Current and universal scaling in anomalous transport

0.16 MB·English

by  I. Goychuk

Current and universal scaling in anomalous transport
I. Goychuk,1 E. Heinsalu,1,2,3 M. Patriarca,1 G. Schmid,1 and P. Ha¨nggi1
1Institut fu¨r Physik, Universit¨at Augsburg, Universit¨atsstr. 1, D-86135 Augsburg, Germany
2Institute of Theoretical Physics, Tartu University, 4 T¨ahe Street, 51010 Tartu, Estonia
6 3Fachbereich Chemie, Philipps-Universit¨at Marburg, 35032 Marburg, Germany
0 (Dated: February 2, 2008)
0
Anomalous transport in tilted periodic potentials is investigated within the framework of the
2
fractional Fokker-Planckdynamicsandtheunderlyingcontinuoustimerandomwalk. Theanalytical

n solution for the stationary, anomalous current is obtained in closed form. We derive a universal
a scaling law for anomalous diffusion occurring in tilted periodic potentials. This scaling relation is
J
corroborated with precise numerical studies covering wide parameter regimes and different shapes

1 for theperiodic potential, being eithersymmetric or ratchet-like ones.
1
PACSnumbers: 02.50.Ey,05.40.Fb,05.60.Cd

]
h
c
Over recent years we witness an increasing interest ψ (τ) for the residence times τ, considering only jumps
e i
m in dynamical processes that occur in systems exhibiting between nearest-neighborsites ona one dimensional lat-
anomalous diffusive behavior, possessing prominent in- tice {x }, with lattice period ∆x. Such a CTRW is de-
- i
t terdisciplinary applications that range from physics and scribed by a generalized master equation (GME) for the
a
chemistrytobiologyandmedicine[1]. Thebenchmarkof site populations P (t), reading [9, 10, 11]
t i
s anomalous diffusion is the occurrence of a mean square
mat. dDiesppelancdeimngenotnofthteheanfoormmalohδurs2(dti)ffiu∼siontα,exwphoenreentαα6=th1e. P˙i(t) = Z t{Ki+−1(t−t′)Pi−1(t′)+Ki−+1(t−t′)Pi+1(t′)
0
d- pmeortdiioffnusciavne (eαith>er1)b.e subdiffusive (0 < α < 1) or su- − [Ki+(t−t′)+Ki−(t−t′)]Pi(t′)}dt′, (1)
n In the following we focus on the subdiffusive regime. where the Laplace-transform of the kernel K±(t) is re-
o Examplesfor subdiffusive transportare verydiverse,en- lated to the Laplace-transformof the residencei time dis-
c compassing phenomena such as charge carrier transport tribution (RTD) via K˜±(s)=q±sψ˜(s)/[1−ψ˜(s)]. The
[ in amorphous semiconductors, nuclear magnetic reso- quantitiesq± arethespilittingpirobaibilitiestojiumpfrom
2 nance,diffusioninpercolativeandporoussystems,trans- site i to siteii±1, obeying q++q− =1.
v port on fractal geometries and dynamics of a bead in Choosing for the RTD theiMittaig-Leffler distribution,
2
a polymeric network, as well as protein conformational
7
dynamics [1, 2, 3]. Another topic that recently gained
1 d
9 attention is the transport of Brownian particles in the ψi(τ)=− Eα[−(νiτ)α], (2)
0 presence of a periodic force, relevant in Josephson junc- dτ
5 tions, rotating dipoles in external fields, superionic con- one obtains K˜±(s) = q±ναs1−α. Here E (z) =
0 ductors, charge density waves, synchronization phenom- ∞ zn/Γ(nα+i 1) denoteis tihe Mittag-Leffler fαunction
at/ ena, diffusion on crystal surfaces, particle separation by Pannd=t0hequantityν−1isthetime-scalingparameteratsite
electrophoresis, and biophysical processes such as intra- i
m i. The corresponding GME can be recast as a fractional
cellular transport [4, 5, 6].
- With this work, we present intriguing results for master equation (FME) [12, 13], reading
d
anomalous diffusion and transport under the combined
n
o action of a periodically varying spatial force and an ex- P˙i(t) = 0Dˆ1t−α{fi−1Pi−1(t)+gi+1Pi+1(t)
c ternal constant bias F. In particular, we derive a closed
− (f +g )P (t)}, (3)
: form expression in terms of two quadratures for the i i i
v
i fluctuation-assisted current and its corresponding non-
X linear mobility. Furthermore, we establish a universal where the symbol Dˆ1−α stands for the integro-
0 t
r scaling relation for diffusive transport which is valid in differential operator of the Riemann-Liouville fractional
a
tilted, corrugated nonlinear periodic potentials. derivative acting on a generic function of time χ(t), as
We start out by presenting a novel derivation of [1, 2, 14]
the fractional Fokker-Planck equation (FFPE) from a
space-continuous limit of a continuous-time random Dˆ1−αχ(t)= 1 ∂ tdt′ χ(t′) .
walk(CTRW).Ourderivationinvolvesnearestneighbors 0 t Γ(α)∂tZ (t−t′)1−α
0
jumps only; moreover, it provides new insight and com-
plements prior treatments in Refs. [1, 7, 8]. The quantities f =q+να and g =q−να will be referred
i i i i i i
Derivation of the FFPE from the CTRW. We study to as fractional forward and backward rates. Using the
a CTRW characterized by the probability distributions normalizationconditionforthesplittingprobabilitiesone
2
obtains that ν =(f +g )1/α, and q+ =f /(f +g ) and Withrespecttothecaseofnormaldiffusiontheexpres-
− i i i i i i i
q =g /(f +g ), intermsofthe fractionalrates. Foran sionforthemeansquaredisplacementcontainsbesidesa
i i i i
arbitrary potential U(x) one can set thermalcontribution∝tαalsoaballistic-liketerm∝t2α.
Asa consequence,avalueα<1 doesnotnecessarilyim-
fi =(κα/∆x2)exp[−β(Ui+1/2−Ui)], (4a) ply subdiffusivebehavior. Infact,inthe presenceofbias
gi =(κα/∆x2)exp[−β(Ui−1/2−Ui)]. (4b) for 1/2<α<1 superdiffusion takes place.
ForafinitebiasF,theballisticterminEq.(9b)equals
Here Ui ≡ U(i∆x) and Ui±1/2 ≡ U(i∆x±∆x/2), with zero only in the case α = 1, for which normal Brownian
U(x) the total potential, β = 1/k T is the inverse of motion is recovered. From Eqs. (9) one obtains then a
B
temperature, and κ is the anomalous diffusion coeffi- generalized nonlinear Einstein relation, which is nonlin-
α
cient with dimension cm2s−α. The form (4) of the frac- ear in force and valid for a finite space step ∆x,
tional rates ensures that the Boltzmannrelationis satis-
hδx2(t)i−hδx2(0)i
fied, fi−1/gi =exp[β(Ui−1−Ui)]. =∆xcoth(Fβ∆x/2). (10)
Byuse ofthe Laplace-transformmethodonecanshow hx(t)i−hx(0)i
that the FME (3) can be brought into the form [15]
In the limit F → 0, Eq. (10) yields the well-known
D∗αPi(t)=fi−1Pi−1(t)+gi+1Pi+1(t)−(fi+gi)Pi(t), (5) Einstein relation (α = 1), κα/µα(0) = 1/β, between the
thermal diffusion coefficient
where the symbol Dα on the l.h.s. denotes the Caputo
∗ hδx2(t)i
fractional derivative [14], κ =Γ(α+1) lim F=0 (11)
α t→∞ 2tα
1 t 1 ∂
D∗αχ(t)= Γ(1−α)Z0 dt′(t−t′)α∂t′χ(t′). manodbitlhiteylµinαe(aFr)m=obvαil(itFy)/µFα(Fbei=ng0r)e,lawteitdhttohtehneoannlionmeaar-
lous current v (see below), i.e.
Let us introduce the finite difference operator ∆/∆x, α
∆P(x,t)/∆x = [P(x+∆x/2,t)−P(x−∆x/2,t)]/∆x,
hx(t)i
which in the limit ∆x → 0 yields the partial derivative µα(F)=Γ(α+1)tl→im∞ Ftα . (12)
operator ∂/∂x. Using the fractional rates (4) the FME
(5) can now be rewritten as ThesameEinsteinrelationisvalidalsobetweenthesub-
mobility and the subdiffusion coefficient for any α < 1
DαP(x ,t)=κ ∆ e−βU(xi) ∆ eβU(xi)P(x ,t) , (6) [17], as the ballistic term in the mean square displace-
∗ i α∆x(cid:18) ∆x i (cid:19) ment (9b) vanishes for F =0.
However, a relation analogous to the generalized non-
where P(xi,t) = P(i∆x,t) = Pi(t)/∆x. By taking the linear Einstein relation (10) ceases to be valid for α< 1
continuous limit in Eq. (6) one obtains the FFPE, for any finite F, as the mean square displacement be-
comesdominatedbytheballisticcontributioninthelong
∂ ∂
DαP(x,t)=κ e−βU(x) eβU(x)P(x,t) , (7) timelimit. Instead,fromEqs.(9)oneobtainsthefollow-
∗ α
∂x(cid:18) ∂x (cid:19) ing asymptotic scaling relation,
which can be rewritten in the well-known form with the hδx2(t)i 2Γ2(α+1)
Riemann-Liouvillefractionalderivativeonther.h.s[1,7], lim = −1. (13)
t→∞ hx(t)i2 Γ(2α+1)
∂P(x,t)=κ Dˆ1−α ∂ e−βU(x) ∂ eβU(x)P(x,t) . (8) This result no longer contains the fractional transition
∂t α 0 t ∂x(cid:18) ∂x (cid:19) rates and holds true independent of the strength of the
biasF andthetemperatureT. Therelation(13)hasbeen
Biased diffusion. For a constant force F the potential obtainedin[18]foracontinuous-timerandomwalkerthat
reads U(x) = −Fx and the fractional rates (4) become is exposed to a constant force. As a main finding of this
site-independent,fi ≡f andgi ≡g,satisfyingtheBoltz- work we prove below that this very relation holds true
mannrelation,i.e.,f/g=exp(βF∆x)foranyfinitevalue universally for the nontrivial case of tilted nonlinearly
of ∆x. Using the Laplace-transform one finds the solu- corrugatedpotentials.
tions of Eq. (5) for the mean particle position and the Tilted periodic potentials: fractional Fokker-Planck
mean square displacement of anomalous biased Brown- current. We next study the case of a periodic poten-
ian motion [16], tial with period L in the presence of a constant force F.
Towardsthis goal,it is convenientto considerthe FFPE
hx(t)i=hx(0)i+∆x(f −g)tα/Γ(α+1), (9a)
(7), in which the Caputo derivative appears only on the
l.h.s. AnalogouslytothecaseofnormalBrownianmotion
hδx2(t)i = hδx2(0)i+∆x2(f +g)tα/Γ(α+1) (9b) [4, 5, 19] one obtains that the probability flux
+ 2 − 1 ∆x2(f −g)2t2α. J (x,t)=−κ e−βU(x) ∂ eβU(x)P(x,t) (14)
(cid:18)Γ(2α+1) Γ2(α+1)(cid:19) α α ∂x
3
reaches asymptotically the stationary current value, i.e., 1
k T =0.01
∗ B
Dαhx(t)i=LJα =vα(F)=µα(F)F . (15) 0.75 k T =0.1
αα B
κκ
The anomalous current vα(F) is given in closed form by //αα 0.50 kBT =0.5
µµ
κ L[1−exp(−βFL)] TT
α BB
vα(F)= L x+L . (16) kk 0.25
dx dy exp(−β[U(x)−U(y)])
0 x
R R 0
This result constitutes a first main result: It is the
anomalous counterpart of the current known for normal 0.05 0.1 0.2 0.5 1 2
diffusion. It obeys a form that mimics the celebrated FF//FF
ccrr
Stratonovich formula for normal diffusion with α = 1
[4, 19, 20]. From Eq. (15) the mean particle position
FIG. 1: (color online). The scaled nonlinear mobility
follows as kBTµα(F)/κα is depicted for the case of a tilted cosine po-
tentialversus F/Fcr. Thenumericsfordifferenttemperatures
hx(t)i=hx(0)i+v (F)tα/Γ(α+1). (17)
α T anddifferentα-values,varyingbetween0.1−1, fitsthean-
alyticprediction(16)(continuouslines) withinthestatistical
Universal scaling in washboard potentials. We next show
errors.
that the relation in (13) is valid as well for anomalous
transportinwashboard-likepotentials. We provethis by
mapping the dynamics onto an equivalent CTRW; i.e., 22ii 1 Eq. (13)
))
tt
we consider a discrete state reduction of the continuous x(x( U (x)
diffusion process x(t): To this aim, we introduce a lat- /h/h 0.75 1
tice with sites {xˆ = jL}, located at the minima of the 22)i)i U2(x)
j tt
periodic part of the potential, and study the RTD ψˆj(τ) hδx(hδx( 0.50 U3(x)
for the hopping process between sites {xˆ }. For such a ∞∞
system, the ratio qˆ+/qˆ− = exp(βFL) equjals that of the →→ 0.25
j j mmtt
constant force case, due to the choice ∆x =L. Further- lili 0
more, the analogy between the solutions (9a) and (17), 0 0.2 0.4 0.6 0.8 1
both exhibiting an asymptotic power law ∝ tα, implies αα
the same form ψˆ (τ) ∝ 1/τ1+α for τ → ∞. In fact, for
j
ψˆ (τ) ∼ ανˆ−α/Γ(1−α)τ1+α, with some suitable scal- FIG. 2: (color online). Universal scaling: Asymptotic values
injgcoefficienjtsνˆ ,thecorrespondingkernelsoftheGME oftheratio hδx2(t)i/hx(t)i2 as afunction of theparameter α
j
obey K˜±(s) = qˆ±νˆαs1−α in the limit s → 0. Therefore, for anomalous diffusion. All the points corresponding to the
j j j same α but different values of F and T match the function
by making use of Tauberian theorems for the Laplace-
givenin Eq.(13)(solid line) within thestatistical errors. We
transform [11], it follows that the asymptotic solution
use three different temperatures: T = 0.01 with the bias F
(t→∞)isofthe form(9),beingdeterminedonlybythe ranging between 0.9 − 2.0; correspondingly, T = 0.1,F =
asymptotic power law behavior of the RTD [21], despite 0.7−2.0 and T = 0.5,F = 0.4−2.0. The open triangles
the fact that the values of fˆand gˆdepend onthe chosen correspond to the cos-potential U1(x), the filled squares to
shape for the periodic potential. Because the result in the double-hump potential U2(x) and the open circles to the
(13) is independent of fˆand gˆ, the scaling relation thus asymmetric ratchet potential U3(x),see in text.
still holds true. It is universal in the sense that it holds
independentlyofthedetailedshapeofthewashboardpo-
tential, the temperature T and the bias strength F. maxima. For a given temperature T, all values of µα
Numerical verification of universal scaling. We have with α taken from the interval, 0.1− 1, coincide with
numericallytestedthescalingrelation(13)andthegener- Eq. (16) (continuous lines). For tilting forces that ex-
alized Stratonovich formula (16) through the simulation ceed F/Fcr = 2 the dynamics approaches the behavior
of the fractional CTRW in tilted washboard potentials of a free CTRW being exposed to a constant bias. We
of various shapes and for different parameter values for further note that the regime of linear response at low
α, F, and dimensionless T. In doing so, we not only in- temperatures is numerically not accessible. This is so
vestigatethearchetypecaseofasymmetricsimplecosine because in this parameter regime the corresponding es-
potentialU (x)=cos(x),butaswellasymmetricdouble cape times governing the anomalous fluctuation-assisted
1
hump potential U (x) = cos(x)+cos(2x) and an asym- transport become far too large [20].
2
metric,ratchet-likepotentialU (x)=sin(x)+sin(2x)/3. The universalscaling in tilted corrugatedperiodic po-
3
In Fig. 1 we depict the scaled nonlinear mobility tentialsisillustratedwithFig.2,inwhichtheasymptotic
k Tµ (F)/κ defined by Eq. (12) for the cosine poten- ratiohδx2(t)i/hx(t)i2 isplottedversusαforthethreedif-
B α α
tial. TheforceisinunitsofthecriticaltiltF ,whichcor- ferent periodic potentials mentioned above. For a given
cr
responds to the disappearance of potential minima and α various data are presented, corresponding to different
4
potential shapes and values of F and T. As one can de- as averagesover 104 trajectories.
duce, these points overlap, demonstrating that the ratio
Conclusions. We have investigated the CTRW with
is independent of bias and temperature, as well as the
power-law distributed residence times in a periodic po-
specific shape of U(x). At the same time, the data fit
tentialinthepresenceofanexternalbias. Thefractional
verywellwith the analyticalexpression(13)(continuous
Fokker-Planck dynamics has been derived from the cor-
line).
respondingspace-inhomogeneousCTRWinanovelman-
Asdetailedabove,thelongtimebehaviorofthesystem
ner. ThecelebratedStratonovichsolutionforthestation-
is determined only by the tail of the RTD. Since we are
ary currentin a tilted periodic potential has been gener-
interested in the asymptotic behavior (t →∞), we have
alized to the case of anomalous transport. Moreover,we
usedinthenumericalsimulationstheParetodistribution
haveproventhatthereexistsauniversalscalinglaw(13)
(0<α<1),
— relating the mean square displacement and the mean
particle position in washboard potentials — that does
αbν d 1
ψ (τ)= i =− . (18) not involve the exact form of the periodic potential, the
i (1+bν τ)1+α dτ (1+bν τ)α
i i applied bias F, and the temperature T. This universal
scaling has been verified by numerical simulations.
This distribution, with b = Γ(1 −α)1/α, has precisely
Ourfindings forthe currentandthe meansquarefluc-
the sameasymptotic formas the Mittag-Leffler distribu-
tuations can readily be applied to the diverse physical
tion (2). For α=1 correspondingto a normalBrownian
situations mentioned in the introduction. The versatile
process we have employed the exponential RTD ψ (τ)=
i
and widespread use of the grand Stratonovich result for
ν exp(−ν τ). In our simulations we have assumed a
i i
spacestep∆x≪LsuchthatU′′(x)∆x≪2U′(x),ensur- thestationarycurrentincaseofnormaldiffusioncanthus
readily be put to powerful use in all those multifaceted
ingthesmoothnessoftheperiodicpotential. Eachtrajec-
applications where the corresponding transport behaves
tory was assigned the same initial condition x(t ) = x .
0 0
anomalously.
Then a residence time τ was extracted randomly from
the RTD (18), time was increasedto t =t +τ and the We thank E. Barkai, J. Klafter, R. Metzler and S.
1 0
particle was moved either to the right or left site with Denysovforconstructivediscussions. Thisworkhasbeen
respective probabilities q+ and q−. Using this procedure supported by the ESF STOCHDYN project and the Es-
i i
wecomputedthefullrandomtrajectoryoftheBrownian tonianScienceFoundationthroughgrantno. 5662(EH),
particle (103 time steps at least). The mean displace- the DFG via research center, SFB-486, project A10, the
ment and the mean square displacement were obtained Volkswagen Foundation, via project no. I/80424.
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