex or disordered media has achieved a substantial progress during the last years [1,2,3,4,5,6,7]. Anomalous diffusion in physical and biological systems can be formulated in the framework of the generalized Langevin equation (GLE) [1,7,8,9,10,11,12,13,14,15]. If one considers the dynamics of a harmonic oscillator with frequency ω under the influence of a random force modeled as Gaussian colored noise, the corresponding GLE is written as [9,10,16]: where X(t) represents the position of a particle of mass m = 1 at time t, and γ(t) is the frictional memory kernel. The random force ξ(t) is zero-centered and stationary Gaussian that obeys the fluctuation-dissipation theorem [17] where k B is the Boltzmann constant, and T is the absolute temperature of the environment. It is now well established that the physical origin of anomalous diffusion is related to the long-time tail correlations [1,2]. Therefore, in order to model anomalous diffusion process, pure power-law correlation functions are usually employed [1,8,10,16,18,19], which may be written as where Γ(z) is the Gamma function, and C λ is a proportionality coefficient dependent on the exponent λ but independent of time. The exponent λ can be taken as 0 < λ < 1 or 1 < λ < 2, which is determined by the dynamical mechanism of the physical process considered. Viñales and Despósito have introduced a noise whose correlation is proportional to a Mittag-Leffler function [20]. This correlation behaves as a power-law for large times, but is non-singular at the origin due to the inclusion of a characteristic time.
The aim of this work is to investigate the effects of the Mittag-Leffler noise on the behavior of a harmonically bounded particle governed by the GLE (1). This paper is organized as follows. In Section 2 we discuss some characteristics of the Mittag-Leffler noise. In Section 3, we show the formal expressions for the relaxation functions that govern the dynamics of the particle in the case of an arbitrary noise correlation function. Analytical solutions of the GLE for a harmonically bounded particle driven by a Mittag-Leffler noise are obtained in Section 4. The Section 5 is devoted to the analysis of temporal behavior of the relaxation functions, and is compared with the results in the case of a pure power-law noise correlation function. Finally, the conclusions are presented in Section 6.

MITTAG-LEFFLER NOISE
It is well known that if the correlation function (2) is a Dirac delta function the stochastic process is Markovian and its dynamics can be directly obtained [21]. However, in a complex or viscoelastic environment one must take into account the memory effects through a long-time tail noise to describe the effect of the environment on the particle. The non-Markovian dynamics is involved in these physical processes.
Recently, Viñales and Despósito introduced a Mittag-Leffler noise given by [20] where τ acts as a characteristic memory time and 0 < λ < 2. The E α (y) function denotes the Mittag-Leffler function [22] defined through the following series Using the asymptotic behaviors of the Mittag-Leffler function [23] one can easily deduce that, for λ = 1, the correlation function (4) behaves as a stretched exponential for short times and as an inverse power law in the long time regime [23,24].
Setting λ = 1, the correlation function (4) reduces to an exponential form which describes a standard Ornstein-Uhlenbeck process [21]. Moreover, in the limit τ → 0 and from the limit representation of the Dirac delta [25] we get that C(t) = 2 C 1 δ(t) which corresponds to a white noise, nonretarded friction and standard Brownian motion [21]. On the other hand, for λ = 1 the limit τ → 0 of the proposed correlation function (4) reproduce the powerlaw correlation function (3). This behavior is obtained introducing in expression (4) the asymptotic behavior at large y of the Mittag-Leffler function [23] It is worth pointing out that the Mittag-Leffler correlation function (4) is a well defined and no-singular function. From (4), its value at t = 0 is given by C(0) = C λ /τ λ , while for the power-law correlation (3) C(0) diverges. Then, the introduction of the characteristic time τ enables to avoid the singularity of the powerlaw at the origin. Considering that the Mittag-Leffler function is the natural generalization of the exponential function [22], we can also consider the Mittag-Leffler correlation function as a generalization of the power-law correlation, and similarly, the colored noise (6) is considered as a generalization of the white noise.

SOLUTIONS OF THE GENERALIZED LANGEVIN EQUATION
In what follows we consider the Langevin equation (1) with the deterministic initial conditions x 0 = X(0) and v 0 =Ẋ(0). By means of the Laplace transformation to Eq. (1) one can easily obtain a formal expression for the displacement X(t) and the velocity V (t) =Ẋ(t). The displacement X(t) satisfies that where is the position mean value. The relaxation function G(t) is the Laplace inversion of where γ(s) is the Laplace transform of the damping kernel and On the other hand, the velocity V (t) satisfies that where is the velocity mean value and the relaxation function Explicit expressions of the variances can be obtained from Eqs. (8) and (12). Taking into account the symmetry property of the correlation function and Eq.(2), yields [9,10,14,16] where β = 1/k B T . From an experimental point of view, information about the observed diffusive behavior is extracted from the mean square displacement ρ(t). In the long time measurement time, it is related to the relaxation function I(t) as [26] ρ(τ L ) = lim where τ L is the so-called time lag. Alternative information about the dynamics can be extracted from the normalized velocity autocorrelation function C V (t), which is related to the relaxation function g(t) as [16,26] Then, the knowledge of the relaxation functions I(t), G(t) and g(t) allows us to describe the diffusive behavior of the oscillator. In the next section we will give explicit expressions for the relaxation functions in the case of a Mittag-Leffler noise (4) assuming that λ = 1.

ANALYTICAL RELAXATION FUNCTIONS FOR A MITTAG-LEFFLER NOISE
From relation (2), the memory kernel γ(t) corresponding to the Mittag-Leffler noise (4) can be written as where γ λ = C λ /k B T . Taking into account that the Laplace transform of the memory kernel reads [23] γ the relaxation function I(t) can be written as the Laplace inversion of where andγ λ is defined as Following the approach given in Ref. [27] we get where E α,β (y) is the generalized Mittag-Leffler function [23] defined by the series expansion and E (k) α,β (y) is the derivative of the Mittag-Leffler function .
Notice that in the limit ω → 0 only survive the terms with n = 0 in equations (26) and (27). Then, Eq.(25) reduces toγ λ = γ λ , and the expression of the relaxation function I(t) for the free particle case [20] is recovered.
On the other hand, in the limit τ → 0 the function I 1 (t) vanishes and the behavior of I 0 (t) can be achieved introducing the asymptotic behaviors of the generalized Mittag-Leffler function [27] and its derivative in Eq. (26). Then, after some algebra we obtain where we have used thatγ λ → γ λ for τ → 0, according to (25). The expression in series given in Eq.(40) coincides with the expression for the relaxation integral function I(t) corresponding to a pure power-law correlation function, previously obtained in Ref. [16]. Likewise, one can verify that in the limit τ → 0 the relaxation functions G(t) and g(t) are also the same to that in the case of a pure power-law correlation function, given by

TEMPORAL BEHAVIOR OF THE RELAXATION FUNCTIONS
The analytical expressions (30), (32) and (35) are the main result of this work. In the following we will analyze the time behavior of the relaxation functions for different regimes.
The short-time behavior (t ≪ τ ) of the relaxation functions can be obtained using the series expansions (28) and (29). Then which are the expected for a harmonic oscillator driven by a noise with a finite correlation at the origin [9,10,28]. Now we get an expression of the function I(t) for times bigger than the characteristic time τ of the noise, i.e. t ≫ τ . For this purpose we introduce the approximation (39) in Eqs. (26) and (27). After some algebra we get and Let us analyze the behaviors of the relaxation functions I(t), G(t) and g(t) forγ λ t 2−λ ≫ 1. Introducing the approximation (39) in (46) and (47), after some calculations and using (5) one gets Then, from (11) and (14) we get and where we introduced the dimensionless factor The relaxation functions (48) to (50) have the same functional form to the results obtained in the pure powerlaw case [16] but with the presence of the scale factor ν. In the limit τ → 0 is ν = 1 and one recovers the expressions corresponding to a pure power-law noise [16].
It is worth pointing out that these expressions are the same to those that can be obtained directly discarding the inertial term s 2 in (10). Then, Eqs. (48) to (50) represents the solutions in the high friction limit.
The strictly asymptotic behavior of the relaxation functions I(t), G(t) and g(t) can be obtained introducing the asymptotic behavior (7) of the Mittag-Leffler function in Eqs. (48) to (50). Then, for ν ω 2 γ λ t λ ≫ 1 the relaxation functions can be written as As expected, the relaxation functions (52) to (54) behave as a power law in the long-time limit. These results are in agreement with those obtained in Refs. [16,29] due to the fact that the Mittag-Lefler noise decays as a powerlaw for very large times. In the same way, substitution of these asymptotic expansions into Eqs. (15) to (17) give the long-time behavior of the variances of the process, which again coincide with those obtained in Refs. [16,29].

CONCLUSIONS
In this work we have presented an analytically resoluble model for the dynamics of a classical harmonic oscillator in a complex environment, which is valid for all time range. We have shown that an anomalous diffusion process can be generated by a Mittag-Leffler noise deriving exact expressions for the relaxation functions of the oscillator in terms of the generalized Mittag-Leffler function and its derivatives. Moreover, in the appropriate limits the results for a harmonic oscillator driven by a power-law noise is recovered. However, differences in relation to the usually employed pure power-law noise appear in the interval of short and intermediate times. For times shorter than the characteristic time of the noise the relaxation functions include a correction due to the presence of the characteristic time τ . In the range of intermediate times, the relaxation functions have a similar functional form to the previously obtained for a pure power-law noise [16], but with the inclusion of a scaling dimensionless parameter. Finally, in the strictly asymp-totic limit we recover the anomalous behavior of an harmonically bounded particle driven by a power-law noise, which is in agreement with the previous results given in Refs. [16,29].