Cent. Eur. J. Phys. • 10(2) • 2012 • 382-389DOI: 10.2478/s11534-012-0008-0

Central European Journal of Physics

The fractional oscillator as an open system

Research Article

Vasily E. Tarasov1∗

1 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

Received 07 December 2011; accepted 17 January 2012

Abstract: A dynamical system governed by equations with derivatives of non-integer order, such as the fractionaloscillator, can be considered as an open (non-isolated) system with memory. Fractional equations of motionare obtained from the interaction between the system and the environment with power-law spectral density.

PACS (2008): 45.10.Hj,45.05.+x, 03.65.Yz

Keywords: fractional derivatives • open systems • fractional oscillator© Versita Sp. z o.o.

1. Introduction

Fractional calculus [1] and fractional equations [2, 3] havefound many applications in recent studies in mechanicsand physics (for example, see books [4–14]) The interestin fractional equations has been growing continually dur-ing the last few years. More often fractional equations fordynamics or kinetics appear as some phenomenologicalmodels. Recently, the method to obtain fractional ana-logues of equations of motion was considered for prob-lems related to sets of coupled particles that interact bya long-range power-law [15–19].In this paper, we describe the interaction between simpleclassical systems and some environments. These systemscan be considered as open systems with memory. Under-standing dissipative dynamics of open systems remains achallenge in mathematical physics. This problem is rele-vant in various areas of fundamental and applied physics.As a model of the environment, we consider an infinite∗E-mail: tarasov@theory.sinp.msu.ru

set of harmonic oscillators coupled to the system. Thisis an independent-oscillator model, since the oscillatorsare not interacting with each other. Power-law spectraldensities (PLSD) for the environment lead us to power-laws for memory functions. Note that a power-law memoryhas been detected for fluctuation within a single proteinmolecule [20]. We obtain fractional equations from theinteraction between the system and the PLSD environ-ment. It allows us to use fractional differential equationsfor non-Markovian dynamics of a wide class of open andnon-Hamiltonian systems. The linear fractional oscilla-tor (LFO) is considered as an open system, i.e. a systemthat interacts with the PLSD environment. Note that theLFO has been the subject of numerous investigations [35–51, 61] because of its different applications. The LFO canbe considered as a simple model of fractional generaliza-tion of classical mechanics [52–60].

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Vasily E. Tarasov

2. Interaction between system andenvironmentAs a model of environment, we consider an infinite set ofharmonic oscillators coupled to a system. This model iscalled the independent-oscillator model, since the oscil-lators do not interact with each other. Let Q and P be thecoordinate and momentum of the system respectively, andqk and pk describe those of the environment. The usualPoisson brackets are

Q,P = 1, qn, pk = δnk , (1)and all other brackets vanish.The total Hamiltonian H of the system and the environ-ment is composed of the system part

Hs = P22M + V (Q), (2)the environment Hamiltonian

He = N∑n=1(p2n2mn

+ mnω2nq2

n2), (3)

and the interaction partHi = −Q N∑

n=1 Cnqn +Q2 N∑n=1

C 2n2mnω2

n, (4)

where Cn are the coupling constants between the systemand the environment. As a result, the total HamiltonianH = Hs +He +Hi has the formH = P22M+V (Q)+ N∑

n=1[p2n2mn

+ mnω2n2(qn −

Cnmnω2

nQ)2]

.

(5)Note that the function (5) with V (Q) = (MΩ2/2)Q2 iscalled the Caldeira-Legget Hamiltonian. The infinity ofchoices for mn and ωn give this model its great generality.3. Equations of motion for opensystemsUsing (5) we can derive Hamiltonian equations of motionfor the system and the environment. For the system theseare

dQdt = Q,H = M−1P,

dPdt = P,H = −V ′(Q) + N∑

n=1(Cnqn −

C 2n

mnω2nQ). (6)

The equations for the environment aredqndt = qn, H = m−1

n pn,

dpndt = pn, H = −mnω2

nqn + CnQ. (7)Eliminating the momenta variables P and pn, n = 1, ..., N ,we can write Eqs. (6) and (7) in the form

Md2Qdt2 + V ′(Q) = N∑

n=1(Cnqn −

C 2n

mnω2nQ), (8)

mnd2qndt2 +mnω2

nqn = CnQ. (9)The solution of Eq. (9) has the form

qn(t) =qn(0) cos(ωnt) + pn(0)mnωn

sin(ωnt)+Cnmnωn

∫ t

0 Q(τ) sinωn(t − τ)dτ, (10)where qn(0) and pn(0) are the initial coordinate and mo-mentum of the environment n-th oscillators. We supposethat for t = 0 the environment is not perturbed, i.e.,qn(0) = 0 and pn(0) = 0. Integration by parts of thelast term in Eq. (10) gives∫ t

0 Q(τ) sinωn(t − τ)dτ = 1ωn

∫ t

0 Q(τ)d cosωn(t − τ) == 1ωnQ(t)− 1

ωnQ(0) cos(ωnt)−∫ t

0dQ(τ)dτ cosωn(t−τ)dτ.

Setting Q(0) = 0 for simplicity, we obtainqn(t) = Cn

mnω2nQ(t)− Cn

mnω2n

∫ t

0dQ(τ)dτ cosωn(t − τ)dτ.(11)Substituting (11) into (8) gives the equation

Md2Qdt2 + ∫ t

0 M(t − τ)dQ(τ)dτ dτ + V ′(Q) = 0, (12)

whereM(t) = N∑

n=1C 2n

mnω2n

cos(ωnt) (13)383

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The fractional oscillator as an open system

is a memory kernel. As a result, we have the non-Markovian equation of motion for the system. The inter-action between the system and the independent-oscillatorenvironment allows us to describe the dynamics with amemory.In general (qn(0) 6= 0, pn(0) 6= 0, Q(0) 6= 0) the right-handside of equation (12) is not zero. It is equal to the functionF (t) of the form

F (t) = N∑n=1(Cnqn(0) cos(ωnt) + Cnpn(0)

mnωnsin(ωnt)−

C 2n

mnω2nQ(0) cos(ωnt)) . (14)

The function (14) can be interpreted as a stochastic force.The stochastic interpretation of the function can be ap-plied since the initial states of the environment are uncer-tain and it can be determined by a distribution of initialstates qn(0) and pn(0). For example, if the initial stateof the environment is an equilibrium state, then the dis-tribution of environment oscillators, described by F (t), isGaussian. In this paper, we assume that the initial statesare equal to zero for simplicity.4. Memory and fractional calculusLet us consider the evolution of a dynamical system inwhich some quantity K (t) is related to Q′(t) through amemory function M(t):

K (t) = ∫ t

0 M(t − τ)Q′(τ)dτ, (15)where Q′(τ) = dq(τ)/dτ . Equation (15) means that thevalue K (t) is related with Q′(t) by the convolution opera-tion

K (t) =M(t) ∗Q′(t).Equation (15) is a typical non-Markovian equation ob-tained in the study of systems coupled to an environment,with environmental degrees of freedom being averaged.We consider special cases of Eq. (15). We can con-sider the memory effects and limiting cases widely used inphysics: (a) absence of the memory, (b) complete memory,and (c) power-like memory.(A) For a system without memory, we have the Markovprocesses, and the time dependence of the memory func-tion is

M(t − τ) = δ(t − τ), (16)where δ(t − τ) is the Dirac delta-function. The absenceof the memory means that the function K (t) is defined

by Q′(t) only at instant t. For this limiting case, thesystem loses all its states except for one with infinitelyhigh density. Using (15) and (16), we haveF (t) = ∫ t

0 δ(t − τ)Q′(τ)dτ = Q′(t). (17)Expression (17) corresponds to the process with completeabsence of memory. This process relates all subsequentstates to previous states through the single current stateat each time t.(B) If memory effects are introduced into the system thedelta-function turns into some function, with the time in-terval during which Q′(t) influences the function K (t). LetM(t) be the step function

M(t − τ) = t−1, 0 < τ < t;0, τ > t.(18)

The factor t−1 is chosen to get normalization of the mem-ory function to unity:∫ t

0 M(τ)dτ = 1.Then in the evolution process the system passes throughall states continuously without any loss. In this case,

K (t) = 1t

∫ t

0 Q′(τ)dτ,and this corresponds to complete memory.(C) Let us consider the power-like memory function

M(t − τ) = A (t − τ)−β . (19)The function indicates the presence of the fractionalderivative or integral. Substitution of (19) into (15) givesthe temporal fractional derivative of order β:K (t) = λΓ(1− β)

∫ t

0 (t − τ)−βQ′(τ)dτ, (0 < β < 1),(20)where Γ(1−β) is the Gamma function, and λ = Γ(1−β)A.The parameter λ can be regarded as the strength of theperturbation induced by the environment of the system.The physical interpretation of the fractional derivative isan existence of a memory effect with power-like memoryfunction. The memory determines an interval t duringwhich the function Q′(τ) affects the function K (t).384

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Vasily E. Tarasov

Equation (15) is a special type of equations for K (t) andQ′(t), where K is directly proportional toM(t) ∗Q′(t). Inour case, we have equation (12) of the form

MQ′′(t) +M(t) ∗Q′(t) + V ′(Q) = 0. (21)Equation (21) is a fractional equation for the dynamicalsystem. This equation describes the processes with mem-ory. As a result, we can use the fractional calculus [2] todescribe dynamics of classical and quantum systems withmemory.5. Memory function for open sys-temsTo describe dissipative systems by Eq. (12), we must de-fine the conditions for the frequencies ωn, the number ofenvironment oscillators N , and the coupling constants Cn.The memory function M(t) describes dissipation if M(t)is positive definite and decreases monotonically. Theseconditions are achieved if N → ∞ and if C 2

n /(mnω2n) and

ωn are sufficiently smooth functions of the index n. ForN →∞, we replace the sum in Eq. (13) by the integral

M(t) = ∫ ∞0 g(ω)C (ω) cos(ωt)dω, (22)where g(ω) is a density of states. We assume that theoscillator environment contains an infinite number of os-cillators with a continuous spectrum.The spectral density J(ω) is related with the memory func-tionM(t) by

J(ω) = ω∫ ∞

0 M(t)cos(ωt)dt (23)or by the inverse equation

M(t) = 2π

∫ ∞−∞

J(ω)ω cos(ωt)dω. (24)

Using equations (22) and (23), we haveJ(ω) = πω2 g(ω)C (ω). (25)

For the spectral densityJ(ω) = πω2 N∑

n=1C 2n

mnω2nδ(ω − ωn), (26)

equation (24) gives the memory function (13). If we con-sider the Cauchy distributionJ(ω) = a

ω2 + λ2 , (27)then Eq. (24) gives the exponential memory kernel

M(t) = aλ e−λt . (28)

For a wide class of environments, we can consider apower-law for the spectral density.6. Fractional equations for opensystemsTo derive a fractional differential equation for the opensystem, we consider a power-law for the spectral density(PLSD):

J(ω) = Aωβ , 0 < β < 1, (29)where A > 0. Equation (29) can be achieved by a varietyof combinations of coupling coefficients C (ω) and densityof states g(ω) in Eq. (25). The density (29) leads to thepower-law for the memory function M(t) ∼ t−β .Let us obtain an explicit form of equation (12) for thespectral density (29). Using the Fourier cosine-transform(see Sec. 1.3.1. in [62])∫ ∞0 x−α cos(xy)dx = π2Γ(α) cos(πα/2)yα−1,

(0 < α < 1), (30)and equations (29), (24), we have

M(t) = AΓ(1− β) cos(π(1− β)/2) t−β . (31)Substitution of (31) into (12) gives

Md2Qdt2 + Acos(π(1− β)/2) 0Dβ

t Q + V ′(Q) = 0. (32)This equation is a fractional equation with the Caputofractional derivative

0Dβt Q(t) =0I1−βt D1

tQ(t) = 1Γ(1− β)∫ t

0Q′(τ)dτ(t − τ)β ,(0 < β < 1), (33)

where D1tQ(t) = Q′(t) = dQ(t)/dt. As a result, we ob-tain a fractional differential equation from the interactionbetween the system and the environment with power-lawspectral density. Note that fractional equations of motioncan be derived from the variational principle that has beensuggested by Agrawal in [64, 65].

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7. Fractional friction for open sys-temsWe can consider an environment with an interaction withsystem different to that considered in (4). The simplestHamiltonian for the interaction between the system andthe environment has the bilinear form

H ′i = −Q N∑n=1 Cnqn. (34)

This is Hamiltonian (4) without the quadratic term. Weconsider the total Hamiltonian of the system and the en-vironmentH ′ = Hs +He +H ′i ,where Hs is defined by (2) and the environment Hamilto-nian He is (3). In this case, the equations of motion forthe system are

dQdt = Q,H = M−1P,

dPdt = P,H = −V ′(Q) + N∑

n=1 Cnqn. (35)Eliminating the momentum P , we have

Md2Qdt2 + V ′(Q) = N∑

n=1 Cnqn. (36)The equations for the environment are (7). The corre-sponding solutions are presented by (10). If we integrateby parts the last term in Eq. (10), thenqn(t) = Cn

mnω2nQ(t)−

Cnmnω2

n

∫ t

0dQ(τ)dτ cosωn(t − τ) dτ +Dn(t), (37)

whereDn(t) =qn(0) cos(ωnt) + pn(0)

mnωnsin(ωnt)−

Cnmnω2

nQ(0) cos(ωnt). (38)

If we assume that for t = 0 the environment is not per-turbed, i.e., qn(0) = 0, pn(0) = 0, and Q(0) = 0 for sim-plicity, then Dn(t) = 0. Substituting (37) into (36) givesMd2Qdt2 + ∫ t

0 M(t − τ)dQ(τ)dτ dτ + U ′(Q) = F (t), (39)

whereU(Q) = V (Q)− MΩ202 Q2, (40)

Ω20 = 1M

N∑n=1

C 2n

mnω2n, (41)

F (t) = N∑n=1 CnDn(t). (42)

The memory kernelM(t−τ) in Eq. (39) is defined by Eq.(13). Equation (39) is the integro-differential equation ofthe non-Markovian dynamics of the system. We note thatthis equation depends on the environment variables onlythrough their initial values qn(0) and pn(0) that occur inthe function F (t). If we assume that for t = 0 the environ-ment is not perturbed, i.e., qn(0) = 0, and pn(0) = 0, thenequation (39) has no dependence on the environment vari-ables. The memory kernel M(t) and the potential U(Q)depend on the strength of the environment coupling pa-rameters Cn and the spectrum of frequencies ωn of theenvironment oscillators. If F (t) 6= 0, then Eq. (39) can beconsidered as a generalized Langevin equation which issuggested by Mori [21] and Kubo [22].For N → ∞, we replace the sum in Eq. (13) by theintegral (22). Using the spectral density (29) equations(39) and (31) giveMd2Qdt2 + Acos(π(1− β)/2) 0Dβ

t Q + U ′(Q) = F (t). (43)This equation defines systems with a fractional frictionterm of order β. It is used to describe damping processeswith a fractional damping term [23, 24]. If F (t) 6= 0, thenEq. (43) can be considered as a fractional Langevin equa-tion [25–33] which is a generalized Langevin equation (39)with a power-law memory kernel. Equation (43) is frac-tional differential equation of the open dynamical system.As a result, fractional equations can be obtained from theinteraction between the system and the environment withpower-law spectral density. Fractional calculus is a pow-erful instrument for the description of open classical andquantum systems.8. The linear fractional oscillator asan open systemThe equations of motion for the linear fractional oscillatorcan be derived from equations for open system (32) and(43). If V (Q) = 0, then Eq. (32) gives

D2tQ + Ω2 0Dβ

t Q = 0, 0 < β < 1, (44)386

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Vasily E. Tarasov

where Ω2 = AM cos(π(1− β)/2) .

In general, the operations D1 and I1−β in (33) do not com-mute:D1t 0I1−βt Q(t) = 0I1−βt D1

tQ(t) + t−βΓ(1− β)Q(0). (45)For Q(0) = 0, we get

0Dβt Q(t) = D1

t 0I1−βt Q(t) = 0I1−βt D1tQ(t). (46)

Performing the fractional integration of order β in Eq.(44), and using Iα1t I

α2t = Iα1+α2

t , we obtain0D2−β

t Q + Ω2 Q = 0, (0 < β < 1). (47)This equation defines the linear fractional oscillator(LFO). As a result, the fractional oscillator can be con-sidered as a free system that interacts with the PLSDenvironment. Note that the LFO has been the subjectof numerous investigations [35–39, 42, 43, 45–51, 61] be-cause of its different applications. As a result, the frac-tional oscillator can be considered as an open system.Equation (43) with V (Q) = 0 gives

D2tQ + Ω2

c 0Dβt Q = 1

MF (t), (0 < β < 1), (48)where Ω2

c = Ω2 −Ω20.and Ω0 is defined by (41). Performing the fractional inte-gration of order β in Eq. (48), we obtain0D2−β

t Q + Ω2c Q = f (t), (0 < β < 1), (49)

wheref (t) = 1

M 0I1−βt F (t) + Ω2cQ(0).

We note that the function f (t) depends on the strengthof the environment coupling constants Cn, the spectrumof frequencies ωn of the environment oscillators, and ini-tial values qn(0), pn(0), Q(0). Equation (49) describes afractional oscillator driven by the force f (t).The linear fractional oscillator (47) can be defined by theequation0Dα

t Q(t) + Ω2Q(t) = 0, 1 < α < 2, (50)

where α = 2 − β, and 0 < β < 1. The fractional Caputoderivative 0Dαt allows us to use the usual initial conditions[3].The exact solution of Eq. (50) is

Q(t) = Q(0)Eα,1(−Ω2tα ) + tQ′(0)Eα,2(−Ω2tα ), (51)where

Eα,β (z) = ∞∑k=0

zkΓ(αk + β) (52)is the generalized two-parameter Mittag-Leffler function.The decomposition of (51) is [35]:

Q(t) =Q(0) [fα,0(Ω2/α t) + gα,0(Ω2/α t)]+Ω2/α tQ′(0) [fα,1(Ω2/α t) + gα,1(Ω2/α t)] . (53)Here

fα,k (t) = (−1)kπ

∫ ∞0 e−rt rα−1−k sin(πα)

r2α + 2rα cos(πα) + 1dr,gα,k (t) = 2

α et cos(π/α) cos [t sin(π/α)− πk/α ] , (54)

where k = 0, 1. We note that this function exhibits os-cillations with angular frequency ω(α) = sin(π/α) andan exponentially decaying amplitude with rate λ(α) =| cos(π/α)|. The functions fα,k (t) exhibit an algebraic de-cay as t →∞.To describe this algebraic decay, we consider the integralrepresentation for the generalized Mittag-Leffler functionof the form

Eα,β (z) = 12πi∫Ha

ξα−βeξξα − z dξ, (55)

where Ha denotes the Hankel contour, a loop, which startsfrom −∞ along the lower side of the negative real axis,encircles the circular disc |ξ| ≤ |z|1/α in the positive direc-tion, and ends at −∞ along the upper side of the negativereal axis. By the replacement ξα → ξ Eq. (55) transformsinto [3, 66]:Eα,β (z) = 12πiα

∫γ(a,δ)

eξ (1−β)/αξ − z dξ, (1 < α < 2), (56)

where πα/2 < δ < minπ, πα. The contour γ(a, δ) isdefined by two rays S±δ = arg(ξ) = ±δ, |ξ| ≥ a, anda circular arc Cδ = |ξ| = 1,−δ ≤ arg(ξ) ≤ δ. Let387

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The fractional oscillator as an open system

us denote the region on the left from γ(a, δ) as G−(a, δ).Then [66]:Eα,β (z) = − ∞∑

n=1z−nΓ(β − αn) , z ∈ G−(a, δ), (|z| → ∞),

(57)and δ ≤ | arg(z)| ≤ π. In our case, z = −Ω2tα , arg(z) =π, and

Eα,1(−Ω2tα ) ≈ t−αΩ2Γ(1− α) ≈ −β 1Ω2 t−2+β . (58)In a similar way, we get

Eα,2(−Ω2tα ) ≈ t−αΩ2Γ(2− α) ≈ β 1Ω2 t−2+β . (59)We can use the asymptotic behavior of the generalizedMittag-Leffler function (52) for the exact solution of (51).Then, substitution of (58) and (59) into (51) givesQ(t) ≈ −βQ(0)Ω2 t−2+β + βQ

′(0)Ω2 t−1+β , (βt 1). (60)As the result, we arrive to the asymptotic (60), which ex-hibits an algebraic decay for t →∞. This algebraic decayis the most important effect of the non-integer derivativein the considered fractional equations, contrary to the ex-ponential decay of the usual damped-oscillation and re-laxation phenomena.9. ConclusionIn this paper, we consider simple open systems in an en-vironment with power-law spectral density (PLSD). Thesesystems can be described by fractional differential equa-tions. The environment is defined as an infinite set of inde-pendent harmonic oscillators coupled to a system. Theseoscillators do not interact with each other. The power-lawspectral density of the environment leads us to a power-law for the memory function. Equations of motion withthis memory are differential equations with fractional timederivatives. The fractional differential equations describenon-Markovian motion of open systems. The fractionalderivatives in the equations describe the interaction withthe PLSD environment. As a result, fractional differentialequations allow us to consider open and non-Hamiltoniansystems with memory.Fractional calculus can be a powerful instrument to de-scribe a wide class of open classical and quantum systems.An open quantum system is a quantum system which is

found to be in interaction with an external quantum sys-tem, the environment. The open quantum system can beviewed as a distinguished part of a larger closed quantumsystem, the other part being the environment. We hopethat fractional differential equations can find many appli-cations in the non-Markovian dynamics of quantum openand non-Hamiltonian systems [67–72].References

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Vasily E. Tarasov

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