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: January 31, 2015)

Systems living in complex non equilibrated environments often exhibit subdiffusion characterizedby a sublinear power-law scaling of the mean square displacement. One of the most common modelsto describe such subdiffusive dynamics is the continuous time random walk (CTRW). Stochastictrajectories of a CTRW can be described mathematically in terms of a subordination of a normaldiffusive process by an inverse Levy-stable process. Here, we propose a simpler Langevin formulationof CTRWs without subordination. By introducing a new type of non-Gaussian noise, we are able toexpress the CTRW dynamics in terms of a single Langevin equation in physical time with additivenoise. We derive the full multi-point statistics of this noise and compare it with the noise drivingscaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive behaviour.Interestingly, these two noises are identical up to the level of the 2nd order correlation functions, butdifferent in the higher order statistics. We extend our formalism to general waiting time distributionsand force fields, and compare our results with those of SBM.

I. INTRODUCTION

Many systems in nature live in complex non-equilibrated or highly crowded environments, thus ex-hibiting anomalous diffusive patterns, which deviate fromthe well known Fick’s law of purely thermalized systems[1–3]. Their distinctive feature is the power-law scalingof the mean-square displacement (MSD) [1–5]:

E[(Y (t)− Y0)2

]∼ tα, (1)

where E[ · ] indicates the ensemble average over differ-ent realizations of the stochastic process Y (t) describingthe dynamics, usually either a velocity or a position. Y0

is its initial condition and α ∈ R+. While Fick’s lawis recovered by setting α = 1, thus predicting for nor-mal diffusion the typical liner scaling of the MSD, wecan distinguish between different types of anomalous be-haviour. Indeed, we define subdiffusion if 0 < α < 1 andsuperdiffusion if α > 1, which correspond to processesdispersing with a slower or faster pace than Brownianmotion, respectively. Examples of such anomalous pro-cesses were first found in physical systems, such as chargecarriers moving in amorphous semiconductors, particlesbeing transported on fractal geometries or diffusing inturbulent fluids/plasma or in heterogeneous rocks (see[6] and references therein). However, with the recent im-provements of experimental techniques in biology, jointposition-velocity datasets have been obtained, which arerevealing the existence of many more examples in liv-ing systems. On the one hand, cells have been foundto move often superdiffusively [7–12], whereas biologi-cal macromolecules, like proteins and chromosomal loci,show subdiffusive scaling while moving within the cyto-plasm or on the cells’ membrane, due to the viscoelastic

∗ Correspondence to: a.baule@qmul.ac.uk

properties of such media [13, 14]. Furthermore, thesesystems often exhibit a richer dynamical behaviour, suchas non linear MSDs, showing crossover between differentscaling regimes [15–19] at different timescales, and/or adependence of the corresponding diffusion coefficients onenergy-driven active mechanisms [15, 18–21].

Considering this wide, though not exhaustive, varietyof different anomalous behaviours, one needs to have atool-kit of well studied models with which trying to fitthe experimental data and infer the specific microscopicprocesses underlying the observed dynamics. Here wefocus on subdiffusive processes, for which many modelshave been introduced so far, which are capable of repro-ducing the characteristic scaling of Eq. (1), while stillshowing distinct features if we look at other properties,like the multipoint correlation functions [22–28]. Amongthe most commonly applied to data analysis, we find thecontinuous-time random walk (CTRW) [2, 29] and thescaled Brownian motion (SBM) [30–33].

In the seminal paper [29], the CTRW was introducedas a natural generalization of a random walk on a lattice,with waiting times between the jumps and their size be-ing sampled from general and independent probabilitydistributions. Only later, a convenient stochastic rep-resentation of these processes was derived in terms ofsubordinated Langevin equations [34], which provided asuitable formalism to derive their multipoint correlationfunctions [22, 24, 25]. Although the focus was first onpower-law distributed waiting times, which indeed pro-vided Eq. (1) exactly for all times, recent works adoptedmore general distributions [35–38], thus being able tomodel the crossover phenomena so often occurring in bi-ological experiments.

On the other hand, the SBM has been recently intro-duced as a Gaussian model of anomalous dynamics [30],providing the same scaling of Eq. (1) for all its dynam-ical evolution. If B(t) is a usual BM, its scaled versionis defined by making a power-law change of time withexponent α: B(tα). Although being commonly used to

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fit data [13, 31, 39], it has recently been shown to be anon stationary process with paradoxical behaviour underconfinement, i.e. in the presence of a linear viscous-likeforce, as the corresponding MSD unboundedly decreasestowards zero. This is suggested to be ultimately causedby the time dependence of the environment, either of thetemperature or of the viscosity. As a consequence, it hasbeen ruled out as a possible alternative model of anoma-lous thermalized processes [33].

In this paper, we derive a new type of noise, which al-lows us to express a free diffusive CTRW in terms of a sin-gle Langevin equation in physical time. We provide thefull characterization of its multipoint correlation func-tions and we compare them with those of the noise driv-ing a SBM. Both purely power-law waiting times and gen-eral waiting time distributions are discussed [38, 40]. Wefind that the correlation functions are identical up to thetwo point ones, but different for higher orders: The noisedriving SBM is a Gaussian noise, while our new noisedriving a CTRW is clearly non-Gaussian. Here, all oddcorrelation functions vanish, as for Gaussian noise, butthe even ones do not satisfy Wick’s theorem [41, 42]. Thenewly defined noise enables us to define a class of CTRWlike processes with forces acting for all times, which aredifferent from the corresponding standard CTRWs. Fur-thermore, we revisit the behaviour of the SBM under con-finement and show that its MSD correctly converges to aplateau as it is typical of confined motion [43], providedthat we use more general time changes with truncatedpower-law tails. This suggest that the anomaly observedin [33] is mainly due to the localizing effect of the exter-nal linear force, which is able to trap the particle in thezero position if we allow for infinitely long waiting timesbetween the jumps to eventually occur in the long timelimit.

A. NOTATION

We use the following notation throughout the paper.

• Fourier transform: g denotes the Fourier trans-form of a function g(x) defined on the real line;

g(k) = F {g(x)} (k) =

∫ +∞

0

eikxg(x) dx. (2)

• Laplace transform: f denotes the Laplace trans-form of a function f(t) defined on the positive halfline;

f(λ) = L{f(t)} (λ) =

∫ +∞

0

e−λtf(t) dt. (3)

• Convolution: ϕ1 ∗ ϕ2 denotes the convolution oftwo functions ϕ1 and ϕ2, defined on the positivehalf line;

(ϕ1 ∗ ϕ2)(t) =

∫ t

0

ϕ1(t− τ)ϕ2(τ) dτ. (4)

The corresponding definitions for functions of mul-tiple variables follows straightforwardly.

II. CTRWS, SCALED BM ANDGENERALIZATION TO ARBITRARY WAITING

TIMES’ DISTRIBUTION AND TIMETRANSFORMATIONS

We provide here a brief review of the free diffusiveCTRW and SBM, which will be useful later in the dis-cussion. Our main interests are their stochastic Langevinformulation and both the Fokker-Planck (FP) equationand the MSD of the corresponding integrated processes.We then generalize these results to the case of arbitrarywaiting times distribution or time transformations forCTRW and SBM respectively.

A. CTRW

A Langevin representation of a CTRW was first pro-posed in [34], where the method of the stochastic time-change of a continuous-time process is used. Its set-upconsists in introducing two auxiliary processes X(s) andT (s), which we assume for now to be purely diffusive andLevy stable with parameter α (0 < α ≤ 1) respectively.They both depend on the arbitrary continuous parame-ter s and have dynamics described in terms of Langevinequations:

X(s) =√

2σ ξ(s) (5a)

T (s) = η(s) (5b)

where ξ(s) and η(s) are two independent noises. ForX(s)to be a normal diffusion, we require ξ(s) to be a whiteGaussian noise with E[ξ(s)] = 0 and E[ξ(s1)ξ(s2)] =δ(s2 − s1). On the other hand, η(s) is a stable Levynoise with parameter α [44]. The anomalous CTRW isthen derived by making a randomization of time, i.e. byconsidering the time-changed (or subordinated) process:Y (t) = X(S(t)), with S(t) being the inverse of T (s),defined as a collection of first passage times:

S(t) = infs>0{s : T (s) > t}. (6)

The process Y (t) is easily shown to satisfy Eq. (1) exactlyfor all its time evolution, by recalling that the probabilitydensity function (PDF) of S(t) reads in Laplace trans-

form as h(s, λ) = λα−1e−sλα

[22] and that E[X2(s)

]=

2σ s. Indeed, we obtain in Laplace space:

E[Y 2(λ)

]=

∫ +∞

0

E[X2(s)

]h(s, λ) ds =

2σ

λ1+α, (7)

whose inverse transform confirms its anomalous scaling:

E[Y 2(t)

]=

2σ

Γ(1 + α)tα. (8)

3

As expected, this same MSD is obtained by takingthe diffusive limit of the microscopic formulation of theCTRW, which is given by a generalized random walk,where we allow for asymptotically power-law distributedwaiting times between the jumps of the walker, whosesizes are drawn from a distribution with finite variance[2]. In the diffusive limit, this model also provides a frac-tional diffusion equation for the PDF of Y (t):

∂

∂tP (y, t) = Dα

∂2

∂y2D1−αt P (y, t), (9)

where Dα is a generalized diffusion coefficient and

D1−αt f(t) = 1

Γ(α)∂∂t

∫ t0

(t− τ)α−1

f(τ) is the Riemann-

Liouville time-integral operator, which makes the nonMarkovian character of the CTRW evident. It is thennatural to investigate if the set of Eqs. (5) can give thissame FP equation. This has been proved in [40, 45, 46],with the specification: Dα = σ

Γ(1+α) , thus confirming the

equivalence in the diffusive limit of the original discretemodel and of the subordinated Langevin Eqs. (5).

B. SCALED BM

If instead of a stochastic time change, we consider thedeterministic time transformation t → t∗ = tα in thenormal diffusive process X(t) (now in the physical timet), we obtain the SBM Y∗(t) = X(t∗). Its equivalentLangevin equation is given by [30–33]:

Y∗(t) =√

2ασ tα−1 ξ(t), (10)

with ξ(t) being a white Gaussian noise (with the sameproperties as before, but in the physical time t). By usingEq. (10) we can prove straightforwardly that the MSD ofY∗(t) is the same as Eq. (8) and that the correspondingFP equation is given by:

∂

∂tP (y, t) = ασ tα−1 ∂

2

∂y2P (y, t), (11)

which has time dependent diffusion coefficient [47]. Thisprocess preserves all the properties of Brownian motion[30]: it is indeed Gaussian with time-dependent varianceand Markov, as the monotonicity of the time change pre-serves the ordering of time. Furthermore, Y∗(t) is self-similar and it has independent increments for non over-lapping intervals. However, differently from Brownianmotion, it is strongly non stationary [33]. Furthermore,Y∗(t) turns out to be the mean-field approximation of theCTRW, as it describes the motion of a cloud of randomwalkers performing CTRW motion in the limit of a largenumber of walkers [32].

C. ARBITRARY WAITING TIMES’DISTRIBUTION AND TIME

TRANSFORMATIONS

In this section, we first focus on the generalization ofEqs. (5) to arbitrary waiting time distributions of theunderlying random walk [35–38, 40, 48]. This extensionis obtained naturally by choosing a different process T (s)with the only assumption of it being non decreasing inorder to preserve the causality of time. Thus, we considerη(s) in Eq. (5b) to be an increasing Levy noise with pathsof finite variation and characteristic functional [49]:

G[k(τ)] = E[e−

∫ +∞0

k(τ)η(τ) dτ]

= e−∫ +∞0

Φ(k(τ)) dτ .

(12)Here Φ(k(s)) is a non negative function with Φ(0) = 0and strictly monotone first derivative, while k(τ) is a testfunction. We recall that for Φ(s) = sα we recover theCTRW model. Under these assumptions, the integratedprocess T (s) is a a one-sided increasing Levy process withfinite variation. Furthermore, we assume η(s) to be in-dependent on the realizations of ξ(s) in Eq. (5a). As aconsequence of the finite variation and the monotonicityof the paths of T (s) respectively, S(t) has continuous andmonotone paths, with this second property implying thefundamental relation [22]:

Θ(s− S(t)) = 1−Θ(t− T (s)). (13)

Similarly to Eq. (7), we can derive the corresponding

MSD by recalling that h(s, λ) = Φ(λ)λ e−sΦ(λ) [38, 40]:

E[Y 2(t)

]= 2σ

∫ t

0

K(τ) dτ, (14)

for K(t) being related in Laplace space to Φ(s) by:

K(λ) =1

Φ(λ). (15)

Furthermore, the PDF of Y (t) is obtained by solving thegeneralized FP equation [40]:

∂

∂tP (y, t) = σ

∂2

∂y2

∂

∂t

∫ t

0

K(t− τ)P (y, τ) dτ, (16)

whose solution in this particular case can be found forgeneral Φ(s) in Laplace space:

P (y, λ) =1

λ

√Φ(λ)

2σe−

√Φ(λ)2σ |y|. (17)

We look as an example at the case of a tempered stableLevy noise with tempering index µ and stability index α[50], which is obtained by setting Φ(λ) = (µ + λ)α −µα, e.g. K(t) = e−µ t tα−1Eα,α((µ t)α) [51]. As alreadypointed out, the CTRW case is recovered by setting µ =0, meaning that we do not truncate the long tails of thedistribution, thus accounting for very long waiting times

4

with a power-law decaying probability of occurrence. Weplot in Fig. 1 the numerical Laplace inverse of Eq. (17)(main) and the corresponding MSD (inset) at a fixed timet = 1000 (dotted line in the inset), which is given by [52]:

E[Y 2(t)

]=

2σ

µα

[−1 +

∞∑n=0

γ(µt;αn)

Γ(αn)

], (18)

with γ(x; a) =∫ x

0e−tta−1 being an incomplete gamma

function, leading to the asymptotic behaviour [40, 52]:

E[Y 2(t)

]∼{ 2σ

Γ(1+α) tα t << 1(

2σα µ

1−α) t t >> 1(19)

We remark that Eq. (19) does not apply to the long timescaling of CTRWs, for which it would predict a vanishingMSD. In fact, CTRWs do not exhibit a crossover fromsubdiffusive to normal behaviour, but their MSD scalesas a power-law for all times. As expected, for µ = 0 werecover the typical non Gaussian shape of the PDF of afree diffusive CTRW [2]. However, for increasing valuesof µ, the PDF of Y (t), although still being non Gaussian,broadens, thus getting closer to a Gaussian. This has alsoevident consequences on the dynamical behaviour of theMSD, which for increasing values of µ goes from a puresubdiffusive scaling to a normal one (inset).

We now discuss the corresponding extension of theSBM to arbitrary time transformations involving thekernel K(t) obtained by Laplace inverse transform ofEq. (15). We then generalize Eq. (10) by adopting K(t)as the time dependent coefficient of the white noise:

Y∗(t) =√

2σK(t) ξ(t) = ζ(t), (20)

where we define the correlated noise ζ(t) with E[ζ(t)] =0 and two-point correlation function: E[ζ(t1)ζ(t2)] =2σK(t1)δ(t1 − t2). This explicit time dependence clearlysignals that ζ(t) is a non stationary noise. It is easilyshown that the MSD of Y∗(t) is identical to the one ofY (t) given by Eq. (14). However, even if they share thesame MSD, Y (t) and Y∗(t) provide different PDFs. In-deed, Y∗(t) corresponds to a time rescaled Brownian mo-tion X(t∗) with transformation:

t∗ =

∫ t

0

K(τ) dτ. (21)

In the case of the usual Brownian motion the correspond-ing diffusion equation has a Gaussian solution: P (y, t) =

1√4πσt

e−(y−y0)2

4σt for the initial condition P (y, 0) = δ(y −y0). Since Y∗(t) is just Brownian motion in the rescaledtime t∗, we obtain similarly a Gaussian solution, providedwe choose the same initial condition:

P (y, t) =1√

4πσt∗e−

(y−y0)2

4σt∗ , (22)

with t∗ as in Eq. (21). We see that P (y, t) is a solutionof the diffusion equation:

∂

∂tP (y, t) = σK(t)

∂2

∂y2P (y, t), (23)

−20 −15 −10 −5 0 5 10 15 2010

−3

10−2

10−1

100

y

P(y,t)

10−2

100

102

10410

−5

10−3

10−1

101 〈Y 2(t)〉/t

µ = 0µ = 10−4

µ = 10−3

µ = 10−2 ∼ tα−1

Figure 1. PDF (main) and MSD normalized to t (inset) ofan anomalous process Y (t) obtained by subordination of apure diffusive process with a tempered stable Levy processof tempering index µ and stability parameter α = 0.2. ThePDF is obtained by numerical Laplace inversion of Eq. (17)at t = 1000 (black dotted lines in the inset) [53]. Thesmooth transition from the non-Gaussian PDF typical ofCTRWs (µ = 0) and the Gaussian one of normal diffusion(µ→ +∞) is evident, along with the corresponding transitionfrom anomalous to normal scaling of the MSD for increasingµ at a fixed time. Simulations, obtained with the algorithmsof [54, 55], agree perfectly with the analytical results.

with the time dependent diffusion constant: D(t) =σK(t). We remark that Eq. (10) can be recoveredfrom these general results by setting Φ(λ) = λα, i.e.K(t) = tα−1/Γ(α) and t∗ = tα/Γ(1 + α). However, inorder to have exact equivalence, we need to neglect theconstant multiplicative factors in both K(t) and t∗ andmake the following substitution: σ → ασ.

III. LANGEVIN FORMULATION OFANOMALOUS PROCESSES IN PHYSICAL TIME

A. DEFINITION OF THE NOISE

We proceed in this section to derive a Langevin de-scription of the process Y (t) defined in Eqs. (5) directlyin physical time. Starting from the explicit integral ex-

pression: Y (t) =∫ S(t)

0ξ(τ) dτ , we can write the follow-

5

ing:

Y (t) =

∫ +∞

0

δ(s− S(t))

[∫ s

0

ξ(τ) dτ

]ds

=

∫ +∞

0

(− ∂

∂sΘ(t− T (s))

)[∫ s

0

ξ(τ) dτ

]ds

=

∫ +∞

0

Θ(t− T (s))ξ(s) ds, (24)

where the fundamental relation of Eq. (13) is used to ob-tain the second equality and we then get the third onewith an integration by parts. We remark that the bound-

ary term[−Θ(t− T (s))

∫ s0ξ(τ) dτ

]∣∣+∞0

is zero trivially

for s = 0, but it vanishes also for s→ +∞ because T (s)is increasing, thus always being bigger than any fixed(and finite) time t. Written as in Eq. (24), Y (t) is a dif-ferentiable (although in a generalized sense) function oftime, so that we can take its derivative and obtain theequivalent Langevin equation:

Y (t) = ξ(t) (25)

with the newly defined noise:

ξ(t) =

∫ +∞

0

ξ(s)δ(t− T (s)) ds, (26)

whose properties are fully determined by the choice ofthe waiting times distribution, i.e. equivalently of thefunction Φ(s) in Eq. (12). If we recall the independenceof ξ(s) and η(s), we can show that ξ(t) has zero averageand two point correlation function:

E[ξ(t1)ξ(t2)

]= 2σK(t1)δ(t1 − t2), (27)

with K(t) being specified by Eq. (15). In Laplace space,indeed, E

[ξ(t1)ξ(t2)

]is an integral of the two point char-

acteristic function of T (s), which can then be computed

with Eq. (12): E[ξ(λ1)ξ(λ2)

]= 1

Φ(λ1+λ2) . By making

its inverse Laplace transform, Eq. (27) follows straight-forwardly. Consequently, the character of the noise ξ(t)significantly depends on the choice of the function Φ(s)in Eq. (12). Thus, Eq. (25) defines a new Langevin modeldriven by a generalized and typically non Gaussian noise,except possibly for particular choices of the memory ker-nel K(t), which is able to describe free diffusive anoma-lous processes with arbitrary waiting times distributionequivalently to the subordinated Eqs. (5).

B. CHARACTERIZATION OF THEMULTIPOINT CORRELATION FUNCTIONS

The definition in Eq. (26) enables us to derive acomplete characterization of the multipoint correlationstructure of ξ(t). As a preliminary step, we de-rive the Laplace transform of the multipoint charac-teristic function of T (s), i.e. Z(t1, s1; . . . ; tN , sN ) =

E[∏N

m=1 δ(tm − T (sm))]∀N ∈ N. This is obtained by

using the definition of Eq. (5b) as:

Z(λ1, s1; . . . ;λN , sN ) = E

[N∏m=1

e−λm∫ sm0

η(s′m) ds′m

].

(28)Let us first assume an ordering for the sequence of times:s1 < s2 < . . . < sN and compute the correspondingEq. (28). If we rearrange the exponent by separatingsuccessive intervals, we obtain:

Z(λ1, s1; . . . ;λN , sN ) =

= E[e−λN

∫ sNsN−1

η(s′N ) ds′N−...−(λN+...+λ1)∫ s10 η(s′1) ds′1

]= E

[e−

∑N−1m=0[(

∑Nn=m+1 λn)]

∫ sm+1sm η(s′m) ds′m+1

]=

N−1∏m=0

e−(sm+1−sm)Φ(∑Nn=m+1 λn), (29)

where we define s0 = 0 to simplify the notation andwe exploited the independence of the increments of T (s)to factorize the ensemble average. Furthermore, theirstationarity together with Eq. (12) is then used to getEq. (29). However, in the general case where no a-prioriordering is assumed, we need to consider all the possibleordered sequences. We then introduce the group of per-mutations of N objects SN , whose elements act on the se-quence: s = (s1, . . . , sN ). When we make a permutationof s, we obtain a new sequence with permuted indices:s′ =

(sσ(1), . . . , sσ(N)

). All the possible orderings of s

are thus obtained by summing over all the permutationsin SN . If we assume that σ(s0) = 0, ∀σ ∈ SN , e.g. theinitial time is kept fixed by the permutations, and we usethe result of Eq. (29), we derive:

Z(λ1, s1; . . . ;λN , sN )=∑σ∈SN

N−1∏m=0

Θ(sσ(m+1) − sσ(m)

)×

× e−(sσ(m+1)−sσ(m))Φ(∑Nn=m+1 λσ(n)) (30)

with the ordering of the permuted sequence being speci-fied by the product of Heaviside functions. By factorizingout the first term, we obtain the fundamental result:

Z(λ1, s1; . . . ;λN , sN )=∑σ∈SN

e−sσ(1)Φ(∑Nm=1 λm)× (31)

×N−1∏m=1

Θ(sσ(m+1)−sσ(m)

)e−(sσ(m+1)−sσ(m))Φ(

∑Nn=m+1 λσ(n))

As an example, we recover the two-point case:

Z(λ1, s1;λ2, s2) [40]. If we put N = 2 in Eq. (31)and we consider the two possible permuted sequences:s = (s1, s2) and s′ = (s2, s1), we obtain:

Z(λ1, s1;λ2, s2)=Θ(s2 − s1) e−s1Φ(λ1+λ2)e−(s2−s1)Φ(λ2)

+ Θ(s1 − s2) e−s2Φ(λ1+λ2)e−(s1−s2)Φ(λ1). (32)

6

We now use Eq. (31) to compute the correlation func-tions of ξ(t). Indeed, we obtain from Eq. (26) ∀N ∈ N:

E[ξ(t1) . . . ξ(t2N )

]=

[2N∏m=1

∫ +∞

0

dsm

]×

×E

[2N∏m=1

ξ(sm)

]E

[2N∏m=1

δ(tm − T (sm))

], (33)

where the average is factorized due to the independenceof the noises. If we recall the Wick theorem holding forthe white noise ξ(s) [41, 42]:

E

2N∏j=1

ξ(tj)

= 1

N2N

∑σ∈S2N

N∏j=1

E[ξ(tσ(2N−j+1)

)ξ(tσ(j)

)]=

1

N2N

∑σ∈S2N

N∏j=1

δ(tσ(2N−j+1) − tσ(j)

)(34)

and we substitute it in Eq. (33), we can derive:

E[ξ(t1) . . . ξ(t2N )

]=

(2σ)N

N2N

∑σ∈S2N

[2N∏m=1

∫ +∞

0

dsm

]×

×N∏j=1

δ(sσ(2N−j+1) − sσ(j)

)E

[2N∏i=1

δ(ti − T (si))

]

=(2σ)N

N2N

∑σ∈S2N

[N∏m=1

∫ +∞

0

dsσ(m)

]× (35)

×E

N∏j=1

δ(tσ(2N−j+1) − T

(sσ(j)

))δ(tσ(j) − T

(sσ(j)

))with N integrals being solved by using the delta functionsobtained from E[ξ(s1) . . . ξ(s2N )]. If we make a Laplacetransform of Eq. (35), we obtain an expression involving

Z(λ1, s1;λ2, s2; . . . ;λN , sN ):

E

2N∏j=1

ξ(λj)

=(2σ)N

N2N

∑σ∈S2N

[N∏m=1

∫ +∞

0

dsm

]×

× Z(λσ(1)+λσ(2N), s1; . . . ;λσ(N)+λσ(N+1), sN

), (36)

which can thus be further simplified with Eq. (31). Bysubstitution and by making a further permutation of theindices, we obtain:

E

2N∏j=1

ξ(λj)

=(2σ)N

N2N

∑σ∈S2N

∑σ′∈SN

[N∏m=1

∫ +∞

0

dsσ′(m)

]×

× e−sσ′(1)Φ(∑Nm=1 λm)

N−1∏m=1

[Θ(sσ′(m+1) − sσ′(m)

)× (37)

×e−(sσ′(m+1)−sσ′(m))Φ(∑Nn=m+1(λσ(σ′(n))+λσ(2N−σ′(n)+1)))

],

where the N integrals can then be solved by making suit-able changes of variables. This leads to the followingresult for the Laplace transform of even multipoint func-tions of ξ(t):

E[ξ(λ1) . . . ξ(λ2N )

]=

(2σ)N

N2NΦ(∑2N

m=1 λm

) ∑σ∈S2N

× (38)

×∑σ′∈SN

N−1∏m=1

1

Φ(∑N

n=m+1

(λσ(σ′(n)) + λσ(2K−σ′(n)+1)

)) .We remark that odd multipoint correlation functions arezero; indeed, if we make the substitution 2N → 2N+1 inEq. (33), we obtain an expression depending on the oddmultipoint correlation functions: E[ξ(s1) . . . ξ(s2N+1)],which vanish ∀N ∈ N. The corresponding quantities intime are derived by making the inverse Laplace trans-form of Eq. (38), which can be written as a 2N−foldconvolution:

E[ξ(t1) . . . ξ(t2N )

]=

(2σ)N

N2NK(t1)

N−1∏i=1

δ(ti+1 − ti) ∗2N g(t1, t2, . . . , t2N−1, t2N ) (39a)

g(t1, t2, . . . , t2N−1, t2N ) = L−1

∑σ∈S2N

∑σ′∈SN

N−1∏m=1

1

Φ(∑N

n=m+1(λσ(σ′(n)) + λσ(σ′(2N−σ′(n)+1)))) (39b)

with K(t) being the memory kernel defined in Eq. (15).The set of Eqs. (39) can then be used to compute all themultipoint correlation functions of ξ(t) and consequentlyof Y (t). It is straightforward to recover the two point caseof Eq. (27), whereas we provide below as an example thefour point function. First, we need to compute Eq. (39b)

in time space:

g(t1, t2, t3, t4) = [K(t1)δ(t2 − t1)δ(t3)δ(t4)

+K(t1)δ(t1 − t3)δ(t2)δ(t4) +K(t2)δ(t2 − t4)δ(t1)δ(t3)

+K(t1)δ(t1 − t4)δ(t2)δ(t3) +K(t2)δ(t2 − t3)δ(t1)δ(t4)

+K(t3)δ(t3 − t4)δ(t1)δ(t2)] , (40)

7

and then solve the (2N)!N !N2N

∣∣∣N=2

= 6 convolution integrals

of Eq. (39a). This can be done explicitly, so that wederive:

E[ξ(t1)ξ(t2)ξ(t3)ξ(t4)

]= 4σ2 [K(min(t1, t2))×

×K(|t1 − t2|)δ(t4 − t1)δ(t3 − t2) +K(min(t1, t3))××K(|t1 − t3|)δ(t4 − t3)δ(t2 − t1) +K(min(t1, t4))××K(|t1 − t4|)δ(t3 − t1)δ(t4 − t2)] . (41)

We verified that the same similar structure of the timedependent coefficients is shared by the six point corre-lation function. Considering the recursive structure evi-dent from Eqs. (39), we conjecture the following formulafor the even correlation functions in time space (witht0 = 0 kept fixed by the permutations):

E

2N∏j=1

ξ(t)

=(2σ)N

N2N

∑σ∈S2N

N∏m=1

δ(tσ(2N−m+1) − tσ(m)

)×

×∑σ′∈SN

Θ(tσ(σ′(m)) − tσ(σ′(m−1))

)×

×K(tσ(σ′(m)) − tσ(σ′(m−1))

). (42)

C. COMPARISON WITH THE SCALED BM

Once the underlying noise structure of the CTRW isrevealed by Eqs. (39-42), we found that a comparisonwith the corresponding multipoint correlation functionsof the noise ζ(t) of the SBM reveals important commonfeatures of these two processes. Indeed, the correlationfunctions of ζ(t) are obtained straightforwardly by us-ing the definition of Eq. (20) and the Wick theorem ofEq. (34):

E

2N∏j=1

ζ(tj)

=σN

N

∑σ∈S2N

N∏m=1

K(tσ(m)

)×

× δ(tσ(2N−m+1) − tσ(m)

). (43)

Odd correlation functions of ζ(t) are zero as for ξ(t). Asan example to better clarify our discussion, we providethe four point correlation function:

E[ζ(t1)ζ(t2)ζ(t3)ζ(t4)]=4σ2K(t1)K(t2)δ(t1−t3)δ(t2−t4)

+ 4σ2K(t1)K(t3)δ(t1 − t2)δ(t3 − t4)

+ 4σ2K(t2)K(t4)δ(t1 − t4)δ(t2 − t3). (44)

A first remark has to be done when we set N = 2, thusstudying the two point correlation function. Indeed, thisis found to be the same for both the noises ξ(t) and ζ(t)and equal to Eq. (27), thus explaining why the corre-sponding integrated processes Y (t) and Y∗(t) share thesame MSD. On the contrary, differences are evident onlyif we look at the higher order correlation functions. Thus,

the two integrated processes are distinguishable only bylooking at quantities dependent on these higher order cor-relation functions, e.g. the PDFs or the correspondinghigher order correlation functions of the integrated pro-cesses. Furthermore, by comparing Eqs. (42-43), we canobserve the same similar structure of the delta functions,typical of Gaussian processes, but with a different corre-lated and mainly not factorizable time structure of thecoefficients in the case of ξ(t), which depend on the dif-ference between successive time in the ordered sequence.This indeed causes its non Gaussian typical character.

IV. MODELS WITH EXTERNAL FORCES

We now consider models of anomalous processes inthe presence of external forces [6, 45, 56–59]. Origi-nally the external fields were introduced directly into theLangevin equation of the parent process X(s), thus mod-ifying Eqs. (5) into [45, 59]:

X(s) = F (X(s)) + σξ(s), (45a)

T (s) = η(s), (45b)

with the function F (x) satisfying standard conditions[60]. With this definition, the forces are implicitly as-sumed to act on the subordinated process Y (t) only atthe jump times. Instead, they do not affect the dynam-ics when the moving particle hits an obstacle and theprocess gets trapped. However, the characterization ofthe noise ξ(t) provided by Eqs. (39), or equivalently byEq. (42), enables us to relax this hypothesis and define anew class of models where these forces are exerted on thesystem also during the trapping events. This is obtainedby defining the Langevin equation:

Y (t) = F (Y (t)) +√

2σ ξ(t). (46)

The difference between these two models is clearly ob-served when we look directly at their simulated trajecto-ries. In Fig. 2 we plot the simulated paths of Y (t) ob-tained both via subordination of Eqs. (45) (panel b) andvia integration of Eq. (46) (panel a) for a linear viscous-like force F (x) = −γx with γ positive real constant. Weclearly observe that during the time intervals where Y (t)is constant in the subordinated dynamics (red arrows,panel b), meaning that the particle is immobilized, theforce is instead exerted on the system in the dynamicsgenerated by Eq. (46), so that Y (t) is rapidly dampedtowards zero (red arrows, panel a). Thus, while externalforces act only during the jump times in the subordi-nated case, in the other one they affect the dynamicsof the system for all times. We mention that a differ-ent way of including external fields acting throughoutthe all dynamical evolution of the system is proposedin [61], where however, it is assumed that these forcesmodify the underlying waiting time distribution of therandom walk. This is not the case for Eq. (46), which is

8

−2

−1

0

1

2

Y(t

)

0 20 40 60 80 100−2

−1

0

1

2

t

Y(t

)

(a)

(b)

Figure 2. Simulated trajectories of a CTRW with a linearviscous-like force acting along its all time evolution (panel a,Eq. (46)) or acting only during the jumps (panel b, subordi-nated Eqs. (45)). Numerical algorithms are adapted from [54].The difference on how the force affects the dynamics duringtrapping events is evident (red double arrows): (a) the forceacts on the particle, thus damping Y (t) towards zero; (b) theforce does not act, so that the particle gets physically stuckand Y (t) is kept constant.

thus not suitable to provide a Langevin representation ofthese different processes. In the following, we present acomparison of the MSD obtained from Eqs. (45-46) fora tempered stable subordinator as in Sec. II C and fordifferent choices of the external force F (x). Except whenexplicitly stated we assume zero initial condition, so thatthe MSD coincides with the second order moment. Werecall that the model of Eq. (46) defined with the timescaled noise ζ(t) instead of ξ(t) provides the same MSD.

A. CONSTANT FORCE CASE

We first look at the case of a constant homogeneousforce field: F (Y (t)) = F with F ∈ R+, for which Eq. (46)becomes:

Y (t) = F +√

2σ ξ(t). (47)

This equation can be solved formally for the exact tra-jectory of Y (t):

Y (t) = F t+

∫ t

0

ξ(τ) dτ (48)

and then used, together with Eq. (27), to derive the MSD:

E[Y 2(t)

]= F 2 t2 + 2σ

∫ t

0

K(τ) dτ (49)

or equivalently in Laplace transform as a function of Φ(s):

E[Y 2(λ)

]=

2F 2

λ3+

2σ

λΦ (λ). (50)

In the subordinated case, the MSD is computed with thesame technique of Eq. (14) but with the different varianceE[X2(s)

]=(F 2 s2 + 2σ s

). In Laplace space we obtain:

E[Y 2(λ)

]=

2F 2

λ (Φ(λ))2 +

σ2

λΦ(λ). (51)

The Laplace inverse transform of both Eqs. (50-51) isplotted, together with their corresponding scaling behav-iors, in Fig. 3 (main panel and inset respectively). Inthe small time limit, we find that both share the samepower-law scaling of Eq. (19). However, they differ be-tween themselves and with Eq. (19) when we look at thescaling for long times. On the one hand, Eq. (50) pro-vides the long time scaling: E

[Y 2(t)

]∼ F 2 t2. Hence,

the constant force in this limit induces a crossover fromsubdiffusive to ballistic dynamics. Examples of this non-linear behaviour have been recently discovered in the dy-namics of chromosomal loci, which exhibit rapid ballisticexcursions from their fundamental subdiffusive dynam-ics, caused by the viscoelastic properties of the cytoplasm[14, 19]. Furthermore, it is evident that the exponentialdumping of the waiting times’ distribution does not affectthe long time scaling, differently from the correspondingscaling of Eqs. (51), which turns out to be (Fig. 3, inset):

E[Y 2(t)

]∼

(Fµ1−α

α

)2

t2 µ 6= 0

2F 2

Γ(1+2α) t2α µ = 0

(52)

Thus, we find the same crossover to ballistic diffusionwhen µ 6= 0, but with different µ-dependent scaling coef-ficients, whereas in the CTRW case (µ = 0) this crossoverpattern is lost and the subdiffusive scaling is conserved,although with a different exponent.

B. HARMONIC POTENTIAL CASE

We now consider an external harmonic potential, lead-ing to a friction-like force: F (Y (t)) = −γY (t) with γ realpositive constant. Thus, Eq. (46) provides the following:

Y (t) = −γY (t) +√

2σ ξ(t). (53)

As before, we can solve formally Eq. (53) for the trajec-tory of Y (t) and use it together with Eq. (27) to computethe Laplace transform of the corresponding MSD:

E[Y 2(λ)

]=

2σ

(λ+ 2 γ) Φ (λ). (54)

9

10−2

100

102

104

100

102

104

106

108

t

⟨

Y2(t)

⟩

µ = 0µ = 0.1µ = 1µ = 10µ = 100

10−2

100

102

104

100

106

1012

∼ tα

∼ t2

∼ t2

∼ tα

Figure 3. MSD (colored lines) and corresponding scaling be-havior (black lines) of an anomalous process with temperedstable (α = 0.2) distributed waiting times in the presenceof a constant force acting throughout the all dynamical evo-lution (main panel) or only during the jump times (inset),obtained by numerical Laplace inverse transform of Eqs. (50-51) respectively. The different long time scaling is evident:(1) force acting only during jump times induces µ-dependentscaling coefficient (inset); (2) force acting for all times removesthis dependence and all curves asymptotically converge to thesame one (main).

On the contrary, in the subordinated case we can pro-ceed as in Eq. (14) by substituting: E

[X2(s)

]=

σγ

(1− e−2 γ s

). One can thus obtain the result below:

E[Y 2(λ)

]=

σ

λ [2γ + Φ(λ)]. (55)

We plot in Fig. 4 the numerical inverse transform ofEqs. (54-55) (main panel and inset respectively), alongwith their asymptotic behavior for small times (blacklines). While the small time scaling is in both cases thesame as in Eq. (19), we observe a very different behaviorin the long time limit. Indeed, we find for Eq. (54) thefollowing scaling laws:

E[Y 2(t)

]∼

{µ1−α

γα µ 6= 0σ

γΓ(α) tα−1 µ = 0

(56)

Thus, in the CTRW case the MSD decreases as a power-law towards zero. If we recall that this process is equalto the SBM up to the MSD, this is the same anomaly al-ready reported in [33]. However, we also show that Y (t)correctly converges to a plateau for µ 6= 0, this being theexpected dynamical behavior of confined diffusion. By

10−5

10−3

10−1

101

10310

−2

10−1

100

101

102

103

t

〈Y 2(t)〉

µ = 0µ = 0.1µ = 1µ = 10µ = 100

10−5

10−1

10310

−1

100

∼ tα

∼ tα

Figure 4. MSD (colored lines) and corresponding scaling be-havior (black lines) of an anomalous process with temperedstable (α = 0.2) distributed waiting times for a linear vis-cous like force, obtained by numerical Laplace inversion ofEqs. (54-55) (main panel and inset respectively). While forsmall times we observe subdiffusive scaling, the long time be-havior depends both on how the force is exerted on the systemand on the tails of the waiting times’ distribution. When theforce act for all times (main), the MSD decreases to zero inthe CTRW case (µ = 0), while it converges to µ-dependentplateaus for µ 6= 0. In the subordinated case (force acting dur-ing the jump times only, inset) instead, all curves converge tothe same plateau.

looking at this process, the interpretation of the men-tioned anomaly is clear. Indeed, the truncation of thelong tails of the waiting time distribution is fundamen-tal to let the system find a stationary state, so that theMSD then converges to the characteristic plateau. Infact, in the CTRW case, no damping of the tails is done,so that very long trapping events may still happen withnon zero, but small probability. Thus, if we wait longenough, e.g. in the long time limit, these events eventu-ally occur. However, Eq. (53) establishes that the systemis affected by the external linear force also during suchevents, which then kills all the movements of the sys-tem. This clearly implies that the MSD should decreaseto zero, because the system is not able to disperse andgets immobilized in Y = 0. On the contrary, in the sub-ordinated case the effect of the external force is stoppedduring the trapping events, so that the system does notget trapped in the zero position in the long time limit.Indeed, the MSD for different values of µ share the samelong-time plateau: E

[Y 2(t)

]∼ σ

γ .

10

V. CONCLUSION

In this work, we identified the underlying noise struc-ture of a free diffusive CTRW with an arbitrary wait-ing time distribution and we defined its correspondingstochastic force. This enables us to write a new Langevinequation, which describes its dynamics directly in physi-cal time and equivalently to the original formulation de-rived with the subordination technique. We then deriveda general formula, both in Laplace space and in phys-ical time, providing all its multipoint correlation func-tions, which, although presenting the same time struc-ture of Gaussian processes, have time dependent coeffi-cients with a non factorizable dependence on the memorykernel generated by the corresponding subordinator ofthe equivalent time-changed formulation. Thus, exceptfor specific choices of the kernel recovering their factoriz-ability, the noise is both non Gaussian and non Markov.By using this noise, we defined a new class of CTRW

like processes with external forces being exerted on thesystem for all times. This differs from the original subor-dinated model where they only affect the dynamics dur-ing the jump times, so that during the trapping eventsthe system gets immobilized and the corresponding pro-cess becomes constant. Furthermore, we found that theseprocesses have the same MSD of those obtained with thecharacteristic noise of the SBM with time dependent dif-fusion coefficient being a function of their memory kernel.This relation indeed both provides a better interpretationfor the anomaly reported in [33] and show that the cor-rect scaling of the MSD typical of confined motion can beobtained by choosing more general kernels, which preventan unbounded decay of the diffusion coefficient.

ACKNOWLEDGMENTS

This research utilized Queen Mary’s MidPlus compu-tational facilities, supported by QMUL Research-IT andfunded by EPSRC grant EP/K000128/1.

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