Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors

Abstract

The dynamical phase diagram of the fractional Langevin equation is investigated for harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) $\alpha_c=0.402\pm 0.002$ marks a transition to a non-monotonic under-damped phase, (ii) $\alpha_R=0.441...$ marks a transition to a resonance phase when an external oscillating field drives the system, (iii) $\alpha_{\chi_1}=0.527...$ and (iv) $\alpha_{\chi_2}=0.707...$ marks transition to a double peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over-damped, under-damped regimes, motion and resonances, show behaviors different from normal. Comment: 18 pages, 15 figures

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Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors.
S. Burov, E. Barkai
Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel
The dynamical phase diagram of the fractional Langevin equation is investigated for harmonically
bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of
the system. Four different critical exponents are found. (i) αc = 0.402±0.002 marks a transition to
a non-monotonic under-damped phase, (ii) αR = 0.441... marks a transition to a resonance phase
when an external oscillating field drives the system, (iii) αχ1= 0.527... and (iv) αχ2= 0.707... marks
transition to a double peak phase of the “loss” when such an oscillating field present. As a physical
explanation we present a cage effect, where the medium induces an elastic type of friction. Phase
diagrams describing over-damped, under-damped regimes, motion and resonances, show behaviors
different from normal.
PACS numbers: 02.50.-r,05.10.Gg,05.70.Ln,45.10.Hj
I.INTRODUCTION
In this paper we investigate the phenomenological
description of stochastic processes using a Fractional
Langevin Equation (FLE) [1, 2, 3, 4, 5, 6, 7]. While in
simple systems the memory friction kernel is an exponen-
tially decaying function or a delta function, in complex
out of equilibrium systems the picture is in some cases
different. Namely the relaxation is of a power law type,
and the particle may exhibit anomalous diffusion and re-
laxation [8]. Mathematically such systems are modeled
using fractional calculus, e.g.
recent experiment on protein dynamics of the group of
Xie [9, 10]. There anomalous dynamics of the coordinate
x, describing donor-acceptor distance was recorded, and
a FLE (see Sec. II) was found to describe the experimen-
tal data. The motion of x is bounded by a harmonic force
field, and the equation of motion for the average ?x? is
¨
?x? + ω2?x? + γdα?x?
where 0 < α < 1, ω is the harmonic frequency and γ > 0.
For the case α = 1 we get the usual damped oscilla-
tor [11]. For such a normal case two types of behav-
iors: the under-damped and the over-damped motions
are found. In the under-damped case ?x? is oscillating,
and crossing the zero line, while for over-damped case
?x? is monotonically decaying with no zero crossing. For
α = 1, there exist a critical frequency ωc =
separates these two types of motion. Here we explore
a similar scenario for the fractional oscillator, and find
rich types of physical behaviors. It is known [12] that
in the long time limit all solutions ?x? (i.e. any 0 < α,
0 < γ and 0 < ω ) decay monotonically, some what like
the over-damped behavior of the usual oscillator, how-
ever now the decay is of power law type. The interesting
Physics occurs at shorter times where the solution may
exhibit different types of relaxation and oscillations. We
find that for α < αc the solution is non-monotonic for
any set of parameters (γ,ω > 0). Thus we find a critical
α which marks a dynamical transition in the behavior of
the system.
d1/2
dt1/2. An example is the
dtα
= 0(1)
γ
2which
We also investigate the response of a system described
by Eq. (1) to an external oscillating force F0cos(Ωt). For
the regular case of α = 1, a resonance is present if the
frequency ω is larger than the critical value γ/√2. The
behavior for 0 < α < 1 is quite different and we dis-
cover that the transition between α → 1 and α → 0 is
not smooth. In particular we find another critical expo-
nent αR, where for α < αRa resonance is always present.
Other critical exponents are found for the imaginary part
of complex susceptibility. Our goal is to clarify the na-
ture of solution to Eq. (1), investigate the meaning of
fractional critical frequency with and without external
oscillating force and provide a mathematical tool box for
finding and plotting solutions of Eq. (1). Our finding
that critical αs mark dynamical transitions is very sur-
prising and couldn’t be obtained without our mathemat-
ical treatment.
The paper is organized as follows. In Sec. II we present
the FLE. In Sec. III we present two different methods for
solution for first order moments and examples are solved
in Sec. IV. In Sec. V we present different definitions
for over-damped and under-damped motion and find the
critical exponent αc. We also interpret our results from a
more Physical point of view and discuss the cage effect as
a viscoelastic property of the medium. In Sec.VI we in-
troduce external oscillating force into the system and find
the response for such force for the free (Sec. VIA) and
harmonically bounded (Sec. VIB) particles. In Sec. VIC
we investigate the properties of the “loss” - the imaginary
part of complex susceptibility. A summary is provided in
Sec. VII, and the three Appendixes deal with some tech-
nical aspects. A brief summary of some of our results
was published [13].
II.THE MODEL
We consider the FLE
md2x(t)
dt2
+ ¯ γ
?t
0
1
(t −`t)α
dx
d`td`t = F(x,t) + ξ(t)(2)
arXiv:0802.3777v1 [cond-mat.stat-mech] 26 Feb 2008

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where ¯ γ > 0 is a generalized friction constant (γ =
1
m¯ γΓ(1 − α)), 0 < α < 1 is the fractional exponent and
ξ(t) is a stationary, Fractional Gaussian noise [14, 15]
satisfying the fluctuation-dissipation relation [16]
?ξ(t)? = 0,
?ξ(t)ξ(`t)? = kbT¯ γ|t −`t|−α.
(3)
F(x,t) is an external force. We follow experiment [10]
and assume F(x,t) = −mω2x, later in Sec. VI we will
have F(x,t) = −mω2x+F0cos(Ωt). In Laplace Space it
is easy to show, using the convolution theorem
ˆ x(s) =
s +
msβ(s) + ω2x0+
1
mβ(s)
s2+
1
1
s2+
1
msβ(s) + ω2v0
+
1
s2+
1
msβ(s) + ω2ξ(s)
(4)
where x0and v0are initial conditions and
β(s) = ¯ γΓ(1 − α)sα−1.
(5)
All along this work the variable in the parenthesis de-
fines the space we are working in (e.g.
Laplace Transform of x(t)). Eq. (2) with α =
F(x,t) = −mω2x, describes single protein dynamics [10].
An experimentally measured quantity is the normalized
correlation function
Cx(t) =?x(t)x(0)?
ˆ x(s) is the
1
2and
?x(0)2?
.
(6)
In what follows, thermal initial conditions are assumed
?ξ(t)x(0)? = 0, ?x(0)2? = kbT/mω2and ?x(0)v(0)? = 0.
From Eq. (4) we find
ˆCx(s) =
s + γsα−1
s2+ γsα+ ω2.
(7)
It is easy to show that Cx(t) satisfies the following
fractional Eq.
¨Cx(t) + ω2Cx(t) + γdαCx(t)
dtα
= 0,
(8)
with the initial conditions Cx(0) = 1 and ˙Cx(0) = 0,
where the fractional derivative is defined in the Caputo
sense [17, 18]
dαf(t)
dtα
=0Dα−1
t
?df(t)
dt
?
(9)
and 0Dα−1
tor [17, 18]
t
is the Riemann-Liouville fractional opera-
0Dα−1
t
f(t) =
1
Γ(1 − α)
?t
0
(t −`t)−αf(`t)d`t.
(10)
Note that another way to write Eq. (2) is
¨ x + γdαx
dtα+ ω2x = ξ(t)(11)
hence the name Fractional Langevin equation is justified.
For the force free particle (F(x,t) = 0 in Eq. (2)) ?x2? ∝
tα[6, 19], this is a sub-diffusive behavior since 0 < α < 1.
The FLE in general and Eq. (8) in particular can be de-
rived from the Kac-Zwanzig model of a Brownian particle
coupled to a Harmonic bath [20]. Eq. (8) can be derived
also from the Fractional Kramers Equation [21]. The
use of fractional-differential equations like Eqs. (8,11) be-
came quite common in recent years [8, 22], especially
in the context of anomalous diffusion [8, 23, 24]. Sev-
eral other fractional oscillator equations were considered
in the literature [25, 26, 27] and general solutions for
fractional-differential equations of the type Eq. (8) are
studied in the mathematical literature [18, 28, 29]. In
the next section we present a practical recipe, a tool box,
for the solution of Eq. (8), and show how to plot its
solution. Our methods are general and can be applied
to other linear fractional differential equations. From a
more physical point of view the following questions are
addressed in the next sections: (i) When is Cx(t) pos-
itive? (i.e. similar to over-damped behavior, α = 1).
(ii) When is Cx(t) non-monotonic? (similar to under-
damped motion for α = 1) (iii) What is the analogue to
the critical frequency ωc=
(iv) Does a critical exponent αcexist and if so, what is
it’s value?
γ
2found for the α = 1 case.
III. THE GENERAL SOLUTION
In this section we will present a recipe for an analytical
solution of Eq. (8).
The formula for Cx(t) in Laplace space is Eq. (7)
and if α was an integer then performing an Inverse
Laplace Transform would be simple using analysis of
poles [30, 31], because then the denominator of Eq. (7) is
a polynomial. We assume α is of the form α =p
q > p > 0 are integers and
to some other
q, where
p
qis irreducible (i.e not equal
nwhere l < p and n < q are integers).
l
A. Method A
We rewrite Eq. (7) as
ˆCx(s) =
s + γs
s2+ γs
p
q−1
p
q+ ω2
ˆQ(s)
ˆQ(s)
=
?
s + γs
p
q−1?
ˆP(s)
ˆQ(s)
(12)
whereˆP(s) is a polynomial in s. According to a mathe-
matical theorem [18] we can always findˆQ(s) such that
the denominator of Eq. (12)
ˆP(s) = (s2+ γs
p
q+ ω2)ˆQ(s)(13)
is a regular polynomial in s. The polynomialˆQ(s) is
called the complementary polynomial. The task of find-
ingˆQ(s) is simple,ˆQ(s) is a polynomial in s
1
qof degree

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2q(q − 1)
ˆQ(s) =
2q(q−1)
?
m=0
Bms
m
q
(14)
with B2q(q−1)= 1. The coefficients Bm are found by
equating all the coefficients in the expansion of the prod-
uctˆQ(s)(s2+γs
zero. This produces a linear system of 2q(q−1) equations
for 2q(q − 1) variables, which in principle is a solvable
problem.
We also assume that all 2q zeros ofˆP(s) are distinct.
The generalization to the case where the zeros are not
distinct will be treated later, such a behavior is related
to a critical frequency of the system. We use the partial
fraction expansion
p
q+ω2), with non-integer powers of s, to
1
ˆP(s)
=
2q
?
k=1
Ak
s − ak
(15)
where akare the solutions ofˆP(s) = 0 and Akare con-
stants defined as
Ak=
1
dˆP(s)
ds
|s=ak
.
(16)
It can be shown [18] that
sm
ˆP(s)
=
2q
?
k=1
am
kAk
s − ak
, m = 0,1,...,2q − 1.
(17)
The numerator of Eq. (12) is written using the expansion
ˆQ(s)
?
s + γs
p
q−1?
=
2q−1
?
m=0
q−1
?
j=0
˜Bmjsm−j
q.
(18)
Hence using Eqs. (12,15,18)
ˆCx(s) =
2q−1
?
m=0
q−1
?
j=0
2q
?
k=1
am
k˜BmjAk
s − ak
s−j
q.
(19)
So finally we reduced the problem of calculating the In-
verse Laplace Transform of Eq. (7) to performing Inverse
Laplace Transform for
1
s − ak
and
1
sceakt
(20)
s
j
q(s − ak)
q,akt) is a tabulated Incomplete Gamma Func-
tion [32]. Using the series expansion for γ(a,x)
s
c
eakt
Γ(j
q)ak
j
qγ(j
q,akt)(21)
where γ(j
γ(a,x) = Γ(a)xae−x
∞
?
n=0
xn
Γ(a + n + 1)
we can write Eq. (21) by the means of generalized Mittag-
Leffler function [33]
1
s
j
q(s − ak)
sct
j
qE1,1+j
q(akt) .
(22)
The Mittag-Leffler function satisfies
Eη,µ(y) =
∞
?
n=0
yn
Γ(ηn + µ)
(23)
with
Eη,µ(y) ∼ −
y−1
Γ(µ − η)
y → ∞.
(24)
Actually we have finished our task, we can now perform
an Inverse Laplace Transform of expressions like Eq. (7)
and even more general expressions which can be pre-
sented as a fraction of polynomials with fractional pow-
ers. To summarize, we need to follow four steps (i) Cal-
culateˆQ(s), which is equivalent to the diagonalisation
of a matrix of size (2q(q − 1))2. (ii) Find the zeros of
ˆP(s), ak, and the coefficients of partial fractions expan-
sion ,Ak of Eq. (16). (iii) Find the coefficients˜Bmj for
Eq. (18) and writeˆCx(s) as the sum in Eq. (19). (iv) Use
Eqs. (20,21,22) to inverse Laplace transform Eq. (19), we
find
Cx(t) =
2q−1
?
m=0
q−1
?
j=0
2q
?
γ(j
Γ(j
k=1
am−j/q
k
˜BmjAkeakt
Γ(j
q)
γ(j
q,akt) (25)
where for j = 0,
q,akt)
q)
= 1, or using Eq. (22)
Cx(t) =
2q−1
?
m=0
q−1
?
j=0
2q
?
q(akt) = eakt. Now we have a prac-
k=1
am
k˜BmjAkt
j
qE1,1+j
q(akt) (26)
and for j = 0 E1,1+j
tical tool for finding an explicit analytical solution for
the fractional damped harmonic oscillator. Since γ(a,x)
is tabulated in programs like Mathematica, the solution
which is a finite sum of such incomplete gamma func-
tions, can be plotted.
B.Method B
The task of findingˆQ(s) is sometimes difficult since as
described in Sec. IIIA, generally one must solve a linear
system of 2q(q − 1) equations with 2q(q − 1) variables.
But for the special case of Eq. (7) we will provide a simple
method for findingˆQ(s) andˆP(s). By writing
Cx(t) =
s + γs
p
q−1
(s2+ ω2)q+ (−1)q−1γqsp
??s2+ ω2?q+ (−1)q−1γqsp
s2+ γs
p
q+ ω2
?
(27)

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FIG. 1:
γ = 1, versus t. Three types of solutions are presented (a)
ω = 0.3 and the function decays monotonically. (b) ω =
ωz ≈ 1.053 the transition between motion with and without
zero crossing, Cx(t) = 0 at a single point in time, and Cx(t)
does not cross the zero line. (c) ω = 3 the under-damped
regime. Note that for large t, Cx(t) > 0.
The short time behavior for Cx(t) with α =
1
2and
we can write
ˆQ(s) =
?s2+ w2?q+ (−1)q−1γqsp
s2+ γs
p
q+ w2
(28)
and
ˆP(s) =
?
s2+ γs
p
q+ w2?
ˆQ(s) =?s2+ ω2?q+(−1)q−1γqsp.
(29)
One easily sees that indeedˆP(s) is a polynomial in s.
As forˆQ(s), it is found by standard method of division
of two polynomials in s1/qand it is also a polynomial in
s1/q. Since the fraction of two polynomials
?y2q+ ω2?q+ (−1)q−1γqyqp
is also a polynomial in y, we see that any solution of
y2q+ γyp+ ω2= 0 is also a solution of
(−1)q−1γqyqp= 0, and by performing a substitute y =
s
in s1/q. After findingˆQ(s) andˆP(s) explicitly we can
return to method A and use Eqs. (15-26) in order to
write down the final solution.
y2q+ γyp+ ω2
(30)
?y2q+ w2?q+
1
qinto Eq. (30) we find that indeedˆQ(s) is a polynomial
IV.EXAMPLES α =1
2AND α =3
4.
A.α =1
2
The α =1
[10], and so it will be our first illustration for the method.
From Eq. (7) we get
2case was recently measured in experiment
ˆCx(s) =
s + γs−1
s2+ γs
2
1
2 + w2.
(31)
FIG. 2:
γ = 1,versus t, on a loglog scale. Three types of solutions are
presented (a) ω = 0.3 over-damped limit, monotonic decay.
(b) ω = ωz ≈ 1.053 the transition between motion with and
without zero crossing. (c) ω = 3 the under damped regime,
for large t, Cx(t) > 0 (for short time Cx(t) < 0, not shown).
Notice that non-monotonic decay in the under-damped case
is observed only for short times, while for long-times Cx(t) ∝
t−1
The long time behavior for Cx(t) with α =
1
2and
2.
The first step is to find the complementary polynomial
of the denominator on the right hand-side of Eq. (31).
Using method A of the previous section one can easily
see that the coefficients of the complementary polynomial
ˆQ(s) are
B4= 1, B3= B2= 0, B1= −γ, B0= ω2
and
ˆQ(s) = s2− γs
1
2+ ω2
(32)
since
ˆP(s) =ˆQ(s)
?
s2+ γs
1
2+ ω2?
= s4+ 2ω2s2− γ2s + ω4.
(33)
We rewrite Eq. (31) as
ˆCx(s) =s3+ sω2− γ2+ γω2s−1
(s2+ ω2)2− γ2s
and notice that the degree of the denominator of Eq. (34),
i.e.
ˆP(s), is 4. Its zeros are easily found using Ferrari
formula [32], and we call them ak, k = 1,...,4. The
coefficients of the partial fraction expansion Akare given
by Eq. (16)
2
(34)
Ak=
1
4ak(a2
k+ ω2) − γ2.
(35)
The partial fraction expansion is found using Eq. (19)
ˆCx(s) =
3
?
m=0
1
?
j=0
4
?
k=1
am
k˜BmjAk
s − ak
s−j
q
(36)

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and the˜Bmjin Eq. (18) are found using the numerator
of Eq. (34)
˜B30= 1,˜B10= ω2,˜B00= −γ2,˜B01= γω2
(37)
and other elements of the matrix˜Bmjare equal 0. Using
Eq. (25) the solution is
Cx(t) =
4
?
k=1
??−γ2+ ω2ak+ a3
k
?Akeakt+ γω2Akt
1
2E1,3
2(akt)
?
.
(38)
FIG. 3:
γ = 1, versus t. Three types of solutions are presented (a)
ω = 0.3 and the function decays monotonically. (b) ω =
ωz ≈ 0.965 the transition between motion with and without
zero crossing, Cx(t) does not cross the zero line. (c) ω = 3
the under-damped regime.
The short time behavior for Cx(t) with α =
3
4and
By using a series expansion and some algebra we find
akand Eq. (38) has the following asymptotic behavior


We see that Cx(t) for long times decays as a power-law,
which is the signature of slow relaxation and anomalous
diffusion. The same asymptotic results are found by ap-
plying Tauberian Theorems [34] to Eq. (31).
Cx(t) =





1 −1
2ω2t2+γω2
γ
ω2Γ(1
Γ(9
?γ
2)t
7
2
t → 0
2)t−1
2−
w2
?3
t−3
2Γ(1
2
2)
t → ∞
.


(39)





,
The asymptotic expansions Eq. (39) provides the be-
havior for long (and short) times, but the intermediate
behavior is not obvious. Using the exact solution Eq. (38)
we plot Cx(t) for various values of γ and ω in Fig. 1 and
Fig. 2. Three types of behaviors exist (i) Monotonic
decay of the solution - Fig. 1(a).(ii) Non-monotonic de-
cay in the non-negative half of the plane, Cx(t) ≥ 0, -
Fig. 1(b). (iii) Oscillations of the solution, where Cx(t)
also takes negative values - Fig. 1(c). These are typ-
ical behaviors of the solution which we found also in
other parameter set (not shown). From Fig. 1 we identify
ω = ωz= 1.053 as a fractional critical point, in the sense
that if ω > ωz we have zero crossings for Cx(t). For
α =
ωm = 0.426 where for ω < ωm Cx(t) is monotonically
decaying - Fig. 1(a). We will soon discuss ωz and ωm
more generally.
1
2there exist also another fractional critical point
B.α =3
4
In this subsection we demonstrate the solution for α =
4using method B. Our goal is to invert Eq. (7).
3
ˆCx(s) =
s + γs
s2+ γs
3
4−1
4 + w2
3
.
(40)
We find the complementary polynomialˆQ(s) andˆP(s),
using Eq. (28) and Eq. (29)
ˆP(s) =?s2+ ω2?4− γ4s3
(41)
and
ˆQ(s) =
?s2+ ω2?4− γ4s3
s2+ γs
3
4 + ω2
=s6− γs
19
4 + 3ω2s4+ γ2s
7
2− 2ω2γs
11
4 − γ3s
9
4+ 3ω4s2+ ω2γ2s
3
2− γω4s
3
4+ ω6
(42)
and so we can write Eq. (40) as
ˆCx(s) =s7+ 3ω2s5+ γω2s
15
4 + 3ω4s3− γ2ω2s
5
2 − γ4s2+ 2γω4s
(s2+ ω2)4− γ4s3
7
4 + γ3ω2s
5
4 + ω6s − γ2ω4s
1
2 + γω6s−1
4
.
(43)