Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

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by  Eli Barkai

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
E. Barkai1
1Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel∗
Weinvestigatestatisticsofoccupationtimesforanover-dampedBrownianparticleinanexternal
6 forcefield. AbackwardFokker-PlanckequationintroducedbyMajumdarandComtetdescribingthe
0 distributionofoccupation timesissolved. Thesolution givesageneralrelationbetweenoccupation
0 timestatisticsandprobabilitycurrentswhicharefoundfromsolutionsofthecorrespondingproblem
2 offirstpassagetime. Thisgeneralrelationship betweenoccupation timesandfirstpassage times,is
validfornormalMarkoviandiffusionandfornon-Markoviansub-diffusion,thelattermodeledusing
n
the fractional Fokker-Planck equation. For binding potential fields we find in the long time limit
a
J ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking
isfound,inagreement withpreviousresultsofBelandBarkaionthecontinuoustimerandomwalk
8
on a lattice. For non-binding potential rich physical behaviors are obtained, and classification of

] occupationtimestatisticsismadepossibleaccordingtowhetherornottheunderlyingrandomwalk
h is recurrent and the averaged first return time to the origin is finite. Our work establishes a link
c between fractional calculus and ergodicity breaking.
e
m PACSnumbers:
-
t
a
I. INTRODUCTION Statistics of occupation times is of-course not limited to
t
s Brownian motion and diffusion, and it is a topic of wide
.
t investigation [1], for example in the context of renewal
a Consider the trajectory of a single Brownian parti-
m cle. The total time the particle spends in a given do- processes [9], theory and experiments of blinking quan-
tum dots [10, 11], weak ergodicity breaking of dynamics
- mainis calledthe residencetime orthe occupationtime.
d generatedusingdeterministicmaps[12],andworkdistri-
The well known example is P. L´evy’s arcsine law [1, 2].
n bution functions of a single spin [13].
Consider a Brownian motion x˙(t) = η(t) where η(t) is
o
Gaussian white noise with zero mean. L´evy investigated The problemof occupationtimes ofa Brownianparti-
c
[ the residence time of the particle in the domain x > 0, cleinthepresenceofexternalfieldwasconsiderrecently,
which we call T+, when the motion is unbounded and byMajumdarandComtet[14]. UsingtheKacformalism
1 the totalobservationtime is t. Naive expectationis that [1]theyfoundabackwardFokker–Planckequationwhose
v
T+/t=1/2withsmallfluctuationswhent ,namely solution yields statistics of occupation times. In [14] the
3
4 theparticleoccupiesthedomainx>0forh→alf∞ofthetime problem of occupation time statistics of a particle per-
1 of observation. However, instead the probability density forming a random walk on a random walk i.e. the Sinai
1 function of T+/t is given by the well known arcsine law, model was investigated. It was shown that statistics of
0 f(T+/t)=[π (T+/t)(1 T+/t)] 1with0 T+/t 1. occupation times are drastically changed when averages
6 − − ≤ ≤ over random disorder are made.
ThisprobabilitydensityhasaUshape,whichmeansthat
0 for a typical rpealization of the Brownian trajectory, the In the first part of this manuscript we consider the
/
t particlespendsmostofthetime inonehalfofspace(say problem of occupation times for normal Brownian mo-
a
m x>0) and not in the other (x<0). tioninanexternalfields. We solveexactlythe backward
Fokker–Planckequationgivenin[14]. Thissolutiongives
- Many extensions of this well known result are found
d in the literature. Darling and Kac [3, 4] found the lim- ageneralrelationbetweenoccupationtimeandfirstpas-
n sage time statistics. Besides the theoretical interest in
iting distribution of the time spent in a domain in two
o such a relation, the solution is used to classify very gen-
dimensions, and this line of investigation was extended
c
eral behaviors of occupation times based on the corre-
: tothreedimensionbyBerezhkovskiietal[5]. Lamperti’s
v sponding properties of the first passage times. The later
[6]limittheoremgivesaverygeneralmathematicalfoun-
i are investigated in great detail in the literature [2], and
X dation for occupation time statistics (see more details
wecanusethis knowledgeto solveanalyticallythe prob-
in the manuscript). Recently in [7] Pearson’s type of
r
a ballistic motion with random reorientation was consid- lem of occupation times at-least for some simple cases.
For example we show that in the limit of long measure-
ered, instead of the usual assumption of an underlying
ment times, and for binding force fields, statistics of oc-
continuum process. The basic mathematical theory for
cupation times is determined by Boltzmann’s statistics,
thecalculationofoccupationtimestatisticsforBrownian
namely the underlying dynamics is ergodic, as expected.
motion was developed by Kac, and is usually based on
the Feynmann-Kac formula (see [1, 8] and Ref. therein). Statistics of occupation time is important from a fun-
damental point of view, since if we are able to calcu-
late statistics of occupation times from some underlying
dynamics, one can check the validity of the ergodic hy-
∗Electronicaddress: [email protected] pothesis and its possible extensions. A trivial example
2
is Gaussian Brownian motion in a system of finite size teracting particles is
0 < x < L, in the absence of external force fields. Then
it is easy to show that the residence time in half of the ∂c(x,t) ∂2 ∂ F(x)
=D c(x,t), (1)
system i.e. in the domain (0,L/2), is in statistical sense ∂t ∂x2 − ∂x k T
(cid:20) b (cid:21)
halfoftheobservationtime,whentheobservationtimeis
where T is the temperature and D is the diffusion coef-
long, as expected. For dynamics described by fractional
ficient. As well known the equilibrium of the ensemble
kinetic equations [15, 16], we show that such a simple
of particles is the Boltzmann equilibrium, provided that
ergodic picture does not hold.
the force field is binding.
Inthesecondpartofthepaperweconsidertheproblem
Consider a single particle, which at time t = 0 is on
of a particle undergoing an anomalous diffusion process.
x , the observation time of the stochastic dynamics is t.
WemodelthisbehaviorusingthefractionaltimeFokker- 0
The randomvariableweinvestigatehereis T+, the total
Planck equation [17, 18]. This fractional framework is
time the particle occupies the region x>0. In principle
based on fractional calculus e.g. d1/2/dt1/2, which is
during the observation time the particle may cross the
brieflyintroducedinthemanuscript. Weshowforexam-
point x = 0 many times, and then the occupation time
ple, that the general relation between occupation times
T+ is composed of many sojourn times in x>0.
andfirstpassagetimeswefindinthefirstpartofthepa-
Let P (T+) be the probability density function
perisstillvalid,evenforthenon-Markoviansub-diffusive x0,t
(PDF) of T+. The double Laplace transform
case. Similar to normal diffusion case a classification of
typical behaviors of occupation times is found, and ana-
lyticalsolutionsprovided. For dynamicsin binding force Px0,s(u)= ∞ ∞e−ste−uT+Px0,t(T+)dtdT+, (2)
fields we find weak ergodicity breaking. In the conclu- Z0 Z0
sions we compare our results on occupation times found isdefinedsothatsandtanduandT+ areLaplacepairs.
here using the fractional framework, and recent results
MajumdarandComtet[14]foundtheequationofmotion
ofBeland Barkai[19, 20]onstatistics of residencetimes for P (T+) in double Laplace space
for continuous time random walks. x0,t
For applications, residence times are of interest in the ∂2 F (x ) ∂
0
D + P (u) [s+Θ(x )u]P (u)= 1.
contextofchemicalreactions[21, 22,23]andrathergen- ∂x2 k T ∂x x0,s − 0 x0,s −
erally for statistical analysis of experimental data. Res- (cid:20) 0 b 0(cid:21) (3)
idence times are very important in the context of single Where Θ(x ) is the step function: Θ(x ) = 1 if x > 0
0 0 0
molecule dynamics [24, 25]. It is now possible to fol- otherwise it is zero. This type of equation is called a
lowdynamicsofsinglemoleculesembeddedincondensed backward Fokker–Planck equation, the operator on the
phaseenvironments,usingopticaltechniques. Forexam- left hand side depends on the initial condition x . Eq.
0
pledynamicsofsinglemoleculesincellsorinsolutionare (3) is solved for the matching boundary conditions
usedtofollowchemicalreactionsinrealtime,withoutthe
problem of ensemble averaging found in usual measure- P (u) =P (u) ,
x0,s |x0=0+ x0,s |x0=0−
ments. A typical experiment uses a laser to investigate
thedynamicsofaparticle. Inmanycasesandundercer-
∂P (u) ∂P (u)
tain conditions [24] if a particle or a reaction coordinate x0,s = x0,s . (4)
isinafinitedomain,thesystemmayemitphotons,while ∂x0 |x0=0+ ∂x0 |x0=0−
when the particle is out of the domain the system does
We will re-derive Eq. (3) later as a special limiting case
notemit. Verybriefly,thedomainwidthcanbeimagined
of a more general non-Markoviandynamics.
asthe widthofthe laserbeamin singlemolecule fluores-
To prepare for the solution of Eq. (3) we define the
cence experiments whenthe particle comesin andoutof
following survival probabilities. The probability that a
resonance with the exciting laser field, due to its diffu-
particle starting at x with x <0, to remain in the do-
sioninspace e.g. [26], or itcouldbe the Fo¨ster radiusin 0 0
mainx<0withoutleavingitevenonce,duringthetime
fluorescence resonance energy transfer measurement e.g.
t is the survival probability W (t). Let W (s) be the
[27]. Thus the total time the photons are emitted is ap- x−0 x−0
Laplace transform of W (t) and similarly the Laplace
proximately the residence time, which is proportional to x−0
transformofthesurvivalprobabilityinthedomainx>0
the numberofemittedphotons,whichisgenerallyaran-
is W+(s) for x > 0. The key to the solution of the
dom variable. For other sources of fluctuations in single x0 0
problem of occupation times in half space, is to recall
molecule experiments see [24].
Tachiya’s equation for the survival probability of a par-
ticle in half space [28, 29]
II. NORMAL DIFFUSION ∂2 F (x ) ∂
0
D + W (s) sW (s)= 1 x <0,
∂x2 k T ∂x x−0 − x−0 − 0
(cid:20) 0 B 0(cid:21)
Consider a one dimensional over-damped Brownian (5)
motioninanexternalforcefieldF(x). TheSmoluchowski and a similar equation holds for x > 0. The bound-
0
Fokker–Planck equation for the concentration of non in- ary conditions for Eq. (5) are the standard conditions
3
usedforthecalculationofsurvivalprobabilities. Namely, previously in the literature [5, 8, 20]. Finally, while we
W (s) = 0, means that the particle reaches the consideredthe occupationtime inhalfspace,occupation
x−0 |x0=0
boundary on x =0 instantaneously if the particle starts times in a finite domain are also obtained in a similar
very close to the absorbing boundary and if x way,and itis straightforwardto extended our resultsto
0
→ −∞
survival is unity. higher dimensions.
The solution of Eq. (3) for the occupation time is Majumdar and Comtet [14] classify statistics of occu-
pationtimes accordingto behavior ofthe potential field,
Px0,s(u)=Wx−0(s)+ 1−sWx−0(s) Gs(u) inparticulartheyconsidermotioninstable,unstableand
flat potential fields. Here the relation between survival
if x <0, (cid:2) (cid:3)
0 currents and statistics of occupations times, Eq. (7) can
P (u)=W+(s+u)+ 1 (s+u)W+(s+u) G (u) be used to characterized certain very general and new
x0,s x0 − x0 s behaviors of occupation times.
(6)
(cid:2) (cid:3) Survival probabilities in a finite and infinite domain
if x > 0. From Eq. (6) the physical meaning of
0
exhibit three well known typical physical behaviors [2],
G (u) becomes clear, it is the double Laplace trans-
s
form of G (T+) the PDF of the random variable T+ weconsidertherightrandomwalk(i.e. x0 >0)andsimi-
t
for a particle starting on x = 0. The PDF G (T+) larclassificationholdsfortheleftrandomwalk. Laterwe
0 t
will classify behaviors of residence times based on these
contains the information on the problem of occupation
three behaviors of first passage times.
times,whilethesurvivalprobabilitywasinvestigatedpre-
Case 1 The random walk is recurrent, and the average
viously by many authors, hence in what follows we in-
vestigate G (T+). Using Eq. (5) the reader can verify first passage time from x0 to 0 is finite. Such cases cor-
t
respond to diffusion in a system of finite size, when the
that Eq. (6) is indeed the general solution of the prob-
particle cannot escape to infinity, e.g. the driving force
lem of occupation times Eq. (3). Using the boundary
condition Wx−0=0(s) = Wx+0=0(s) = 0 and the solution fiCealdseis2biTnhdeinrga.ndom walk is transient, i.e. the survival
Eq. (6) it is easy to see that the boundary condition
probability in x > 0 is finite in the limit of long times.
P (u) = P (u) = G (u) in Eq. (4) is
x0,s |x0=0+ x0,s |x0=0− s Such cases happen when the external force drives the
satisfied.
particlefarfromthe origin,andthesystemisinfinite. In
The second matching boundary condition in Eq. (4),
that case in the limit of small s
on the derivatives of P (u) yields G (u) using Eq. (6)
x0,s s
ǫ+
Gs(u)= (s+Ju+)J(s++(su+)−u)J−s(sJ) (s) (7) Wx+0 ∼ sx0 (9)
− −
whereǫ+ isthesurvivalprobabilityoftheparticlestart-
where the currents are x0
ing on x , without reaching x=0,when t . Similar
0
→∞
J+(s+u)= ∂Wx+∂0(xs+u)|x0=0+, J−(s)= ∂W∂xx−0(s)|x0=0−. Cnoatsaetio3nRiasnudsoedmfworaltkhsealreeftrercaunrdroemnt,wtahloku,gwhitthheǫ−xa0v.erage
0 0
(8) firstpassagetimeisinfinite. Aparticularlycommonsitu-
Eqs. (6, 7) are the main results so far since they yield ationisthecasewhenthesurvivalprobabilitydecayslike
a general relation between statistics of occupation times t−1/2 for long times. This happens if the non-diverging
and survival probability currents. From Eq. (8) we see external field F(x) = 0 for x > xc and the system is in-
that the solution of the problem of occupation times is finite, namely when diffusion controls the long time dy-
foundintermsoftwosolutionsofthe correspondingfirst namics. For such a case [2]
passagetimeproblems,thefirstforaparticlestartingon
A+
x0 > 0 and absorbed on x = 0 (i.e. J+) and the second W+ x0, s 0, (10)
for a particle starting on x < 0 and absorbed on x = 0 x0 ∼ s1/2 →
0
(i.e. J ). Thus the problem of residence times is solved
− where A+ > 0 depends of-course on the details of the
in three steps: x0
force field.
(i) Find solutions of two first passage time problems for,
We now consider certain generalproperties of the statis-
x >0 and x <0 in Laplace space.
0 0 tics of occupation times for the three cases.
(ii) Use Eq. (7) to find the solution of the problem of
Case 1 The longtime behaviorofG (T+)is nowcon-
residence times in double Laplace space. t
sidered. When the left and right random walks, starting
(iii) And then use a two dimensional inverse Laplace
at x < 0 or x > 0, respectively, are recurrent and the
transform to get G (T+) from G (u). 0 0
t s averagefirstpassagetimeisfinite. Forthiscasethesmall
Sincethereexistsavastliteratureonthesolutionsofthe
s limit yields
problemoffirstpassagetime[2],therelationshipEq. (7)
itsiovnertyimuesse.fuWlefonrottheethcaatlcsuolamteiocnononfescttaitoinstsicbsetowfeoecncufiprast- Wx±0(s=0)=ht±x0i, (11)
passage times and occupation times, which are different where t is the average time for the particle starting
h ±x0i
and in our opinion less general than Eq. (8), appeared on x < 0 (or x > 0) to reach the origin for the first
0 0
4
time. The small s and u limit, their ratio arbitrary, of Transformingtothetimedomainwefindtheasymmetric
Eq. (7) gives the long t and T+ behavior of G (T+), we arcsine PDF [6]
t
find
1 T+
1 G (T+) f (20)
Gs(u)∼ ∂ht+x0i . (12) t ∼ t (cid:18) t (cid:19)
s+u ∂x0 |x0=0
∂ht+x0i ∂ht−x0i where
∂x0 |x0=0− ∂x0 |x0=0
1
The differential equation for t+ is well known and is f(x)= R . (21)
h x0i πx1/2(1 x)1/2[ 2(1 x)+x]
obtained from the small s expansion of Eq. (5) − R −
∂2 F(x ) ∂ When =1wefindthearcsinelaw. NotethatthePDF
D t+ + 0 t+ = 1. (13) Eq. (2R0) diverges on T+/t = 1 and T+/t = 0, hence
∂x2h x0i k T ∂x h x0i −
(cid:20) 0 B 0 (cid:21) events where the particle always occupies (or hardly
Solving this equation, using a similar equation for t , never occupies)the domain x>0 have a significantcon-
and inverting Eq. (12) to the time domain we findh −xt0hie tribution.
expected ergodic behavior Another general result obtained from the small u ex-
pansion of Eq. (7) is for the average occupation time
G (T+) δ T+ P+t , (14)
t ∼ − B J+(s)
where PB+ is Boltzmann’s pr(cid:0)obabilityo(cid:1)foccupying x>0 hT+i=L−s→1t(cid:26)s2[J+(s)−J−(s)](cid:27) (22)
U(x)
PB+ = 0∞e−ZkBTdx, (15) twehnetriaelLfi−se→l1dtiissbtihnediinngvearnsedLthaeplraacnedtormanwsfaolrkmi.s rIefctuhrerepnot-,
R
then T+ P+t. Similar relations between higher or-
tZio=n a−n∞∞d eUx(px−)UkiBs(xT)thdex ibsinthdeinngorpmotaelniztiinagl,pwaritthitioFn(xfu)nc=- dperorbmahboimliteiine∼tssaorBef tohbetaoincceudpiantiaonsitmimilaers wanady.thIne Ssuecrv.ivIaVl
R
dU(x)/dx. we consider several particular examples, which explain
−
Case 2 We consider the case where both the left and in greater detail the meaning of the general results ob-
the right random walks are non recurrent. The survival tained in this section. First we generalize our results for
probabilities in the two domains are ǫ+x0 and ǫ−x0, in the fractional dynamics.
longtimelimit. ThenusingEqs. (7,9)wefindfort
→∞
Gt(T+)∼α−δ(T+)+α+δ(T+−t) (16) III. ANOMALOUS DIFFUSION
where
Anomalous diffusion and relaxation is modeled based
∂ǫ+
x0 on the fractional time Fokker–Planck equation (FFPE)
α+ = ∂x0 |x0=0 (17) [17, 18], the concentration of non-interacting particles
∂∂ǫx+x00|x0=0− ∂∂ǫx−x00|x0=0 obeys
and α− = 1 α+. Since the particle always manages ∂αc(x,t) ∂2 ∂ F(x)
to escape eith−er to the left or to the right, eventually ∂tα =Dα ∂x2 − ∂x k T c(x,t), (23)
the particle will either reside in the left domain or the (cid:20) b (cid:21)
right domain forever, hence the delta functions in Eq. where D is a generalized diffusion coefficient and 0 <
α
(16). The weights of these delta functions are given by α < 1. A brief mathematical introduction to the FFPE
the derivatives of the survival probabilities only. is given in Appendix A. We recall physical properties
Case 3 We now consider a case where both the left of the FFPE. (i) when F(x) = 0 and for free bound-
and the right random walks are recurrent, though the ary conditions we have the fractional diffusion equation
average first return time from x0 to x = 0 is infinite, in [30, 31, 32, 33, 34] with anomalous diffusion x2
such a way that Eq. (10) is valid. Then in the small s tα. (ii) In the presence of a binding time indephendien∝t
and u limit force field the equilibrium is the Boltzmann distribution
s 1/2+ (s+u) 1/2 [17, 18]. (iii) Generalized Einstein relations are satis-
− −
G (u) R (18) fied in consistency with linear response theory [17, 18].
s ∼ s1/2+ (s+u)1/2
R (iv) Relaxation of modes follows the Mittag Leffler de-
where the asymmetry parameter is cay, related for example to Cole-Cole relaxation[17, 18].
(v) In the limit α 1 we recover the standard Smolu-
∂A+x0 chowski Fokker-Pla→nck equation. The FFPE is derived
= ∂x0 |x0=0. (19) from the continuous time random walk [18]. Its mathe-
R −∂A−x0 matical foundation is P. L´evy’s generalized central limit
∂x0 |x0=0
5
theorem applied to the number of steps in the underly- and a similar equation holds for x < 0. Eq. (25) is
0
ing random walk [35, 36]. A very general solution of the the fractional generalization of Tachiya’s Eq. (5). The
FFPE in terms of the solution of the standard α = 1 derivation of Eq. (25) is based on results obtained in
Fokker-Planck equation was given in [35] (i.e., subordi- [35] and is simple once the sub-ordination trick is used
nation, and the inverse L´evy transform). Recently there (see some details in Appendix A). Now using Eq. (25)
issomecontroversyonhowtoapplyboundaryconditions it is easy to verify that Eqs. (6,8) are solutions of the
[37, 38, 39] for the anomalous case. Applications of frac- fractional Eq. (24).
tionaldiffusionmodelinginclude: Scher-Montrolltimeof
flight transport of charge carriers in disordered medium
[35], dynamics of ion channels [40], relaxation processes A. Derivation of Fractional Equation for
in proteins [41], anddielectric relaxation[42], and deter- Occupation Times
ministicdiffusion[32]. Forareviewandapopulararticle
on fractional kinetics see [15, 16]. In this subsection we derive our main result Eq. (24)
Similar to the normal diffusion case, we define using the assumption that the underlying dynamics is
P (T+) as the PDF of T+ and P (u), its double described by the fractionalFokker-Planckequation(23).
x0,t x0,s
Laplace transform. As we show in next subsection the The latter describes long time behavior of the continu-
differential Eq. for P (u) for the dynamics described ous time randomwalk (CTRW), which is the underlying
x0,s
by the FFPE Eq. (23) is random walk process we have in mind. In the CTRW a
particle performs a one dimensional random walk on a
∂2 F (x ) ∂
D + 0 P (u) lattice, with jumps to nearest neighbors only and with
α(cid:20)∂x20 kbT ∂x0(cid:21) x0,s − random waiting times between jumps. In CTRW the
waitingtimesbetweenjumpsareindependentidentically
[s+Θ(x0)u]αPx0,s(u)=−[s+Θ(x0)u]α−1. (24) disisrternibeuwteeddaraftnedroemacvharjuiamblpe.s,Tnahme ePlyDFtheofCwTaRitWingprtoimceesss
TheboundaryconditionsforEq. (24)areidenticaltothe is ψ(t). Two classes of CTRWs are usually considered,
normaldiffusion case α=1 givenin Eq. (4). Eq. (24) is the case when the averagewaiting time is finite, and the
importantsinceaspointedoutin[19], fractionaldynam- case when ψ(t) t (1+α) when t and 0 < α < 1.
−
ics is weakly non-ergodic [43], namely occupation time Thelattercasele∝adstoanon-statio→na∞rybehavior,aging,
statisticsisnotdescribedbyBoltzmannequilibriumeven anomalous diffusion and weak ergodicity breaking [43].
in the limit of long time and for binding potential fields. The lattice spacing is ǫ, eventually we will consider the
Thusthe FFPE(23)cannotbeusedtodescribetime av- continuum limit when ǫ is small. On each lattice point
erages of physical observable due to ergodicity breaking, we assign a probability for jumping left and a probabil-
and the interpretation of results derived from the FFPE ity of jumping right. This dynamics in the continuum
must be treated with care. Eq. (24) is a remedy for this limitleadsto behaviordescribedbytheFFPE,whende-
problemsinceaswewillshowityieldsafractionalframe- tailedbalanceconditionsareappliedonthe probabilities
work for the calculationof non-trivialdistribution of oc- for jumping left or right [18] (i.e. probabilities to jump
cupationtimes [i.e. generalizationof Boltzmann’sstatis- left or right are related to external force field and tem-
tics Eq. (14)]. Eq. (24) is afractionalbackwardFokker– perature). In what follows we start with some general
Planck equation in double Laplace space, formally one arguments,assumingonlyarenewalpropertyofthe ran-
may invert it to the time domain using material frac- domwalk,withoutlimitingourselvestoaspecificmodel.
tional derivatives [44], however in practice we solve this The random position of the particle is x(t). The total
equationindoubleLaplacespaceandonlythenmakethe time the particle spends on x 0 is T+, i.e. the occu-
inverse double Laplace transform. pation time of half space. The≥particle starts on x and
0
Interestingly the solution of the fractional Eq. (24) is assumethatx 0,laterwegeneralizeourresultstothe
0
identical to that found for normaldiffusion case, namely case x < 0. W≥e define the PDF of first passage times,
0
our main results Eqs. (6,7,8) are valid also in the non- from x to x = ǫ, as ψ+(t). The PDF of first passage
tMhaerckaolvciualnatdioonmoafinJ±0(<s),αis<th1e.LNapowlacWe xt±r0a(sn)sfnoremedeodf tfhoer wtimithesψfr0+o(mt)xan=d0−[ψt−o(xt)=] rx−e0sǫp(exct=ive−lyǫ.tox=0)isdenoted
survivalprobabilityforthefractionalparticle. ThusEqs. We assume that the first passage times PDFs ψ+(t)
(6,7,8)havesomegeneralvaliditybeyondnormalMarko- and ψ (t) do not depend on x and that sojourn times
− 0
vian diffusion. in domain x > 0 and x < 0 are statistically indepen-
To prove that Eqs. (6,8) are still valid we must first dent. Such assumption holds for Markovian dynamics
find the differential equation for Wx+0(s): the survival but is not obvious otherwise. For CTRW dynamics the
probability of a fractional particle starting on x0 > 0 assumptioniscorrect,sinceasmentionedtheCTRWpro-
in the domain x>0. We can prove that cesses is a renewal process. The process is mapped on a
two state process
∂2 F (x ) ∂
D + 0 W+ (s) sαW+ (s)= sα 1,
α ∂x2 k T ∂x x0 − x0 − − 1 x(t) 0
(cid:20) 0 B 0(cid:21) (25) θx(t)= 0 x(t)≥<0 (26)
(cid:26)
6
and hence T+ = tθ (t)dt. Since either the particle is is renewedonce the particle jumps fromx=0 to x= ǫ
0 x −
in the domain x<0 or not the dynamics is described by or vice versa.
R
a set of sojourn times
τx0,τ1−,τ2+,τ3−,τ4+···. serLveattifoxn0,tti(mTe+)isbte. tLheetPfDF(ouf)Tb+e twhheendotuhbeletoLtaaplloabce-
x0,s
Here the PDF of τx0 is ψx0(t), the PDF of τ1− is ψ−(t), transform of fx0,t(T+). A calculation, using methods of
the PDF of τ+ is ψ+(t), etc. All the sojourn times are renewaltheory,similartotheworkofGodrecheandLuck
2
assumedmutuallyindependent,whichmeanstheprocess [9] yields
1 ψ+(s+u) 1 ψ+(s+u) 1 ψ (s) 1
fx0,s(u)= − sx+0 u +ψx+0(s+u) ψ−(s) − s+u + − s− 1 ψ+(s+u)ψ (s). (27)
(cid:20) (cid:21) − −
Whereψ+(s+u)= texp[ (s+u)t]ψ+(t)dt isthe Laplacetransform. Ifψ+(t)=ψ+(t)=ψ (t)werecoveraresult
x0 0 − x0 x0 −
in [9]. If the particle starts on x <0 then one can show
0
R
fx0,s(u)= 1−ψsx−0(s) +ψx−0(s) ψ+(s+u)1−ψs−(s) + 1−ψs++(su+u) 1 ψ+(s1+u)ψ (s). (28)
(cid:20) (cid:21) − −
We now consider the case when the underlying dynamics is described by the FFPE. By definition the first passage
time PDFs are related to survival probabilities according to
t
W+(t)=1 ψ+(t)dt, (29)
x0 − x0
Z0
or using the convolution theorem in Laplace space
1 ψ+(s)
W+(s)= − x0 . (30)
x0 s
Hence we may rewrite Eq. (27)
1 ψ+(s+u) 1 ψ (s) 1
fx0,s(u)=Wx+0(s+u)+ 1−(s+u)Wx+0(s+u) ψ−(s) − s+u + − s− 1 ψ+(s+u)ψ (s). (31)
(cid:20) (cid:21) − −
(cid:2) (cid:3)
Noticethatf (u)depends onx onlythroughthesur- are identical. Using (27) we have
x0,s 0
vival probability W+(s+u). If we apply the backward
Fokker-Planckoperax0tor 1 ψ+(s+u)
G (u)= − +
s
(s+u)
∂2 F (x ) ∂ (cid:20)
0
D +
α ∂x2 k T ∂x
(cid:20) 0 B 0(cid:21)
1 ψ (s) 1
on this equation and use Eq. (25), namely we assume ψ+(s+u) − − . (33)
s 1 ψ+(s+u)ψ (s)
that the underlying dynamics is described by the FFPE (cid:21) − −
inthecontinuumlimit,weobtainatonceourmainresult Inthecontinuumlimitwehavethefollowingǫexpansion
Eq. (24). Similar method is used for the case x < 0 to
0
complete the proof. ∂ψ
We now derive our main Eq. (7) using the continuum ψǫ−(s)≃ψǫ−=0(s)− ∂xǫ−|ǫ=0ǫ+··· (34)
approximation. We consider the case when the particle
start on x =0 hence we have ψ+ =ψ+ and define andasimilarexpansionholdsforψ+(s). Wherethesub-
0 x0 script inEq. (34)isaddedtoemphǫasizethatthePDFof
ǫ
G (u)=f (u). (32) the first passage time from lattice point ǫ to the origin
s x0=0,s −
0. Notethatψǫ−=0(s)=1,sincethe particleontheorigin
Generally G (u) is not identical to G (u) and our aim isimmediatelyabsorbed. Insertingthe expansion(34)in
s s
now is to find the conditions when these two functions Eq. (33) and using a similar expansion for ψ+(s+u) we
ǫ
7
find that when ǫ 0 where the scaling function is
→
1 ∂ψ+(s+u) 1∂ψ−(s) δ ( ,p)
G (u) (s+u) ∂x0 |x0=0− s ∂x0 |x0=0 (35) α R ≡
s ∼ ∂ψ+(s+u) ∂ψ−(s)
∂x0 |x0=0− ∂x0 |x0=0 sinπα pα−1(1 p)α−1
Using Eqs. (30,35) we obtain our main result Eq. (7) π 2(1 p)2αR+p2α+2− (1 p)αpαcosπα.
andGs(u)isidenticaltoGs(u)inthe continuumlimitof R − R − (41)
ǫ 0.
→ This function is normalized according to
1
δ ( ,p)dp = 1. When α = 1 we find the er-
0 α R
godicbehaviorinEq. (14), while clearlyifα<1we find
IV. EXAMPLES
R
a non-ergodic behavior. The parameter is called the
R
asymmetry parameter. It can be calculated solving Eq.
A. Weak Ergodicity Breaking
(37), we find
tenWtiealnofiweldcoUns(ixd)e,rea.ng.omUa(lxou)s=dyknxa2m/2icswiinthakbin>di0n,gapnod- ∂ ∂τxx+0 α|x0=0+ = D1 ∞e−[U(x′)−U(0)]/kbTdx′, (42)
F(x) = dU(x)/dx. As we showed already the long (cid:0) 0(cid:1) α Z0
time beha−vior of G (T+), for normal diffusion, yields an
esurgbo-ddiiffcubseiohnaviinoraEbtqin.d(i1n4g).poRteesnitdieanlcfieeltdimweasstactoinstsiicdserfeodr ∂ ∂τxx−0 α|x0=0− =−D1 0 e−[U(x′)−U(0)]/kbTdx′.
previouslyin[19],usingthecontinuoustimerandomwalk (cid:0) 0(cid:1) α Z−∞ (43)
modelonalattice. HereweconsiderafractionalFokker- Using Eqs. (39,42,43) we find
Planckequationapproachshowingthatconceptsofweak
ergodicity breaking in [19] are valid also within the frac- P+
= B , (44)
tional framework. R 1 P+
− B
In[35]itwasshownthattherandomwalksinabinding
field are recurrent for α < 1, just like the normal case where P+ is Boltzmann’s probability of finding the par-
B
α=1. Using the subordination trick (see Appendix), or ticle in the domain x > 0 Eq. (15). Eqs. (40,44) were
analyzing Eq. (25) we find that for s 0 found previously in [19] using a different approach. One
→
can show that the averageoccupation time is
α
τ
Wx±0(s)∼ (cid:0)s1x±−0(cid:1)α . (36) hT+i∼PB+t (45)
When α=1 we have τ α = t and the behavior in and fluctuation are very large if α<1
x±0 h ±x0i
Eq. (11). For 0 < α < 1 we have in the time domain
Wx±0(t)∝t−α,reflectin(cid:0)gth(cid:1)elongtαailedtrappingtimesof hT+ 2i−hT+i2 ∼(1−α)PB+ 1−PB+ t2. (46)
theunderlyingCTRW.The τ areamplitudeswhich We inrtoduce a measure for ergodi(cid:0)city brea(cid:1)king the EB
x±0
satisfy parameter
(cid:0) (cid:1)
Dα(cid:20)∂x∂022 + Fk(BxT0)∂∂x0(cid:21) τx±0 α =−1. (37) EB = hT+ h2Ti−+ih2T+i2 ∼(1−α)1−PB+PB+, (47)
(cid:0) (cid:1)
This equation is obtained from the small s expansion of which is zero in the ergodic phase α=1.
Eq. (25). UsingEq. (36)andEq. (7)wefindinthelimit
of small s and u
(s+u)α 1+sα 1 B. Diffusion in an interval
− −
G (u) R (38)
s ∼ (s+u)α+sα
R We consider the case where the particle is free to dif-
where the asymmetry parameter is fuse in an interval of total length L++L . The particle
−
∂(τ+)α is initially on the origin x = 0 and reflecting boundary
R=−∂(∂τx−0)α|x0=0+. (39) cpornopdeitritoinessoafrTe+onthxet=imeL+speanntdinx(0=,L−+L)−ar.enSotwatiisntviceas-l
∂x0 |x0=0− tigated.
Invertingtothetimedomain,weseethatthePDFofT+ The survival probability can be calculated using Eq.
in the long time t limit is described by Lamperti’s limit (25)
theorem [6]
1 cosh Dsαα(L+−x0)
Gt T+ 1δα ,T+ . (40) W+(s)= − co(cid:2)sph DsααL+ (cid:3), (48)
∼ t R t x0 s
(cid:18) (cid:19) (cid:0)p (cid:1)
(cid:0) (cid:1)
8
and when α=1 we recovera text book result [2]. Using breaking is found. We see that the statistics of occupa-
Eq. (7) we find tion times exhibits a transition from a symmetric Lam-
perti PDF with index α/2 when diffusion is dominating
Gs(u)= the dynamics, i.e. for short times, to a generally non-
symmetric Lamperti PDF with index α, for long times
when the particle interacts with the boundaries. Such a
(s+u)α/2−1tanh (s+u√)Dα/α2L+ +sα/2−1tanh sα√/D2Lα− . traInfsLitionisnowthliimleitLe+dtroemfraeeindsifffiunsitioenaadsiwffeersehnotwbelahtaevr-.
(s+u)α/2tanhh(s+u)α/2L+i+sα/2tanh s(cid:16)α/2L− (cid:17) ior is f−ou→nd∞. Now the particle can be found either in a
h √Dα i (cid:16) √Dα (cid:17)(49) domain of finite length 0 < x < L+ or in the infinite
For free boundary conditions, namely in the limit domain < x < 0. Statistically we expect of-course
−∞
where the system size is infinite L+ and L that the particle will reside more in x < 0, though the
−
we find → ∞ → ∞ random walk is recurrent hence, after each sojourn time
in x < 0 the particle is ejected back to x > 0, provided
(s+u)α/2−1+sα/2−1 that we wait long enough. However the average return
G (u)= . (50)
s (s+u)α/2+sα/2 time from a point in x < 0 to some point in x > 0 is
infinite, and this means that simple scaling T+ t does
∼
not not hold. For this case we have
Invertingtothetimedomain,thePDFofT+ isthesym-
metric Lamperti PDF with index α/2 G (u)= (s+u)α/2−1tanh (s+u√)Dα/α2L+ +sα/2−1.
G T+ = 1δ 1,T+ . (51) s (s+u)α/2tanhh(s+u)α/2L+i+sα/2
t t α/2 t √Dα
(cid:18) (cid:19) h i (55)
(cid:0) (cid:1)
Toinvestigatedeviationsfromsimplescalingweconsider
When α= 1, i.e. the case of normal Gaussian diffusion,
moments of the random variable T+, using the small u
we recover the well known arcsine distribution. As α is
expansion of Eq. (55). The average occupation time in
decreasedwearemorelikelytofindtheparticlelocalized
0<x<L+ is
in x > 0 or x < 0 for a time of the order of the obser-
cavoalwmtiaobynisntaoitnmioxen.>oIfn0t)dweooerddTewl+tha=efnu0nαc(pt→iaornts0icwltehitaehlwPTaD+yFs=iontfx(Tp<a+r0tii)sc.lae hT+i=L−s→1t(cid:20)21s2 (cid:18)1−e−2s√α/D2αL+(cid:19)(cid:21), (56)
usiAngdiEffqe.re(n4t9b)ewheavfiinordisfoundforfiniteL+andL−,then pwrheesrseionL−si→s1itnvisertthede uinsvinegrsoenLeaspidlaecdeLt´ervaynssftoarbmle.fuTnhcitsioenxs-,
recall
1 T+ min(L+,L )2 1/α
Gt T+ ∼ tδα/2(cid:18)1, t (cid:19) t<<(cid:20) Dα − (cid:21) . lα/2,2L+/√Dα,1(t)=L−s→1te−2sα/2L+/√Dα, (57)
(cid:0) (cid:1) (52)
For these time scales the particle does not interact with and see [35] and Ref. therein for more mathematical
the boundaries, and G (T+) is the symmetric Lamperti detailsonthis function. Theonesidedstablecumulative
t
PDFwithindexα/2. Inthe longtime limit,correspond- distribution is
ing to small s u limit we find using Eq. (49) t
L (t)= l (t)dt, (58)
α/2,2L+ ,1 α/2,2L+ ,1
(s+u)α−1L++sα−1L− √Dα Z0 √Dα
G (u) (53)
s ∼ (s+u)αL++sαL− and hence
and hence when t 1 t
→∞ hT+i= 2 1−Lα/2,2L+ ,1(t) dt. (59)
1 L+ T+ Z0 (cid:20) √Dα (cid:21)
G T+ δ , . (54)
t α
∼ t (cid:18)L− t (cid:19) For short times hT+i=t/2, since then the particle does
(cid:0) (cid:1) not have time to interact with the boundary, and it
This is in agreement with our more general result Eqs.
spends half of the time in x>0. For long times
(40,44) namely for the case of free diffusion P+ =
B
tLh+e/P(LD+F+ofLT−+) iasnadshexenpceecteRd =nonL-+sy/mLm−.etrIifc,Lr+efl6=ectLin−g T+ L+ t1−α/2 . (60)
h i∼ √D Γ(2 α/2)
the tendency of the particle to reside in the larger inter- α −
val [say (0,L+) if L+ > L ] for longer times compared Weseethatasthe processbecomesslower,namelywhen
−
with the shorter domain. For long times an equilibrium α is decreased, the particle tends to stay more in 0 <
is obtained: for α = 1 an ergodic phase is found where x < L+ i.e. T+ t for α = 0 but T+ t1/2 if
T+/t = L+/(L +L+) while for α < 1 weak ergodicity α=1. We exphlainith∝is result for normalhdiffuisi∝on α=1
−
9
by thinking about the process asa two state process,i.e. namely a symmetric Lamperti PDF with index α/2 de-
the particle is either in x<0 or in x>0. Sojourn times scribestheresidencetimes. Suchbehaviorisindependent
in x > 0 are finite, since the interval 0 < x < L+ is of the drift and can be understood if we notice that for
finite. The PDF of times in state x<0 follow the t 3/2 short times the dynamics is governed by diffusion not
−
power law tail due to usual diffusion. The number of drift. To see this recallthatthe scalingof these twopro-
time the particle will cross zero is n(t) t1/2 at-least cessesisx tα/2 (diffusion)andx tα (drift)andhence
h i ∼ ∼ ∼
for a lattice CTRW process (in the continuum limit this for short times diffusion wins. For long times we use the
question is not well defined). Hence we expect T+ = smalls,uexpansionofEq. (65),G (u) 1/(s+u)which
s
h i ∼
n(t) averagetime in 0<x<L+ t1/2 aswefind. If gives the expected behavior G (T+) δ(T+ t).
t
hα<1i∗weexpect n(t) tα/2andthe∼averagetimeis0< The mean occupation time is ∼ −
x < L+ is propohrtionia∼l to tψ(t)tdt tt (1+α)tdt
ot1f−Tα+, hisence we get hT+i ∝Rt1−α/2. Th≃e Rseco−nd momen≃t hT+i=L−s→1t(cid:26)21s21+√1√+1+4s4αsταατα(cid:27). (68)
For the case of normal diffusion α=1 we find
L+ t2 α/2
T+ 2 4(1 α) − . (61) t t t t
h i∼ − √DαΓ(3−α/2) hT+i= 2 +τ"rπτe−4tτ +(cid:18)2τ −1(cid:19)Erfr4τ#. (69)
Note that we do not have simple scaling and T+ 2
t2 α/2 is not proportional to T+ 2 t2 α, ahnd heinc∝e The long time behavior is
− −
the PDF of T+ does not haveha simipl∝e scaling. 4τ
T+ t τ 1 e−4tτ , (70)
h i∼ − −rπt !
C. Diffusion with Drift theleadingterm T+ tisexpectedsinceasmentioned
h i∼
forlongtimestheparticleisalwaysinx>0whenF >0.
We consider anomalous diffusion in the presence of a For short times
constantdrivingforceF >0,foraninfinite system. The
1/2 3/2
t 2 t 1 t
biaseddiffusionyieldsanetdrift x =DαFtα/[kbTΓ(1+ T+ 1+ +0(t5/2) .
α)]. SinceF >0theparticlewillhesicapetoinfinity,hence h i∼ 2" 3√π (cid:18)τ(cid:19) − 30√π (cid:18)τ(cid:19) #
for a particle starting on x = 0 we expect T+ t when (71)
t is large. ∼ The leading term T+ t shows that at short time
h i ∼ 2
The survival probability in the right half space diffusion not drift is dominating the process, hence from
symmetry half of the time the particle is on x > 0. For
1 exp Fβ+(s)x0 the sub-diffusive case α<1 we investigate the long time
W+ (s)= − − 2kbT , (62) behaviorof T+ usingthe smalls expansionof(68)and
x0 h s i h i
then inverting to the time domain
when x0 > 0. To obtain the survival probability for left 1 (k T)2 τ 2α
randomwalksreplaceβ+(s)inEq. (62)withβ (s), and T+ t 1 b +O . (72)
− h i∼ − Γ(2 α)F2D tα t
(cid:20) − α (cid:16) (cid:17) (cid:21)
β (s)=1 √1+4sατα (63) At short times we use Eq. (68) and Hankel’s contour
±
±
integral, for the Γ(z) function, and find
where
t 1 F√D
(k T)2 T+ 1+ αtα/2+ . (73)
τα = Fb2D . (64) h i∼ 2" 2Γ(2+α/2)(kbT)2 ···#
α
We note that results for the case F < 0 can be easily
Using Eq. (8) obtained from our results for F > 0. The distribution
of times T in x < 0 when F > 0 is equal of-course to
−
β+(s+u) β−(s) the distribution of time T+ in x > 0 when F < 0. Also
Gs(u)= β+((ss++u)u)− βs (s). (65) T++T− =t hence a simple shift ofthe randomvariable
− − yieldsT =t T+,andhencealsostatisticsforthe case
−
−
For sατα >>1 and uατα >>1 F <0.
(s+u)α/2 1+sα/2 1
− −
G (u) . (66)
s ≃ (s+u)α/2+sα/2 D. Diffusion in an Unstable Force Field
Thus for short times t<<τ We consider a particle in an unstable force field
1 T+ F x>0
G (T+) δ 1, (67) F(x)= + (74)
t ≃ t α/2 t F x<0
(cid:18) (cid:19) (cid:26)− −
10
whereF >0andF >0. Forthiscasethe particlewill where F > 0 and F > 0. The random walk is recur-
+ +
eventually escape eit−her to + or , and the random rent. Using an appro−ach similar to one used in previous
∞ −∞
walk is not recurrent. The survival probabilities in left subsection
and right domains are
ξ+(s+u) + ξ−(s)
1 G (u)= s+u s , (83)
Wx±0(s)= s 1−exp ∓F± 1+ 1+4sατ±α x0/(2kbT) . s ξ+(s+u)+ξ−(s)
(75)
(cid:8) (cid:2) (cid:0) p (cid:1) (cid:3)(cid:9) and ξ (s)=F 1 1+4sατα .
where For±small s,u±we−have ±
(k T)2 (cid:0) p (cid:1)
Using Eq. (7) we findτ±α = F±2bDα. (76) Gs(u)∼ (s(+su+)αu−)α1 FFFF−−++ ++ssαα−1, (84)
γ+(s+u) + γ−(s) and hence when t is large
G (u)= s+u s (77)
s
γ+(s+u)+γ (s) 1 F T+
− Gt t+ δα −, . (85)
where ∼ t F t
(cid:18) + (cid:19)
(cid:0) (cid:1)
γ (s)=F 1+ 1+4sατα . (78) This is in agreement with our more general results
± ± ± Eqs. (40, 44). For short times we have G (T+)
Using the definition E(cid:0)q. (9p), the small(cid:1)s behavior of t ≃
Eq. (75) gives the survival probabilities in the + and 1tδα/2 1,Tt+ whichissimilartothe behavioroftheun-
−
domains, x > 0 and x < 0 respectively. In the limit of stable(cid:16)field d(cid:17)iscussed in previous sub-section.
long times
F x
ǫ (x0)=1 exp ± 0 , (79) V. DISCUSSION
± − (cid:18)∓ kbT (cid:19)
where x > 0 for + and x < 0 for . Hence according Statisticsofoccupationtimesforbindingexternalfields
0 0
−
to the rather general Eqs. (16,17) we find exhibits in the limit of long times an ergodic behavior
when the diffusion is normal, or weak ergodicity break-
F F
Gt(T+) + δ T+ t + − δ(T+). (80) ing Eqs. (40,44) when diffusion is anomalous. We estab-
∼ F +F+ − F++F lished a link between weak ergodicity breaking and frac-
− −
(cid:0) (cid:1) tional calculus. The exponent α in the fractionalderiva-
This long time behavior exhibits the same behavior for
tive∂α/∂tα entersinEq. (40)describingthenon-ergodic
the normal diffusion α = 1 as for the anomalous case
properties of the residence times. Since many processes
α<1. Note that Eqs. (16,17) where derived for normal
and systems are modeled today using the fractional cal-
diffusion however one can show that they are valid also
culus approach, it is not out of the question that weak
fortheanomalousdiffusioncase. Toseethisusethesmall
ergodicity breaking has many applications, and is wide
s,u expansion of Eq. (77) which gives
spread. We can say that at-least one must treat with
1 F+ F care, results obtained using fractional kinetic equations,
Gs(u)∼ F +F s+u + s− , (81) since they describe only ensemble averages,not time av-
+ − (cid:18) (cid:19) erages.
whichisthedoubleinverseLaplacetransformofEq. (80)
Forbindingexternalfieldsourresultsareinfullagree-
and is independent of the parameters α,D and T.
α ment with those derived recently, by Bel and the au-
For short times t min(τ ,τ ) we use the large s,u
+ thor [19]. There a continuous time random walk process
≪ −
behavior of Eq. (77) and find G (T+) 1δ 1,T+ . was considered. Technically the methods used to treat
t ≃ t α/2 t
Thus for short times the PDF of T+ is the sy(cid:16)mmetr(cid:17)ic the two problems are different. For the fractionalframe-
work,adifferentialequation,Eq. (24)fortheoccupation
LampertiPDFwhichisindependentofalltheparameters
times is derived and solved, which yields the weakly non
of the problem except for α. At the early stages of the
ergodic properties of the system, while for the CTRW
dynamics the diffusion process not the drift is the most
certainrecursiverelationsmustbe solved[20]. Ourwork
important, and hence forces are not relevant.
showshowthefractionalframework,whichisthecontin-
uum limit of the CTRW (and in this sense simpler) can
E. Diffusion in Binding Force Field be used to obtain statistics of residence times, and for
binding fields weak ergodicity breaking.
We found a general relation between the problem of
We consider a particle in a stable force field
occupationtimes and the problemof first passagetimes,
F x>0 Eq. (7). Mathematically the problem of first passage
F(x)= − + (82)
F x<0 timeisdescribedintermsofadifferentialequationwhose
(cid:26) −