Anomalous Rotational Relaxation: A Fractional Fokker-Planck Equation Approach

0.12 MB·English

by  Ekrem Aydiner

Anomalous Rotational Relaxation:
A Fractional Fokker-Planck Equation Approach
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0 Ekrem Aydıner
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Department of Physics, Faculty of Arts and Sciences
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J Dokuz Eylu¨l University, Tr-35160 Buca, Izmir, Turkey

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1 (Dated: February 2, 2008)

] Abstract
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e In this study we have analytically obtained relaxation function in terms of rotational correlation
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- functionsbasedonBrownianmotionforcomplexdisorderedsystemsinastochasticframework. We
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t found out that rotational relaxation function has a fractional form for complex disordered systems,
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t which indicates relaxation has non-exponential character obeys to Kohlrausch-William-Watts law,
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following the Mittag-Leffler decay.
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o PACS numbers: 05.40.-a,02.50.-r,02.30.-f, 76.20.+q
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Electronic address: [email protected]
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Relaxation for ordered systems is given by Maxwell-Debye law [1, 2] as
Φ(t) = Φ exp(−t/τ), t ≥ 0. (1)
0
Whereas, relaxation in many complex disordered systems such as metallic glasses, spin
glass alloys [3, 4, 5, 6], ferroelectric crystals [7], dielectrics [8] deviates from the classical
exponential Maxwell-Debye pattern and is often described in terms of Kohlrausch-William-
Watts (KWW) (i.e. stretched exponential) law [8, 9],
α
Φ(t) = Φ exp(−t/τ) (2)
0
for 0 < α < 1, or by an asymptotic power-law
−n
Φ(t) = Φ (1+t/τ) (3)
0
withn > 0. The relaxationfunctions inEqs.(1) andEq.(2) arecommonlywritten interms of
the correlation functions which correspond to decay of the fluctuation of a physical quantity
such as magnetization in magnetic materials or polarization in the dielectric materials.
Relaxation function has been derived using rotational relaxation method for some sys-
tem. This method has been firstly used by Debye in the context of dielectric relaxation of
polar molecules [2]. Debye theory is based onthe Smoluchowski equation for the non-inertial
rotational diffusion of the molecules. In his work on dielectric relaxation of an assembly of
non interacting dipolar molecules Debye considered two models of the process, namely (a)
an assembly of fixed axis rotators each having a permanent dipole moment µ and subjected
to Brownian motion torques having their origin in the background or heat bath and (b)
the same assembly however the restriction to fixed axis rotation is removed. The results
in both instances are equivalently the same, if inertial effects are disregarded [10]. The
same picture has been applied to the rotational motion of the magnetization vector of a
super-paramagnetic particle [11], the polarization vector of a polar molecule in a dielectric
[10], and heavy molecules [12, 13] in the liquid and gases. However, Debye theory cannot
explain the experimental data on dielectric relaxation of complex systems, since the inter-
actions between dipoles are ignored. Indeed, the relaxation process in disordered systems is
characterized by the temporal nonlocal behavior arising from the energetic disorder which
produces obstacles or traps which delay the motion of the particles and introduce memory
effects into the motion. Therefore, an important task in relaxation of complex systems as
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well as dielectric relaxation is to extend the Debye theory of relaxation to fractional dy-
namics, so that empirical decay functions, e.g., the stretched exponential of Kohlrausch
[9], Williams and Watts [8], may be justified. Such a generalization of the Debye theory
was given in Refs.[14, 15, 16]. We must remark that other generalizations of the Debye
model in the context of the fractional dynamics have been discussed in recent striking works
[17, 18, 19, 20, 21] as well.
Our main aim, in this study, is obtain relaxation function Eq.(2) for some complex dis-
ordered systems in terms of rotational correlation functions based on rotational Brownian
motion in a stochastic framework.
The most common calculation in which the picture of rotational Brownian motion finds
relevance is that of the rotational correlation function ΦR(t), which can be measured by
l
Infrared (IR) and Raman (R) spectroscopies as well as the neutron and ultrasonic scattering
techniques [22]. Correlation function ΦR(t) measures the correlation in time between the
l
direction of the unit vectors (defining the molecular symmetry axis) u(0) and u(t). There-
fore, ΦR(t) as can be obtained from the IR and Raman spectroscopies may be expressed in
l
the compact form,
ΦR(t) = hP (u(0).u(t))i (4)
l l
where u is the unit vector along the symmetry axis of the molecule, and P is the Legendre
l
polynomial of order l. Argument of P is expressed as u(0).u(t) = cosγ(t), where γ is angle
l
between two different point at the spherical coordinate system. These points are given by
(θ ,φ ) and (θ,φ) which denote two different directions by separated by an angle γ. There
0 0
angles satisfy the trigonometric identity,
cosγ = cosθ cosθ +sinθ sinθcos(φ −φ) (5)
0 0 0
The addition theorem asserts that
l
4π
P (cosγ) = Ym(θ ,φ )Ym∗(θ,φ). (6)
l 2l+1 l 0 0 l
mX=−l
If we inserted Eq.(6) into Eq.(4) rotational correlation function is expressed in general form
using spherical harmonics addition theorem as
l
4π
Φ (t) = hYm(θ ,φ )Ym∗(θ,φ)i. (7)
l 2l+1 l 0 0 l
mX=−l
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Such a theoretical approach allow us to calculate for arbitrary l. Average in Eq.(7) is
calculated using the probability density functions of Brownian particle which refers to end
of point polarization vector or a real particle. The Brownian motion in disordered space in
′
the presence of an external field F(x) = −V (x) leads to Fractional Fokker-Planck Equation
(FFPE) [23, 24, 25, 26, 27]:
∂
W (x ,0 | x,t) = D1−αL W (x ,0 | x,t) (8)
∂t 0 0 t FP 0
∂ V′(x) ∂2
L = +K (9)
FP α
(cid:20)∂x mη ∂x2(cid:21)
α
This equation then characterize sub-diffusion process. The FFPEareclosely relatedgeneral-
izedL´evy-typestatistics [28]andcanbederived fromcontinuoustimerandomwalk(CTRW)
models [29, 30, 31, 32, 33, 34], or from a Langevin equation [35]. In Eq.(8), W (x ,0 | x,t)
0
imply conditional probability for Brownian motion, m denotes mass of the particle, K the
α
diffusion constants associated with the transport process, and the friction coefficient η is a
α
measure for interaction of the particle with its environment. K is a generalization of the
α
Einstein-Stokes-Smoluchowski relation [23, 24, 25, 26, 27] which holds for the generalized
coefficient η , which is defined as K = k T/mη where k is the Boltzman constant, and
α α B α B
T is the temperature. In the Eq.(8) operator D1−α is known fractional Riemann-Liouville
0 t
integro-differential operator [36].
The fractional Riemann-Liouville operator D1−α = d D−α featuring is defined through
0 t dt0 t
t
1 ∂ W (θ ,φ ,0 | θ,φ,t)
D1−αW (θ ,φ ,0 | θ,φ,t) = dt′ 0 0 (10)
0 t 0 0 Γ(α)∂tZ (t−t′)1−α
0
The fractional integro-differentiation operator D1−α contains a convolution integral with a
0 t
slowly decaying power-law Kernel M (t) = tα−1, ensures the non-Markovian nature of the
Γ(α)
sub-diffusion process defined by the fractional diffusion process. Its fundamental property
is the fractional integro-differentiation of a power,
Γ(1+p)
D1−αtp = tp+α−1 (11)
0 t Γ(p+α)
In fact, it can be shown that more general relation
Γ(1+q)
Dptq = tq−p (12)
0 t Γ(1+q −p)
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for any real p, q. Thus, the fractional derivative of a constant,
1
Dq1 = t−q, q > 0 (13)
0 t Γ(1−q)
reproduces an inverse power-law. The special cases of integer order integro-differentiation of
a constant, dn1 = 0, are included through the poles of the Gamma function for q = 1,2,3,....
dtn
′
In the case of V (x) = 0 which means that there is no an external field, the one-
dimensional FFPE can be reduced to diffusive type equation;
∂ ∂2
W (x ,0 | x,t) = D1−αK W (x ,0 | x,t). (14)
∂t 0 0 t α∂x2 0
This equation is called fractional diffusion equation [37, 38, 39] which is a particular form of
the FFPE, which can be represent on the spherical coordinates as a function of the angles
θ and φ as
∂
W (θ ,φ ,0 | θ,φ,t) = D1−αd ∇2W (θ ,φ ,0 | θ,φ,t) (15)
∂t 0 0 0 t α 0 0
where d is referred to as the rotational diffusion constant that is related to translational
α
diffusion constant K and the radius a by d = K /a2 (a = unit)
α α α
1 ∂ ∂ ∂2
∇2 = sinθ sinθ + (16)
sin2θ (cid:20) ∂θ (cid:18) ∂θ(cid:19) ∂φ2(cid:21)
The standard method of solution of Eq.(15) is the separation variables [23, 24, 25, 26,
27]. If we consider separation ansatz, as W = T.Q, where T and Q temporal and spatial
components of the conditional probability function respectively, we will obtain two eigen-
equations as
dT (0 | t)
= −λ2d D1−αT (0 | t) (17)
dt α0 t
∇2Q(θ ,φ | θ,φ) = −λ2Q(θ ,φ | θ,φ) (18)
0 0 0 0
The temporal eigen-equation Eq.(17) is but the fractional relaxation equation, the solution
of which is given in terms of the Mittag-Leffler function [40]
∞ [−d l(l+1)tα]j
T (0 | t) = E [−d l(l+1)tα] ≡ α . (19)
α α
Γ(1+αj)
Xj=0
As can be seen from the series expansion, the exponential form can be recovered in the
Brownian limit α = 1
E d l( l+1) t1 = exp[d l( l+1) t]. (20)
1 1 1
(cid:2) (cid:3)
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This result indicates that for α = 1 Mittag-Leffler relaxation modes leads to Maxwell-Debye
pattern, on the other hand, for interval 0 < α < 1 Mittag-Leffler function indicates non-
Markovian Brownian motion.
On the other hand, a convenient way of expressing solution of the spatial eigen-equation
Eq.(18) is by means of spherical harmonics
∞ l
4π
Q(θ ,φ | θ,φ) = Ym(θ ,φ )Ym∗(θ,φ). (21)
0 0 2l+1 l 0 0 l
Xl=0 mX=−l
As a result, if solutions Eqs.(19) and (21) are combined, conditional probability function
of Brownian particle is obtained in terms of Mittag-Leffler function as
∞ l
4π
W (θ ,φ ,0 | θ,φ,t) = Ym(θ ,φ )Ym∗(θ,φ)E [−d l(l+1) tα] (22)
0 0 2l+1 l 0 0 l α α
Xl=0 mX=−l
Eq.(22) can be written as a compact in terms of the Euler angles Ω (θ ,φ ) and Ω(θ,φ),
0 0 0
hence
∞ l
4π
W (Ω ,0 | Ω,t) = Ym(Ω )Ym∗(Ω)E [−d l( l+1) tα] (23)
0 2l+1 l 0 l α α
Xl=0 mX=−l
Also, Eqs.(22) and (23) should be satisfied
W (Ω ,0 | Ω,0) = δ(Ω −Ω) (24)
0 0
for t = 0.
It is easy give the correlation function Eq.(7) what is by now a familiar interpretation as
l
4π
Φ (t) = w(Ω )Ym(Ω )W (Ω ,0 | Ω,t)Ym∗(Ω)dΩ dΩ. (25)
l 2l+1 Z 0 l 0 0 l 0
mX=−l
In above equation w(Ω ) is a priori probability that the initial orientation is given by Ω ,
0 0
whileW (Ω ,0 | Ω,t)istheconditionalprobability thatthefinalorientationisdetermined Ω.
0
Assuming that the reorientations of the spin (or molecular) symmetry axis may be modelled
as an isotropic rotational Brownian motion, we may write
w(Ω ) = 1/4π (26)
0
and adopt Eq.(22) as the solution for the conditional probability. Hence Eq.(25) yields,
l
4π 1
Φ (t) = E [−d l′( l′ +1) tα]
l 2l+1 (cid:18)2l′ +1(cid:19) α α
mX=−lXl′m′
× Y (Ω )Ym′∗(Ω )dΩ Ym(Ω)Ym′∗(Ω)dΩ (27)
0 l′ 0 0 l l′
Z Z
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Using orthogonal property
Ylm(Ω)Ylm′ ′∗(Ω)dΩ = δll′δmm′. (28)
Z
Hence, rotational correlation function (27) is written in terms of Mittag-Leffler function [40]
Φ (t) = Φ (0)E [−d l(l+1) tα] (29)
l l α α
where normalized factor of Eq.(29) is given as Φ (0) = 4π/(2l+1)2. Eq.(29) states simply
l
that the rotational correlation function, starting from the value unity at t = 0, decay non-
exponentially in time with a relaxation time τ that inversely proportional to rotational
l
diffusion constant d :
α
τ = [d l(l+1)]−1. (30)
α,l α
The rotational correlation function (29) is valid for arbitrary number l, and it has inter-
esting property due to behavior of the Mittag-Leffler functions lies in the observation that
is interpolates between an initial stretched exponential (i.e. KWW) behavior
tα
Φ (t) ≈ exp − (31)
l
(cid:20) τ Γ(1+α)(cid:21)
α,l
and a long-time inverse power-law pattern
τ t−α
α,l
Φ (t) ≈ . (32)
l
(cid:20)Γ(1−α)(cid:21)
In conclusion, it is seen that relaxation functions Eq.(31) and (32) are compatible with
Eqs.(2) and (3) for complex disordered systems, respectively.
While drafting the problem we have disregarded the inertial of free motion governed by
therotationalkinetic energy among thesuccessive collision. Indeed, therotationalBrownian
motionmodelforcomplexsystems makesense onlywhenfrictionη islargei.e. thecollisions
α
are very rapid. There the rotational jumps were imagined to occur by large and arbitrary
angles as opposed to the present instance in which only small-angle jumps are considered.
In this study, we have analytically carried out the rotational relaxation function in terms
of rotationalcorrelation function for complex disordered systems based onrotationalBrown-
ian motion. To obtain rotational correlation function we have introduce generalized Fokker-
Planck equations of fractional order, which generalizes the Stokes-Einstein-Smoluchowski
relation, in consistency with the fluctuation-dissipation theorem. The introduction of the
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Riemann-Liouville operator includes long-range memory effects which are typically found
in complex systems, and consequently a single mode relaxes slowly in time, following the
Mittag-Leffler decay.
Inconclude, we have shown thatrotationalBrownianmotioninthecomplex systems such
asspin glasses ordielectric materials, leads toKWWdecays which indicates non-exponential
relaxation.
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