Langevin formulation of a subdiffusive continuous time random walk in physical time

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by  Andrea Cairoli

Fermilab-Pub-04/xxx-E
Langevin formulation of a subdiffusive continuous time random walk in physical time
Andrea Cairoli and Adrian Baule∗
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, E1 4NS, UK
(Dated: June 5, 2015)
Systemslivingincomplexnonequilibratedenvironmentsoftenexhibitsubdiffusioncharacterized
byasublinearpower-lawscalingofthemeansquaredisplacement. Oneofthemostcommonmodels
to describe such subdiffusive dynamics is the continuous time random walk (CTRW). Stochastic
trajectoriesofaCTRWcanbedescribedintermsofthesubordinationofanormaldiffusiveprocess
5 byaninverseL´evy-stableprocess. Here,weproposeanequivalentLangevinformulationofaforce-
1 freeCTRWwithoutsubordination. Byintroducinganewtypeofnon-Gaussiannoise,weareableto
0 express the CTRW dynamics in terms of a single Langevin equation in physical time with additive
2 noise. Wederivethefullmulti-pointstatisticsofthisnoiseandcompareitwiththescaledBrownian

n motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly,
u these two noises are identical up to the 2nd order correlation functions, but different in the higher
J order statistics. We extend our formalism to general waiting times distributions and force fields
and compare our results with those of SBM. In the presence of external forces, our proposed noise
4
generatesanewclassofstochasticprocesses,resemblingaCTRWbutwithforcesactingatalltimes.

]
h
c I. INTRODUCTION Brownian models, e.g., non Gaussian probability density
e
functions (PDFs) for the position and/or the velocity of
m
Many systems in nature live in complex non- the moving cell and power-law long time scaling of the
t- equilibrated or highly crowded environments, thus ex- velocity auto-correlation functions (see Refs. [6] and ref-
a hibitinganomalousdiffusivepatterns,whichdeviatefrom erences therein). On the other hand, biological macro-
t
s the well known Fick’s law of purely thermalized systems molecules and/or organelles often exhibit a subdiffusive
t. [1–3]. Their distinctive feature is the power-law scaling scaling of the MSD, while moving in molecular crowded
a of the mean-square displacement (MSD) [1–5]: environments. These can be prepared ad hoc for in vitro
m
experiments, e.g., by using solutions of surfactant mi-
- (cid:10)(Y(t)−Y )2(cid:11)∼tα, (1) celles[7]orpolymernetworks[8]tonamejustafewtech-
d 0
niques, or found in vivo, e.g., in the cytoplasm [9] or the
n
o where (cid:104)·(cid:105) indicates the ensemble average over different cells’ membrane [10], whose viscoelastic properties have
c realizations of the stochastic process Y(t) describing the recently been found to play a major role in determining
[ dynamics, usually either a velocity or a position, Y is theanomalousdiffusion[11,12]. Furthermore, biological
0
its initial condition and α ∈ R+. While Fick’s law is systemsmayalsoexhibitarichdynamicalbehavior,such
3
v recovered by setting α = 1, thus predicting for normal as non linear MSDs, showing crossover between different
0 diffusionthetypicallinerscalingoftheMSD,wecandis- scaling regimes [13–19] at different timescales, and/or a
8 tinguishbetweendifferenttypesofanomalousbehaviour. dependenceofthecorrespondingdiffusioncoefficientson
6 Indeed, we define subdiffusion if 0<α<1 and superdif- energy-driven active mechanisms [13, 19, 20]. We refer
6
fusion if α>1, which correspond to processes dispersing theinterestedreaderto[21]forarecentreviewonanoma-
0
with a slower or faster pace than Brownian motion re- lous diffusive systems.
.
1 spectively. Considering this wide, though not exhaustive, variety
0 Examplesofsuchanomalousprocesseswerefirstfound of different anomalous behaviours, one needs to have a
5
in physical systems, such as charge carriers moving tool-kit of well studied models with which trying to fit
1
in amorphous semiconductors, particles being trans- the experimental data and infer the specific microscopic
:
v portedonfractalgeometriesordiffusinginturbulentflu- processes underlying the observed dynamics. Here we
i
X ids/plasma or in heterogeneous rocks (see [2] and refer- focus on subdiffusive processes, for which many models
ences therein). However, with the recent improvements have been introduced so far, which are capable of repro-
r
a of experimental techniques in biology, joint position- ducing the characteristic scaling of Eq. (1), while still
velocitydatasetshavebeenobtained,whicharerevealing showing distinct features if we look at other properties,
the existence of many more examples in living systems. like the multipoint correlation functions [22–25]. Among
On the one hand, cell migration experiments have re- the most commonlyapplied to data analysis, we findthe
vealed a characteristic superdiffusive scaling of the MSD continuous-time random walk (CTRW) [2, 26] and the
along with many more features deviating from standard scaled Brownian motion (SBM) [27–30].
In the seminal paper [26], the CTRW was introduced
as a natural generalization of a random walk on a lat-
tice, with waiting times between the jumps and their
∗ Correspondenceto: [e-mail] size being sampled from general and independent prob-
2
ability distributions. Only later, a convenient stochastic II. CTRWS, SCALED BM AND
representation of these processes was derived in terms GENERALIZATION TO ARBITRARY WAITING
of subordinated Langevin equations [31], which provided TIMES’ DISTRIBUTIONS AND TIME
a suitable formalism to derive their multipoint correla- TRANSFORMATIONS
tion functions [22, 24]. Although the focus was first on
power-law distributed waiting times, which indeed pro- We recall in this first section definitions and proper-
vided Eq. (1) exactly for all times, recent works adopted ties of the free diffusive CTRW and SBM, which will be
more general distributions [32–34], thus being able to useful later in the discussion. We are mainly interested
model the crossover phenomena so often occurring in bi- in their stochastic Langevin formulation and in both
ological experiments. their Fokker-Planck (FP) equation and MSD. We then
generalize these results to the case of arbitrary waiting
times’ distributions or time transformations for CTRW
and SBM respectively.
Throughout the discussion, f(cid:101) indicates the Laplace
On the other hand, the SBM has been recently intro-
transform of a function f(t) defined on the positive half
duced as a Gaussian model of anomalous dynamics [27], line: f(cid:101)(λ) = L{f(t)}(λ) = (cid:82)+∞e−λtf(t)dt. Moreover,
providing the same scaling of Eq. (1) for all its dynam- 0
ϕ ∗ϕ denotes the convolution of two functions ϕ and
ical evolution. If B(t) is a usual BM, its scaled version 1 2 1
ϕ , defined on the positive half line: (ϕ ∗ ϕ )(t) =
is defined by making a power-law change of time with 2 1 2
(cid:82)t
exponent α: B(tα). Although being commonly used to 0 ϕ1(t−τ)ϕ2(τ)dτ. We remark that the correspond-
ing definitions for functions of multiple variables follow
fit data [7, 28, 35], it has recently been shown to be a
straightforwardly.
nonstationaryprocesswithparadoxicalbehaviourunder
confinement, i.e., in the presence of a linear viscous-like
force, as the corresponding MSD unboundedly decreases
A. CTRW
towards zero. This is suggested to be ultimately caused
bythetimedependenceoftheenvironment,eitherofthe
temperature or of the viscosity. As a consequence, it has A Langevin representation of a CTRW was first pro-
been ruled out as a possible alternative model of anoma- posed in [31], where the method of the stochastic time-
lous thermalized processes [30]. change of a continuous-time process is used. Its set-up
consists in introducing two auxiliary processes X(s) and
T(s),whichweassumefornowtobepurelydiffusiveand
L´evy stable with parameter α (0 < α ≤ 1) respectively.
They both depend on the arbitrary continuous parame-
In this paper, we derive a new type of noise, which ter s and have dynamics described in terms of Langevin
allows us to express a free diffusive CTRW in terms of equations [31]:
a single Langevin equation in physical time. We pro- √
videthefullcharacterizationofitsmultipointcorrelation X˙(s)= 2σξ(s) (2a)
functions and we compare them with those of the noise T˙(s)=η(s) (2b)
drivingaSBM.Bothpurelypower-lawwaitingtimesand
general waiting time distributions are discussed [33, 34]. whereξ(s)andη(s)aretwoindependentnoises. ForX(s)
Wefindthatthecorrelationfunctionsareidenticalupto to be a normal diffusion, we require ξ(s) to be a white
the two point ones, but different for higher orders: the Gaussiannoisewith(cid:104)ξ(s)(cid:105)=0and(cid:104)ξ(s )ξ(s )(cid:105)=δ(s −
1 2 2
noise driving SBM is a Gaussian noise, while our new s ). On the other hand, η(s) is a stable L´evy noise with
1
noise driving a CTRW is clearly non-Gaussian. Here, all parameter α (0<α≤1) [38]. The anomalous CTRW is
odd correlation functions vanish, as for Gaussian noise, then derived by making a randomization of time, i.e., by
buttheevenonesdonotsatisfyWick’stheorem[36]. The considering the time-changed (or subordinated) process:
newlydefinednoiseenablesustodefineaclassofCTRW Y(t) = X(S(t)), with S(t) being the inverse of T(s),
like processes with forces acting for all times, which are defined as a collection of first passage times:
different from the corresponding standard CTRWs. Fur-
thermore,werevisitthebehaviouroftheSBMundercon- S(t)= inf {s:T(s)>t}. (3)
s>0
finementandshowthatitsMSDcorrectlyconvergestoa
plateau as it is typical of confined motion [37], provided TheprocessY(t)iseasilyshowntosatisfyEq.(1)exactly
that we use more general time changes with truncated forallitstimeevolution,byrecallingthattheprobability
power-law tails. This suggest that the anomaly observed densityfunction(PDF)ofS(t)hastheLaplacetransform
in [30] is mainly due to the localizing effect of the exter- (cid:101)h(s,λ) = λα−1e−sλα [22] and that (cid:10)X2(s)(cid:11) = 2σs. In-
nal linear force, which is able to trap the particle in the deed, we obtain in Laplace space:
zero position if we allow for infinitely long waiting times
between the jumps to eventually occur in the long time (cid:68)Y(cid:101)2(λ)(cid:69)=(cid:90) +∞(cid:10)X2(s)(cid:11)(cid:101)h(s,λ)ds= 2σ , (4)
limit. λ1+α
0
3
whose inverse transform confirms its anomalous scaling: and measures the arc-length as a function of the physi-
cal time. S(t) thus represents the continuum limit of the
(cid:10)Y2(t)(cid:11)= 2σ tα. (5) randomvariableN(t)thatcountsthenumberofstepsin
Γ(1+α) the renewal picture.
As expected, this same MSD is obtained by taking the
diffusive limit of the microscopic random walk formula-
B. SCALED BM
tion of the CTRW, where we allow for asymptotically
power-law distributed waiting times between the jumps
If instead of a stochastic time change, we consider the
of the walker, whose sizes are drawn from a distribution
deterministic time transformation t → t∗ = tα in the
with finite variance [2]. In this limit, the model also pro-
normal diffusive process X(t) (now in the physical time
videsafractionaldiffusionequationforthePDFofY(t):
t), we obtain the SBM: Y (t) = X(t∗). Its equivalent
∗
Langevin equation is given by [27–30]:
∂ ∂2
P(y,t)=D D1−αP(y,t), (6)
∂t α∂y2 t √
Y˙ (t)= 2ασtα−1ξ(t), (9)
∗
where D is a generalized diffusion coefficient and
α
D1−αf(t) = 1 ∂ (cid:82)t(t−τ)α−1f(τ) is the Riemann- with ξ(t) being a white Gaussian noise (with the same
t Γ(α)∂t 0 propertiesasbefore,butinthephysicaltimet). Byusing
Liouville time-integral operator, which makes the non
Eq. (9) we can prove straightforwardly that the MSD of
Markovian character of the CTRW evident. It is then
Y (t) is the same as Eq. (5) and that the corresponding
∗
natural to investigate if the set of Eqs. (2a-2b) can
FP equation is given by:
give this same FP equation. This has been proved in
[34, 39, 40], with the specification: Dα = Γ(1σ+α), thus ∂ P(y,t)=ασtα−1 ∂2 P(y,t), (10)
confirming the equivalence in the diffusive limit of the ∂t ∂y2
original random walk model and of the subordinated
Langevin Eqs. (2a-2b). which has time dependent diffusion coefficient [41]. This
We remark that the formulation of CTRWs as a sub- process preserves all the properties of Brownian motion
ordinated normal diffusive processes can be considered [27]: it is indeed Gaussian with time-dependent variance
as the continuum limit of the original renewal picture of andMarkov,asthemonotonicityofthetimechangepre-
Montroll and Weiss [26]. Here, the position Y(t) of a serves the ordering of time. Furthermore, Y (t) is self-
∗
CTRW is characterized by two sets of random variables similar and it has independent increments for non over-
{(ξ ,η )} , with N(t) being the number of jumps lapping intervals. However, differently from Brownian
i i i,...,N(t)
up to the time t. The random variables ξ and η spec- motion, it is strongly non stationary [30]. Furthermore,
i i
ify the amplitude of the jumps occurring at the random Y (t)turnsouttobethemean-fieldapproximationofthe
∗
time t , i.e., ξ =Y(t )−Y(t ), and the waiting times CTRW, as it describes the motion of a cloud of random
i i i i−1
between two successive jumps, i.e., η =t −t . Thus, walkers performing CTRW motion in the limit of a large
i i i−1
Y(t) is obtained by summing all the variables ξ : number of walkers [29]. Recent investigation have also
i
shown that SBM exhibits rich aging properties, which
N(t) strongly differentiates it from the standard BM [42].
(cid:88)
Y(t)= ξ . (7)
i
i=1
C. ARBITRARY WAITING TIMES’
In the present discussion, we assume {ξ } and {η } sep-
i i DISTRIBUTION AND TIME
arately to be i.i.d random variables and each ξ to be
i TRANSFORMATIONS
independent of η . On the other hand, we obtain by di-
i
rect integration of Eqs. (2a-2b)
In this section, we first focus on the generalization of
Eqs. (2a-2b) to arbitrary waiting time distributions of
Y(t)=X(S(t))
the underlying random walk [32–34, 43]. This extension
(cid:90) S(t) isobtainednaturallybychoosingadifferentprocessT(s)
= X˙(τ)dτ
with the only assumption of it being non decreasing in
0
√ (cid:90) S(t) ordertopreservethecausalityoftime. Thus,weconsider
= 2σ ξ(τ)dτ. (8) η(s)inEq.(2b)tobeanincreasingL´evynoisewithpaths
0 of finite variation and characteristic functional [44]:
Tinhgetrreafojercet,oFroygoefdtbhye’sraanpdpormoacwhal[k31i]ndtehsecrciobnetsintuhuemrelsiumltit- G[k(τ)]=(cid:68)e−(cid:82)0+∞k(τ)η(τ)dτ(cid:69)=e−(cid:82)0+∞Φ(k(τ))dτ.
by parametrizing both the path of the walker X(·) and (11)
the time elapsed T(·) with an arbitrary continuous arc- Here Φ(k(τ)) is a non negative function with Φ(0) = 0
lengths. ThestochasticprocessS(·)istheinverseofT(·) andstrictlymonotonefirstderivative,whilek(τ)isatest
4
function. We recall that for Φ(s) = sα we recover the
CTRW model. Under these assumptions, the integrated 100
processT(s)isaaone-sidedincreasingL´evyprocesswith 101 hY2(t)i/t
µ=0
finite variation. Furthermore, we assume η(s) to be in-
µ=10−4
dependent on the realizations of ξ(s) in Eq. (2a). As a µ=10−3 10−1
consequence of the finite variation and the monotonicity µ=10−2
ofthepathsofT(s)respectively,S(t)hascontinuousand 10−3
monotone paths, with this second property implying the 10−1
fundamental relation [22]: 10−5
10−2 100 102 104
Θ(s−S(t))=1−Θ(t−T(s)). (12) t)
y,
(
P
Similarly to Eq. (4), we can derive the corresponding
MSD by recalling that (cid:101)h(s,λ)= Φ(λλ)e−sΦ(λ) [33, 34]: 10−2
(cid:90) t
(cid:10)Y2(t)(cid:11)=2σ K(τ)dτ, (13)
0
for K(t) being related in Laplace space to Φ(s) by:
1 10−−230 −15 −10 −5 0 5 10 15 20
K(cid:101)(λ)= . (14) y
Φ(λ)
Furthermore, thePDFofY(t)isobtainedbysolvingthe Figure 1. (Colors online) PDF (main) and MSD normalized
generalized FP equation [34]: to t (inset) of an anomalous process Y(t) obtained by sub-
ordination of a pure diffusive process with a tempered stable
∂ ∂2 ∂ (cid:90) t L´evy process of tempering index µ and stability parameter
P(y,t)=σ K(t−τ)P(y,τ)dτ, (15)
∂t ∂y2∂t α=0.2. ThePDFisobtainedbynumericalLaplaceinversion
0 of Eq. (16) at t=1000 (black dotted lines in the inset) [48].
whose solution in this particular case can be found for The smooth transition from the non-Gaussian PDF typical
of CTRWs (µ=0) and the Gaussian one of normal diffusion
general Φ(s) in Laplace space:
(µ→+∞)isevident,alongwiththecorrespondingtransition
(cid:114) from anomalous to normal scaling of the MSD for increasing
P(cid:101)(y,λ)= 1 Φ(λ)e−(cid:113)Φ2(σλ)|y|. (16) µ at a fixed time. Simulations, obtained with the algorithms
λ 2σ of [49, 50], agree perfectly with the analytical results.
Welookasanexampleatthecaseofatemperedstable
L´evy noise with tempering index µ and stability index α
subdiffusive to normal behaviour, but their MSD scales
[45], which is obtained by setting Φ(λ) = (µ + λ)α −
as a power-law for all times. As expected, for µ = 0 we
µα, i.e., K(t) = e−µttα−1E ((µt)α) [46]. As already
α,α recover the typical non Gaussian shape of the PDF of a
pointed out, the CTRW case is recovered by setting µ=
free diffusive CTRW [2]. However, for increasing values
0, meaning that we do not truncate the long tails of the
ofµ,thePDFofY(t),althoughstillbeingnonGaussian,
distribution, thus accounting for very long waiting times
broadens,thusgettingclosertoaGaussian. Thishasalso
withapower-lawdecayingprobabilityofoccurrence. We
evident consequences on the dynamical behaviour of the
plotinFigure1thenumericalLaplaceinverseofEq.(16)
MSD, which for increasing values of µ goes from a pure
(main)andthecorrespondingMSD(inset)atafixedtime
subdiffusive scaling to a normal one (inset).
t=1000(dottedlineintheinset),whichisgivenby[47]:
We now discuss the corresponding extension of the
(cid:34) ∞ (cid:35) SBM to arbitrary time transformations involving the
(cid:10)Y2(t)(cid:11)= 2σ −1+(cid:88) γ(µt;αn) , (17) kernel K(t) obtained by Laplace inverse transform of
µα Γ(αn) Eq. (14). We then generalize Eq. (9) by adopting K(t)
n=0
as the time dependent coefficient of the white noise:
with γ(x;a) = (cid:82)xe−tta−1 being an incomplete gamma
function, leading0to the asymptotic behaviour [34, 47]: Y˙ (t)=(cid:112)2σK(t)ξ(t)=ζ(t), (19)
∗
(cid:10)Y2(t)(cid:11)∼(cid:26)(cid:0)2Γσ(12µ+σ1α−)αtα(cid:1)t tt><><11 (18) 0whaenredwtweod-epfioninetthceorcroerlaretiloantedfunnocitsieonζ:(t)(cid:104)ζw(itt1h)ζ(cid:104)(ζt2(t))(cid:105)(cid:105) ==
α
2σK(t )δ(t −t ). This explicit time dependence clearly
1 1 2
WeremarkthatEq.(18)doesnotapplytothelongtime signals that ζ(t) is a non stationary noise. It is easily
scalingofCTRWs,forwhichitwouldpredictavanishing shown that the MSD of Y (t) is identical to the one of
∗
MSD. In fact, CTRWs do not exhibit a crossover from Y(t) given by Eq. (13). However, even if they share the
5
same MSD, Y(t) and Y (t) provide different PDFs. In- time, so that we can take its derivative and obtain the
∗
deed,Y (t)correspondstoatimerescaledBrownianmo- equivalent Langevin equation:
∗
tion X(t∗) with transformation:
√
Y˙(t)= 2σξ(t) (24)
(cid:90) t
t∗ = K(τ)dτ. (20)
with the newly defined noise:
0
InthecaseoftheusualBrownianmotionthecorrespond- (cid:90) +∞
ingdiffusionequationhasaGaussiansolution: P(y,t)= ξ(t)= ξ(s)δ(t−T(s))ds, (25)
√ 1 e−(y−4σyt0)2 for the initial condition P(y,0) = δ(y− 0
4πσt whose properties are fully determined by the choice of
y ). Since Y (t) is just Brownian motion in the rescaled
0 ∗
the waiting times’ distribution, i.e., equivalently of the
timet∗,weobtainsimilarlyaGaussiansolution,provided
function Φ(s) in Eq. (11). If we recall the independence
we choose the same initial condition:
of ξ(s) and η(s), we can show that ξ(t) has zero average
P(y,t)= √ 1 e−(y4−σyt0∗)2, (21) and two point correlation function:
4πσt∗
(cid:10) (cid:11)
ξ(t )ξ(t ) =K(t )δ(t −t ), (26)
1 2 1 1 2
with t∗ as in Eq. (20). We see that P(y,t) is a solution
of the diffusion equation: with K(t) being specified by Eq. (14). In Laplace space,
(cid:10) (cid:11)
indeed, ξ(t )ξ(t ) is an integral of the two point char-
1 2
∂ ∂2 acteristic function of T(s), which can then be computed
P(y,t)=σK(t) P(y,t), (22)
(cid:68) (cid:69)
∂t ∂y2 with Eq. (11): (cid:101)ξ(λ )(cid:101)ξ(λ ) = 1 . By making
1 2 Φ(λ1+λ2)
with the time dependent diffusion constant: D(t) = its inverse Laplace transform, Eq. (26) follows straight-
σK(t). We remark that Eq. (9) can be recovered from forwardly. Consequently, the character of the noise ξ(t)
these general results by setting Φ(λ) = λα, i.e., K(t) = significantly depends on the choice of the function Φ(s)
tα−1/Γ(α) and t∗ = tα/Γ(1+α). However, in order to inEq.(11). Thus,Eq.(24)definesanewLangevinmodel
have exact equivalence, we need to neglect the constant drivenbyageneralizedandtypicallynonGaussiannoise,
multiplicative factors in both K(t) and t∗ and make the exceptpossiblyforparticularchoicesofthememoryker-
following substitution: σ →ασ. nel K(t), which is able to describe free diffusive anoma-
lous processes with arbitrary waiting times distribution
equivalently to the subordinated Eqs. (2a-2b).
III. LANGEVIN FORMULATION OF We highlight that so far the standard renewal picture
ANOMALOUS PROCESSES IN PHYSICAL TIME underlyingconventionalCTRWsstillapplies. Eq.(24)is
essentially the time derivative of Eq. (8) expressing the
process in terms of stochastic increments. Differences
A. DEFINITION OF THE NOISE
willappearwhenexternalforcesactonthediffusionpro-
cesses. This case is discussed in Sec. IV.
WeproceedinthissectiontoderiveaLangevindescrip-
tion of the process Y(t) defined in Eqs. (2a-2b) directly
in physical time. Starting from the explicit integral ex-
pression Eq. (8) we can write the following: B. CHARACTERIZATION OF THE
MULTIPOINT CORRELATION FUNCTIONS
√ (cid:90) +∞ (cid:20)(cid:90) s (cid:21)
Y(t)= 2σ δ(s−S(t)) ξ(τ)dτ ds
The definition in Eq. (25) enables us to derive a
0 0
√ (cid:90) +∞(cid:18) ∂ (cid:19)(cid:20)(cid:90) s (cid:21) complete characterization of the multipoint correlation
= 2σ − Θ(t−T(s)) ξ(τ)dτ ds structure of ξ(t). As a preliminary step, we de-
∂s
0 0 rive the Laplace transform of the multipoint charac-
√ (cid:90) +∞ teristic function of T(s), i.e., Z(t ,s ;...;t ,s ) =
= 2σ Θ(t−T(s))ξ(s)ds, (23) (cid:68) (cid:69) 1 1 N N
(cid:81)N δ(t −T(s )) ∀N ∈ N. This is obtained by
0 m=1 m m
using the definition of Eq. (2b) as:
wherethefundamentalrelationofEq.(12)isusedtoob-
tain the second equality and we then get the third one
(cid:42) N (cid:43)
withanint(cid:2)egrationbypa(cid:82)rtss. Were(cid:3)m(cid:12)+ar∞kthatthebound- Z(cid:101)(λ1,s1;...;λN,sN)= (cid:89) e−λm(cid:82)0smη(s(cid:48)m)ds(cid:48)m .(27)
ary term −Θ(t−T(s)) ξ(τ)dτ (cid:12) is zero trivially
0 0 m=1
for s=0, but it vanishes also for s→+∞ because T(s)
is increasing, thus always being bigger than any fixed Letusfirstassumeanorderingforthesequenceoftimes:
(and finite) time t. Written as in Eq. (23), Y(t) is a dif- s < s < ... < s and compute the corresponding
1 2 N
ferentiable (although in a generalized sense) function of Eq. (27). If we rearrange the exponent by separating
6
successive intervals, we obtain: where the average is factorized due to the independence
of the noises. If we recall the Wick theorem holding for
Z(cid:101)(λ1,s1;...;λN,sN)= the white noise ξ(s) [36]:
=(cid:68)e−λN(cid:82)ssNN−1η(s(cid:48)N)ds(cid:48)N−...−(λN+...+λ1)(cid:82)0s1η(s(cid:48)1)ds(cid:48)1(cid:69) (cid:42)2N (cid:43) N
(cid:89) 1 (cid:88) (cid:89)(cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11)
=(cid:68)e−(cid:80)Nm−=01[((cid:80)Nn=m+1λn)](cid:82)ssmm+1η(s(cid:48)m)ds(cid:48)m+1(cid:69) ξ(tj) =N2N ξ tσ(2N−j+1) ξ tσ(j)
j=1 σ∈S2Nj=1
N−1 N
= (cid:89) e−(sm+1−sm)Φ((cid:80)Nn=m+1λn), (28) = N12N (cid:88) (cid:89)δ(cid:0)tσ(2N−j+1)−tσ(j)(cid:1) (33)
m=0 σ∈S2Nj=1
where we define s0 = 0 to simplify the notation and we and we substitute it in Eq. (32), we can derive:
exploited the independence of the increments of T(s) to
factorize the ensemble average. Furthermore, their sta- (cid:10) (cid:11) 1 (cid:88) (cid:34)(cid:89)2N (cid:90) +∞ (cid:35)
tionariness together with Eq. (11) is then used to get ξ(t1)...ξ(t2N) = N2N dsm ×
Eq. (28). However, in the general case where no a-priori σ∈S2N m=1 0
ordering is assumed, we need to consider all the possible N (cid:42)2N (cid:43)
(cid:89) (cid:0) (cid:1) (cid:89)
ordered sequences. We then introduce the group of per- × δ s −s δ(t −T(s ))
σ(2N−j+1) σ(j) i i
mutationsofNobjectsSN,whoseelementsactonthese- j=1 i=1
quence: s=(s1,...,sN). When we make a permutation 1 (cid:88) (cid:34)(cid:89)N (cid:90) +∞ (cid:35)
of s, we obtain a new sequence with permuted indices: = ds × (34)
s(cid:48) = (cid:0)sσ(1),...,sσ(N)(cid:1). All the possible orderings of s N2N σ∈S2N m=1 0 σ(m)
are thus obtained by summing over all the permutations (cid:42) N (cid:43)
in S . If we assume that σ(s ) = 0, ∀σ ∈ S , i.e., the (cid:89) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
N 0 N × δ t −T s δ t −T s
σ(2N−j+1) σ(j) σ(j) σ(j)
initialtimeiskeptfixedbythepermutations,andweuse
j=1
the result of Eq. (28), we derive:
withNintegralsbeingsolvedbyusingthedeltafunctions
(cid:88) N(cid:89)−1 (cid:0) (cid:1) obtained from (cid:104)ξ(s1)...ξ(s2N)(cid:105). If we make a Laplace
Z(cid:101)(λ1,s1;...;λN,sN)= Θ sσ(m+1)−sσ(m) × transform of Eq. (34), we obtain an expression involving
σ∈SNm=0 Z(cid:101)(λ1,s1;λ2,s2;...;λN,sN):
×e−(sσ(m+1)−sσ(m))Φ((cid:80)Nn=m+1λσ(n)) (29) (cid:42)(cid:89)2N (cid:43) 1 (cid:88) (cid:34)(cid:89)N (cid:90) +∞ (cid:35)
with the ordering of the permuted sequence being speci- (cid:101)ξ(λj) = N2N dsm ×
fiedbytheproductofHeavisidefunctions. Byfactorizing j=1 σ∈S2N m=1 0
out the first term, we obtain the fundamental result: ×Z(cid:101)(cid:0)λσ(1)+λσ(2N),s1;...;λσ(N)+λσ(N+1),sN(cid:1), (35)
Z(cid:101)(λ1,s1;...;λN,sN)= (cid:88) e−sσ(1)Φ((cid:80)Nm=1λm)× (30) which can thus be further simplified with Eq. (30). By
substitutionandbymakingafurtherpermutationofthe
σ∈SN
indices, we obtain:
N−1
×(cid:89)Θ(cid:0)sσ(m+1)−sσ(m)(cid:1)e−(sσ(m+1)−sσ(m))Φ((cid:80)Nn=m+1λσ(n)) (cid:42)(cid:89)2N (cid:43) 1 (cid:88) (cid:88) (cid:34)(cid:89)N (cid:90) +∞ (cid:35)
m=1 (cid:101)ξ(λ ) = ds ×
j N2N σ(cid:48)(m)
As an example, we recover the two-point case: j=1 σ∈S2Nσ(cid:48)∈SN m=1 0
aZ(cid:101)n(dλ1w,se1;cλo2n,ssid2)er[3th4]e. twIof wpoesspibulteNper=mut2edinseEqqu.en(c3e0s): ×e−sσ(cid:48)(1)Φ((cid:80)Nm=1λm)N(cid:89)−1(cid:2)Θ(cid:0)sσ(cid:48)(m+1)−sσ(cid:48)(m)(cid:1)× (36)
s=(s1,s2) and s(cid:48) =(s2,s1), we obtain: m=1
×e−(sσ(cid:48)(m+1)−sσ(cid:48)(m))Φ((cid:80)Nn=m+1(λσ(σ(cid:48)(n))+λσ(2N−σ(cid:48)(n)+1)))(cid:105),
Z(cid:101)(λ1,s1;λ2,s2)=Θ(s2−s1)e−s1Φ(λ1+λ2)e−(s2−s1)Φ(λ2)
+Θ(s −s )e−s2Φ(λ1+λ2)e−(s1−s2)Φ(λ1). (31) wheretheNintegralscanthenbesolvedbymakingsuit-
1 2
able changes of variables. This leads to the following
We now use Eq. (30) to compute the correlation func- result for the Laplace transform of even multipoint func-
tions of ξ(t). Indeed, we obtain from Eq. (25) ∀N ∈N: tions of ξ(t):
(cid:10)ξ(t1)...ξ(t2N)(cid:11)=(cid:34)m(cid:89)2N=1(cid:90)0+∞dsm(cid:35)× (cid:68)(cid:101)ξ(λ1)...(cid:101)ξ(λ2N)(cid:69)=N2NΦ(cid:16)(cid:80)12mN=1λm(cid:17)σ∈(cid:88)S2N× (37)
(cid:42) 2N (cid:43)(cid:42) 2N (cid:43) (cid:88) N(cid:89)−1 1
(cid:89) (cid:89) × .
× ξ(sm) δ(tm−T(sm)) , (32) Φ(cid:16)(cid:80)N (cid:0)λ +λ (cid:1)(cid:17)
m=1 m=1 σ(cid:48)∈SNm=1 n=m+1 σ(σ(cid:48)(n)) σ(2N−σ(cid:48)(n)+1)
7
Weremarkthatoddmultipointcorrelationfunctionsare which vanish ∀N ∈ N. The corresponding quantities in
zero;indeed,ifwemakethesubstitution2N →2N+1in time are derived by making the inverse Laplace trans-
Eq. (32), we obtain an expression depending on the odd form of Eq. (37), which can be written as a 2N−fold
multipoint correlation functions: (cid:104)ξ(s )...ξ(s )(cid:105), convolution:
1 2N+1
N−1
(cid:10) (cid:11) 1 (cid:89)
ξ(t )...ξ(t ) = K(t ) δ(t −t )∗ g(t ,t ,...,t ,t ) (38a)
1 2N N2N 1 i+1 i 2N 1 2 2N−1 2N
i=1
 
 (cid:88) (cid:88) N(cid:89)−1 1 
g(t ,t ,...,t ,t )=L−1 (38b)
1 2 2N−1 2N (cid:16) (cid:17)
 Φ (cid:80)N (λ +λ ) 
σ∈S2Nσ(cid:48)∈SNm=1 n=m+1 σ(σ(cid:48)(n)) σ(2N−σ(cid:48)(n)+1)
with K(t) being the memory kernel defined in Eq. (14). responding multipoint correlation functions of the noise
The set of Eqs. (38a-38b) can then be used to compute ζ(t) of the SBM reveals important common features of
all the multipoint correlation functions of ξ(t) and con- these two processes. Indeed, the correlation functions of
sequently of Y(t). It is straightforward to recover the ζ(t) are obtained straightforwardly by using the defini-
two point case of Eq. (26), whereas we provide below as tion of Eq. (19) and the Wick theorem in Eq. (33):
an example the four point function. First, we need to
compute Eq. (38b) in time space: (cid:42)2N (cid:43) N
(cid:89) 1 (cid:88) (cid:89) (cid:0) (cid:1)
ζ(t ) = K t ×
g(t1,t2,t3,t4)=[K(t1)δ(t2−t1)δ(t3)δ(t4) j N2N σ(m)
+K(t )δ(t −t )δ(t )δ(t )+K(t )δ(t −t )δ(t )δ(t ) j=1 σ∈S2Nm=1
1 1 3 2 4 2 2 4 1 3 (cid:0) (cid:1)
×δ t −t . (42)
σ(2N−m+1) σ(m)
+K(t )δ(t −t )δ(t )δ(t )+K(t )δ(t −t )δ(t )δ(t )
1 1 4 2 3 2 2 3 1 4
+K(t )δ(t −t )δ(t )δ(t )], (39)
3 3 4 1 2 Odd correlation functions of ζ(t) are zero as for ξ(t). As
andthensolvethe (2N)!N!(cid:12)(cid:12) =6convolutionintegrals an example to better clarify our discussion, we provide
N2N (cid:12) the four point correlation function:
N=2
of Eq. (38a). This can be done explicitly, so that we
derive:
(cid:104)ζ(t )ζ(t )ζ(t )ζ(t )(cid:105)=K(t )K(t )δ(t −t )δ(t −t )
(cid:10) (cid:11) 1 2 3 4 1 2 1 3 2 4
ξ(t )ξ(t )ξ(t )ξ(t ) =[K(min(t ,t ))×
1 2 3 4 1 2 +K(t )K(t )δ(t −t )δ(t −t )
1 3 1 2 3 4
×K(|t −t |)δ(t −t )δ(t −t )+K(min(t ,t ))×
1 2 4 1 3 2 1 3 +K(t )K(t )δ(t −t )δ(t −t ). (43)
2 4 1 4 2 3
×K(|t −t |)δ(t −t )δ(t −t )+K(min(t ,t ))×
1 3 4 3 2 1 1 4
×K(|t −t |)δ(t −t )δ(t −t )]. (40) A first remark has to be done when we set N = 2, thus
1 4 3 1 4 2
studying the two point correlation function. Indeed, this
We verified that the same similar structure of the time
is found to be the same for both the noises ξ(t) and ζ(t)
dependent coefficients is shared by the six point corre-
and equal to Eq. (26), thus explaining why the corre-
lation function. Considering the recursive structure ev-
sponding integrated processes Y(t) and Y (t) share the
ident from Eqs. (38a-38b), we conjecture the following ∗
same MSD. On the contrary, differences are evident only
formula for the even correlation functions in time space
ifwelookatthehigherordercorrelationfunctions. Thus,
(with t =0 kept fixed by the permutations):
0 the two integrated processes are distinguishable only by
(cid:42)2N (cid:43) N lookingatquantitiesdependentonthesehigherordercor-
(cid:89)ξ(t) = 1 (cid:88) (cid:89)δ(cid:0)t −t (cid:1)× relation functions, e.g., the PDFs or the corresponding
N2N σ(2N−m+1) σ(m) higher order correlation functions of the integrated pro-
j=1 σ∈S2Nm=1
(cid:88) (cid:0) (cid:1) cesses. Furthermore, by comparing Eqs. (41-42), we can
× Θ t −t ×
σ(σ(cid:48)(m)) σ(σ(cid:48)(m−1)) observethesamesimilarstructureofthedeltafunctions,
σ(cid:48)∈SN typical of Gaussian processes, but with a different corre-
(cid:0) (cid:1)
×K t −t . (41) lated and mainly not factorizable time structure of the
σ(σ(cid:48)(m)) σ(σ(cid:48)(m−1))
coefficients in the case of ξ(t), which depends on the dif-
ferencebetweensuccessivetimeintheorderedsequences.
C. COMPARISON WITH THE SCALED BM This ultimately causes its non Gaussian typical charac-
ter. Infact,inthespecificcaseofaconstantmemoryker-
Once the underlying noise structure of the CTRW is nel, for all times or in some scaling limit, the two noises
revealedbyEqs.(38a-38b-41),acomparisonwiththecor- coincide and reduce to a standard Brownian motion.
8
IV. MODELS WITH EXTERNAL FORCES
Wenowconsidermodelsofanomalousprocessesinthe 2
presence of external forces [39, 51–55]. Let us first focus
on the random walk picture of the CTRW and assume 1
that external force fields, which depends on the position )
ofthewalker,onlymodifyitsdynamicsduringthejumps. Y(t 0
In the continuum limit, these forces are then naturally
included in the Langevin equation of the process X(s), −1
thus modifying Eqs. (2a-2b) into [31]: (a)
−2
√
X˙(s)=F(X(s))+ 2σξ(s), (44a) 2
T˙(s)=η(s), (44b)
1
)
where the function F(x) satisfies standard conditions Y(t 0
[56]. However, different scenarios may be observed in
experiments where forces can modify the position of the
−1
walker also during the waiting times between different
(b)
jumps,withoutultimatelychangingtheunderlyingwait-
−2
ing times distribution. For instance, we would expect 0 20 40 60 80 100
t
this situation to occur for the motion of an organelle in-
side the cytoplasm of a cell, which is freely migrating or
Figure 2. (Colors online) Simulated trajectories of a CTRW
drivenbyanexternalfield. Thisdifferentsituationturns
withalinearviscous-likeforceactingalongitsalltimeevolu-
outnottobeeasilydescribedwiththetime-changetech- tion(panela,Eq.(45))oractingonlyduringthejumps(panel
nique, as it is not clear how to modify the Langevin sub- b, subordinated Eqs. (44a-44b)). Numerical algorithms are
ordinated equations in order to take into account these adapted from [49]. The difference on how the force affects
further changes in the position variable. However, the the dynamics during trapping events is evident (dotted ar-
characterization of the noise ξ(t) provided by Eqs. (38a- rows): (a) the force acts on the particle, thus damping Y(t)
38b), or equivalently by Eq. (41), enables us to describe towards zero; (b) the force does not act, so that the particle
gets physically stuck and Y(t) is kept constant.
it by defining a new class of models, defined with the
Langevin equation:
√
Y˙(t)=F(Y(t))+ 2σξ(t). (45) but this formalism does not have an evident connection
with ours.
The difference between the dynamical behaviours gen- Clearly, the inclusion of a force changes the renewal
eratedbythetwomodelsbecomesclearwhenwelookat pictureforthepositionvariableY(t),whichcannolonger
theirsimulatedtrajectories. InFigure2weplotthepaths be expressed as a superposition of i.i.d. position incre-
of Y(t) obtained both via subordination of Eqs. (44a- ments as in Eqs. (7,8). These increments now depend on
44b) (panel b) and via integration of Eq. (45) (panel a) theaccumulatedpositionuptothetimebeforethejump.
for a linear viscous-like force F(x) = −γx with γ posi- However, the process T(s), i.e., the stochastic process of
tiverealconstant. Ontheonehand, inthesubordinated the jump times parametrized by the arc-length still rep-
dynamics(dottedarrows,panelb)weobservetimeinter- resents a renewal process, since the waiting times are
vals where the corresponding anomalous process Y(t) is unaffected by the force.
constant, meaning that the walker, in the corresponding In the following, we present a comparison of the MSD
renewal picture, is waiting for the next jump to occur obtained from Eqs. (44a-44b-45) for a tempered stable
without any force being able to modify its position. On subordinator as in Sec. IIC and for different choices of
the other hand, during these same intervals the process theexternalforceF(x). Exceptwhenexplicitlystatedwe
Y(t) generated by Eq. (45) is rapidly damped towards assume zero initial condition, so that the MSD coincides
zero(dottedarrows,panela),meaningthatthewalkeris withthesecondordermoment. Werecallthatthemodel
being driven by the external force. While indeed exter- ofEq.(45)definedwiththetimescalednoiseζ(t)instead
nalforcesactonlyduringthejumptimesinthestandard of ξ(t) provides the same MSD.
subordinated case, in our case they affect the dynamics
of the system for all times, without intrinsically modify-
ing the waiting times distribution and equivalently the
A. CONSTANT FORCE CASE
relation between the number of steps and the physical
time. We mention that another scenario involving exter-
nalfieldsdirectlymodifyingthewaitingtimedistribution We first look at the case of a constant homogeneous
of the random walk has been recently discussed in [57], forcefield: F(Y(t))=F withF ∈R+,forwhichEq.(45)
9
becomes:
√
Y˙(t)=F + 2σξ(t). (46)
108 1012 ∼t2
This equation can be solved formally for the exact tra- +Y2(t),
6
10
jectory of Y(t):
6
10
0
10
√ (cid:90) t ∼tα
Y(t)=F t+ 2σ ξ(τ)dτ (47)
10−2 100 102 104 ∼t2
0 104
andthenused,togetherwithEq.(26),toderivetheMSD:
2
(cid:90) t 10
(cid:10)Y2(t)(cid:11)=F2t2+2σ K(τ)dτ (48)
0
µ=0
100 µ=0.1
orequivalentlyinLaplacetransformasafunctionofΦ(s):
+Y2(t), µ=1
∼tα µ=10
(cid:68) (cid:69) 2F2 2σ µ=100
Y(cid:101)2(λ) = + . (49)
λ3 λΦ(λ) 10−2 100 102 104
t
Inthesubordinatedcase,theMSDiscomputedwiththe Figure3. (Colorsonline)MSDofananomalousprocesswith
sametechniqueofEq.(13)butwiththedifferentvariance tempered stable (α = 0.2) distributed waiting times in the
(cid:10)X2(s)(cid:11)=(cid:0)F2s2+2σs(cid:1). In Laplace space we obtain: presenceofaconstantforceactingthroughoutthealldynam-
ical evolution (main panel) or only during the jump times
(inset). These two different scenarios are obtained with the
(cid:68) (cid:69) 2F2 σ2
Y(cid:101)2(λ) = + . (50) ξ−drivenprocessorwiththesubordinationtechnique,i.e.,by
λ(Φ(λ))2 λΦ(λ) numerical Laplace inverse transform of Eqs. (49-50) respec-
tively. The different long time scaling is evident: (main) the
The Laplace inverse transform of both Eqs. (49-50) is ξ−drivenprocessexhibitscrossovertoballisticdiffusioninall
cases and without any dependence on the tempering param-
plotted, together with their corresponding scaling be-
eter µ; (inset) the time-changed process exhibits crossover to
haviours,inFigure3(mainpanelandinsetrespectively).
ballistic diffusion with µ-dependent scaling coefficient when
Inthesmalltimelimit,wefindthatbothsharethesame
µ (cid:54)= 0, whereas it still scales as a power-law with exponent
power-law scaling of Eq. (18). However, they differ be-
2α for µ=0.
tween themselves and with Eq. (18) when we look at the
scaling for long times. On the one hand, Eq. (49) pro-
vides the long time scaling: (cid:10)Y2(t)(cid:11)∼F2t2. Hence, the B. HARMONIC POTENTIAL CASE
constantforceinthislimitinducesacrossoverfromsubd-
iffusivetoballisticdynamics. Examplesofthisnonlinear
Wenowconsideranexternalharmonicpotential,lead-
behaviour have been recently discovered in the dynam-
ingtoafriction-likeforce: F(Y(t))=−γY(t)withγ real
ics of chromosomal loci, which exhibit rapid ballistic ex-
positive constant. Thus, Eq. (45) provides the following:
cursions from their fundamental subdiffusive dynamics,
√
caused by the viscoelastic properties of the cytoplasm Y˙(t)=−γY(t)+ 2σξ(t). (52)
[9, 19]. Furthermore, it is evident that the exponential
dumpingofthewaitingtimes’distributiondoesnotaffect As before, we can solve formally Eq. (52) for the trajec-
the long time scaling, differently from the corresponding toryofY(t)anduseittogetherwithEq.(26)tocompute
scaling of Eqs. (50), which turns out to be (Figure 3, the Laplace transform of the corresponding MSD:
inset):
(cid:68) (cid:69) 2σ
Y(cid:101)2(λ) = . (53)
(cid:16) (cid:17)2 (λ+2γ)Φ(λ)
(cid:10)Y2(t)(cid:11)∼ Fµα1−α t2 µ(cid:54)=0 (51) Onthecontrary,inthesubordinatedcasewecanproceed
 2F2 t2α µ=0 asinEq.(13)bysubstituting: (cid:10)X2(s)(cid:11)= σ (cid:0)1−e−2γs(cid:1).
Γ(1+2α) γ
One can thus obtain the result below:
Thus, we find the same crossover to ballistic diffusion (cid:68) (cid:69) σ
whenµ(cid:54)=0, butwithdifferentµ-dependentscalingcoef- Y(cid:101)2(λ) = . (54)
λ[2γ+Φ(λ)]
ficients,whereasintheCTRWcase(µ=0)thiscrossover
patternislostandthepower-lawscalingisconserved,al- We plot in Figure 4 the numerical inverse transform of
though with a different exponent. Eqs. (53-54) (main panel and inset respectively), along
10
with their scaling behaviour for small times. While the
smalltimescalingisinbothcasesthesameasinEq.(18), 103
0
we observe a different behaviour in the long time limit. 10
∼tα
Indeed, we find for Eq. (53) the following scaling laws:
(cid:10)Y2(t)(cid:11)∼(cid:40) σµγ1−αtαα−1 µµ(cid:54)==00 (55) 102 10−1 hY2(t)i
γΓ(α) 10−1 103
1
Thus, in the CTRW case the MSD decreases as a power- 10
law towards zero. If we recall that this process is equal
totheSBMuptotheMSD,thisisthesameanomalyal-
ready reported in [30]. However, we also show that Y(t) 0
10 µ=0
correctly converges to a plateau for µ (cid:54)= 0, this being µ=0.1
the expected dynamical behaviour of confined diffusion. µ=1
By recalling that the waiting times are tempered stable µ=10
distributed,theinterpretationofthementionedanomaly 10−1 µ=100
∼tα
becomes clear. Indeed, the truncation of the power-law
tails of the waiting times’ distribution is fundamental to hY2(t)i
let the system find a stationary state, so that the MSD
−2
10
can converge to a plateau , which is typical of confined −5 −3 −1 1 3
10 10 10 10 10
diffusion. In fact, no damping of the tails is done in the t
CTRWcase,meaningthatverylongtrappingeventsmay
Figure4. (Colorsonline)MSDofananomalousprocesswith
still happen with non zero, but small probability. Thus,
tempered stable (α = 0.2) distributed waiting times in the
if we wait long enough, i.e., in the long time limit, these presence of a linear viscous force acting at all times (main
events eventually occur. However, Eq. (52) establishes panel) or only during the jump times (inset). These two
that the system is affected by the external linear force cases are obtained with the ξ−driven process or with the
also during such events, which then damps all the oscil- subordination technique, i.e., by numerical Laplace inversion
lations of the system. This clearly implies that the MSD of Eqs. (53-54) respectively. Whereas for small times the
should decrease to zero, because the system is not able twoprocessesexhibitsubdiffusivescaling,theirlongtimebe-
haviourclearlydiffers: (main)theMSDoftheξ−drivenpro-
to disperse and gets immobilized in Y =0. On the con-
cess decreases to zero in the CTRW case (µ=0), whereas it
trary, in the subordinated case the effect of the external
converges to a µ-dependent plateau for µ (cid:54)= 0; (inset) in the
force is stopped during the trapping events, so that the
subordinatedcaseallthecurvesconvergetothesameplateau.
system does not get trapped in the zero position in the
long time limit. Indeed, the MSD for different values of
µ share the same long-time plateau: (cid:10)Y2(t)(cid:11)∼ σ.
γ
We then investigated the dynamics exhibited by pro-
cesses driven by the ξ−noise in the presence of external
V. CONCLUSION force fields and compared it with the one observed for
usualsubordinatedprocesses. Ingeneralterms,wefound
that these processes belong to a new class of CTRW-like
In this work, we identified the underlying noise struc-
processeswhereexternalforcesareexertedonthesystem
ture of a free diffusive CTRW with an arbitrary wait-
at all times, i.e., both when the corresponding walker
ing times distribution and we defined its correspond-
jumps or waits for the next jump to occur. Clearly, this
ing stochastic force. This enabled us to write a new
is different from the original subordinated model, where
Langevin equation, describing its dynamics directly in
external forces are implicitly assumed to modify the dy-
physical time and equivalently to the original formu-
namics only during the jump times. Consequently, dur-
lation obtained with the subordination technique. We
ing the typical trapping events of subdiffusive dynamics
then derived a general formula, both in Laplace space
theanomalousprocessY(t)becomesconstantontheone
andinphysicaltime,providingallitsmultipointcorrela-
hand, when it is generated via subordination, or it is de-
tionfunctions,which,althoughpresentingthesametime
terministically driven by the force on the other hand,
structureofGaussianprocesses,havetimedependentco-
when it is driven by the ξ−noise in physical time.
efficientswithanonfactorizabledependenceonthemem-
ory kernel generated by the corresponding subordinator Furthermore, we found that these processes have the
of the equivalent time-changed formulation. Thus, ex- sameMSDofthoseobtainedwiththecharacteristicnoise
cept for the specific choice of a constant kernel, which of the SBM with time dependent diffusion coefficient
recovers the factorizability of these coefficients, but re- being a function of their memory kernel. This rela-
duces the noise to a standard Brownian motion, our new tion indeed both provides a better interpretation for the
ξ−noise was shown to be naturally both non Gaussian anomaly reported in [30] and show that the correct scal-
and non Markov. ing of the MSD typical of confined motion can be ob-