Aging two-state process with Lévy walk and Brownian motion
aa r X i v : . [ phy s i c s . c l a ss - ph ] J un Aging two-state process with L´evy walk and Brownian motion
Xudong Wang, Yao Chen, and Weihua Deng
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems,Lanzhou University, Lanzhou 730000, P.R. China
With the rich dynamics studies of single-state processes, the two-state processes attract moreand more interests of people, since they are widely observed in complex system and have effectiveapplications in diverse fields, say, foraging behavior of animals. This report builds the theoreticalfoundation of the process with two states: L´evy walk and Brownian motion, having been provedto be an efficient intermittent search process. The sojourn time distributions in two states areboth assumed to be heavy-tailed with exponents α ± ∈ (0 , α ± decides the fraction of two states for long times, playing a crucial role in theseMSDs. According to the generic expressions of MSDs, some inherent characteristics of the two-stateprocess are detected. The effects of the fraction on these observables are detailedly presented in sixdifferent cases. The key of getting these results is to calculate the velocity correlation function ofthe two-state process, the techniques of which can be generalized to other multi-state processes. Searching a target is a natural demand in the realworld. At the same time, many physical or biologicalproblems can be regarded as the search processes, de-scribing how a searcher finds a target located in an un-known position. At the macroscopic scale, it is exem-plified as animals searching for food or a shelter [1]. Atthe microscopic scale, one can cite the localization by aprotein of a specific DNA sequence or the active trans-port of vesicles in cells [2]. In these examples, the searchtime is generally a limiting quantity which has to be op-timized by choosing different search strategies. Intermit-tent search strategies have been proved to play a crucialrole in optimizing the search time of randomly hiddentargets [3, 4]. This kind of search behavior could be ex-tended to broader research domains such as the theoryof stochastic processes [5], applied mathematics [6], andmolecular biology [7]; and it also motivates some newinteresting research topics [8].For the intermittent search process, it switches be-tween two phases — local Brownian search phase andballistic relocation phase (L´evy walk). The searcher dis-plays a slow reactive motion in the first phase, duringwhich the target can be detected. The latter fast phaseaims at relocating into unvisited regions to reduce over-sampling, during which the searcher is unable to detectthe target. In the situation of rare targets, it has beenshown that the search process with L´evy distributed relo-cations significantly outperforms that with exponentiallydistributed relocation [4]. While the two-state process ef-fectively models intermittent strategy, it is also observedin the transport of the neuronal messenger ribonucleo-proteins delivered to their target synapses [9], where atype of L´evy walk process is interrupted by the emergingof rest. The rest period can be very long, characterizedby power-law distribution without finite mean. This phe-nomenon becomes a striking feature of the RNA trans-port in neuronal systems.The intermittent strategy has been verified to be opti-mum for searching targets in some specific macroscopicand microscopic situations. But generally it is hard to be- lieve that the intermittent strategy is always the best onein all the foraging behaviors of animals and the intracel-lular transport in microscopic scale. A question naturallycomes up: How about the field of its application? Basedon this motivation, it is necessary to build a complete the-oretical foundation for this kind of two-state processesfor dealing with data observed in experiments. In thisreport, we consider the two-state process mentioned ear-lier (i.e., the standard L´evy walk and Brownian motion)and mainly investigate their statistical behaviors, such asensemble-averaged mean square displacement (EAMSD)and time-averaged mean square displacement (TAMSD).In particular, we carefully examine the aging behaviorsof the two-state process, while the aging continuous-timerandom walk (CTRW) [10], aging renewal theory [11] andaging ballistic L´evy walks [12] have been fully discussed.Since the observation time might not be the beginning ofa process in experiments, aging behavior should be paidsome attention and it may display interesting phenomenain anomalous diffusion processes [13, 14].L´evy walk dynamics describe enhanced transport phe-nomena in many systems. Within the CTRW framework,originally introduced by Montroll and Weiss [15], the sig-nificant feature of L´evy walk is the underlying spatiotem-poral coupling, which penalizes long jumps and leads toa finite EAMSD [16]. While the uncoupled process, L´evyflight [17, 18], has divergent EAMSD. The diffusion be-havior of L´evy walk depends on the exponent α of thepower-law distributed running time. It displays ballisticdiffusion for α < < α <
2. We assume the particle switches betweenL´evy walk phase and Brownian phase, denoted as states‘+’ and ‘ − ’, respectively. The velocities of the two-stateprocess are, respectively, v + ( t ) for L´evy walk and v − ( t )for Brownian motion. The PDF of v + ( t ) is δ ( | v | − v ) / v − ( t ) = √ Dξ ( t ) with ξ ( t ) being a Gaussian whitenoise satisfying h ξ ( t ) i = 0 and h ξ ( t ) ξ ( t ) i = δ ( t − t ).By taking the diffusivity D = 0, the Brownian phase be-comes a trap event and we immediately obtain the pro-cess – L´evy walk interrupted by rest.Let the sojourn times t in the two states ‘ ± ’ be randomvariables obeying power-law distribution: ψ ± ( t ) ≃ a ± | Γ( − α ± ) | t α ± (1)for large t , where a ± are scale factors and Γ( · ) is theGamma function. We assume that the exponents α ± ∈ (0 ,
2) in two states and the sojourn times in two satesare mutually independent. As usual, we apply the ap-proach of Laplace transform ˆ ψ ± ( s ) := R ∞ dte − st ψ ± ( t )and obtain the asymptotic behavior of the sojourn timedistribution for small s :ˆ ψ ± ( s ) ≃ − a ± s α ± for α ± ∈ (0 , , (2)ˆ ψ ± ( s ) ≃ − µ ± s + a ± s α ± for α ± ∈ (1 , , (3)where µ ± is the mean sojourn time in state ‘ ± ’, be-ing finite when α ± ∈ (1 , ± ’ exceeds t is de-fined as Ψ ± ( t ) = R ∞ t dt ′ ψ ± ( t ′ ) with Laplace transformˆΨ ± ( s ) = [1 − ˆ ψ ± ( s )] /s . Note that the dynamic behav-iors of standard L´evy walk are significantly different forexponent less or larger than 1 [16]. We will fully dis-cuss the EAMSD and TAMSD of the two-state processfor different sets of α ± in the following. Although themean sojourn time is finite (i.e., α ± >
1) in most cases,such as the intermittent search process, there are stillsome circumstances presenting scale free dynamics with α ± <
1, for example, the RNA transport in neuronal sys-tems. Here we make uniform discussions with α ± ∈ (0 , Propagator and occupation fraction of two states . Sup-pose that the particles are initialized at the origin. Thepropagator p ( x, t ) represents the PDF of finding the par-ticle at position x at time t . For the two-state process, itis natural to concern which state the particles are locatedin at time t . Here we denote the joint PDF of finding theparticle at position x and state ‘ ± ’ at time t as p ± ( x, t ),which is associated with the propagator by the relation p ( x, t ) = p + ( x, t ) + p − ( x, t ). The subscript ‘ ± ’ will implyan identical meaning for other quantities.The integral equations for p ± ( x, t ) can be similarly ob-tained as the master equations for CTRWs. Besides thesojourn time distribution ψ ± ( t ) and survival probabilityΨ ± ( t ), we introduce the notation G ± ( x, t ) to representthe conditional probability that a particle makes a dis-placement x during sojourn time t at one step in state‘ ± ’. Their expressions are given by G + ( x, t ) = δ ( | x | − v t ) / , (4) G − ( x, t ) = 1 √ πDt exp (cid:18) − x Dt (cid:19) , (5)since the state ‘+’ represents L´evy walk and state ‘ − ’ de-notes Brownian motion, respectively. Then the transportequation governing flux of particles γ ± ( x, t ), which de-fines how many particles leave the position x and change from state ‘ ∓ ’ to state ‘ ± ’ per unit time, satisfies, γ ± ( x, t ) = Z t dt ′ Z ∞−∞ dx ′ ψ ∓ ( t ′ ) G ∓ ( x ′ , t ′ ) γ ∓ ( x − x ′ , t − t ′ )+ p ∓ ψ ∓ ( t ) G ∓ ( x, t ) , (6)where the constant p ± is the initial fraction of two states,that is p ± ( x, t = 0) = p ± δ ( x ). The current density p ± ( x, t ) of particles is connected to the flux γ ± ( x, t ) p ± ( x, t ) = Z t dt ′ Z ∞−∞ dx ′ Ψ ± ( t ′ ) G ± ( x ′ , t ′ ) γ ± ( x − x ′ , t − t ′ )+ p ± Ψ ± ( t ) G ± ( x, t ) . (7)By means of the techniques of Laplace and Fouriertransform, ˜ˆ p ± ( k, s ) = R ∞ dt R ∞−∞ dxe − st e ikx p ± ( x, t ) canbe obtained (see Supplemental Material). Besides, theoccupation fraction of two states p ± ( t ), as the marginaldensity of finding the particles in state ‘ ± ’ at time t , canbe obtained by taking k = 0 in ˜ˆ p ± ( k, s ). The expressionof p ± ( t ) in Laplace space ( t → s ) isˆ p ± ( s ) = p ± + p ∓ ˆ ψ ∓ ( s )1 − ˆ ψ + ( s ) ˆ ψ − ( s ) · − ˆ ψ ± ( s ) s , (8)the normalization of which can be confirmed by verifyingˆ p + ( s ) + ˆ p − ( s ) = 1 /s . EAMSD and TAMSD . If one is eager for more infor-mation of a process, such as the TAMSD, the propaga-tor p ( x, t ) at a single point is not enough. Instead, thetwo-point velocity correlation function h v ( t ) v ( t ) i playsa crucial role. We will calculate it firstly and then showthe generic results of EAMSD and TAMSD for the agingprocess x t a ( t ). The age t a means that this process hasevolved for a time period t a before we start to observe it,and t is the measurement time.Since the model we considered contains two states:L´evy walk and Brownian motion, represented by sym-bols ‘+’ and ‘ − ’, respectively. The velocity correlationfunction could be written as a sum of four possible casesin terms of different states: h v ( t ) v ( t ) i = h v + ( t ) v + ( t ) i + h v − ( t ) v − ( t ) i + h v + ( t ) v − ( t ) i + h v − ( t ) v + ( t ) i . (9)The first term on the right-hand side represents the casethat the velocity process v ( t ) are in L´evy walk phase atboth time points t and t ; other terms stand for simi-lar parts of the correlation function. For the first term,the velocity is correlated only when there is no renewalhappens between t and t . Thus, we have h v + ( t ) v + ( t ) i = v p + ( t ) p + , ( t , t ) , (10)where p + ( t ) has been given in (8) and p + , ( t , t ) is thePDF that no renewal happens between times t and t instate ‘+’. Similarly, the second term on the right handside of (9) is h v − ( t ) v − ( t ) i = 2 Dδ ( t − t ) p − ( t ) p − , ( t , t ) , (11)where p − , ( t , t ) = 1 for t = t , since there must beno renewals within a zero time lag. The two states attimes t and t are different in the last two terms on(9). Therefore, the velocity at t and t are independent.Considering the velocity is unbiased at any time, the lasttwo terms are void.Note that the PDFs p ± ( t ) and p + , ( t , t ) should becalculated firstly to obtain the velocity correlation func-tion in (9). The former one has been given in (8), whilethe double Laplace transform ( t → s, τ → u ) of the latterPDF f + , ( t, τ ) = p + , ( t, t + τ ) is [19]ˆ f + , ( s, u ) = s − u + u ˆ ψ + ( s ) − s ˆ ψ + ( u ) s ( s − u )(1 − ˆ ψ + ( s )) u . (12)It seems not easy to perform the inverse Laplace trans-form on ˆ f + , ( s, u ). Instead, we can obtain the expressionof p + , ( t , t ) in Laplace space ( t → s , t → s ) bysubstituting variables (see Supplemental Material):ˆ p + , ( s , s ) = 1 + ˆ ψ + ( s + s ) − ˆ ψ + ( s ) − ˆ ψ + ( s ) s s (1 − ˆ ψ + ( s + s )) . (13)Taking inverse Laplace transform on (13) becomesdoable. Based on (8) and (13), the velocity correlationfunction h v ( t ) v ( t + τ ) i in (9) can be obtained for differentsojourn time distributions ψ ± ( t ). Noticing the asymp-totic forms of p ± ( t ) and p + , ( t, t + τ ) for large t , the ve-locity correlation function can be rewritten in the scalingform as h v ( t ) v ( t + τ ) i = h v + ( t ) v + ( t + τ ) i + h v − ( t ) v − ( t + τ ) i≃ C t ν − ρ (cid:16) τt (cid:17) + C t ν − δ ( τ ) , (14)where the parameters ν , ν and the scaling function ρ ( · )are determined by p ± ( t ) and p + , ( t, t + τ ). The scal-ing form (14) helps to show different scaling behaviorsof h v ( t ) v ( t + τ ) i for different sojourn time distributions ψ ± ( t ), and brings convenience to give a generic expres-sions of MSDs [20, 21].Now we focus on the aging process x t a ( t ). The EAMSDof this aging process is defined as h x t a ( t ) i = h ( x ( t a + t ) − x ( t a )) i , which can be obtained through the scaling formin (14). For weak aging t a ≪ t and strong aging t a ≫ t cases (see Supplemental Material), it behaves as h x t a ( t ) i ≃ (cid:26) K t ν + K t ν , t a ≪ t,K t βa t ν − β + C t ν − a t, t a ≫ t, (15)where the coefficients K = 2 C /ν R ∞ dt ( t + 1) − ν ρ ( t ), K = C /ν and K = 2 c C [( ν − β − ν − β )] − . Here c depends on the asymptotic form of scaling function ρ ( z ) ≃ c z − δ for small z , and β is the exponent of thevariance of velocity in the L´evy walk phase for large t [20], i.e., h v ( t ) i = v p + ( t ) ∝ t β . (16)When constructing single particle tracking experi-ments, the aging process x t a ( t ) is evaluated in termsof its TAMSD, which is defined as δ t a (∆) = 1 / ( T − ∆) R t a + T − ∆ t a dt [ x ( t + ∆) − x ( t )] with ∆ denoting thelag time and T the total measurement time [22]. TheTAMSD is calculated in the limit ∆ ≪ T to obtain goodstatistics. Weak ergodicity breaking is the common phe-nomenon of a majority of anomalous diffusion. Similarlyto the procedure of calculating EAMSD, we obtain theensemble-averaged TAMSD as (see Supplemental Mate-rial): h δ t a (∆) i ≃ (cid:26) K β T β ∆ ν − β + K T ν − ∆ , t a ≪ T,K t βa ∆ ν − β + C t ν − a ∆ , t a ≫ T. (17)There are at least four findings being worth to reportfrom the observations of the generic results of EAMSDsin (15) and TAMSDs in (17). (i) All the four mentionedformulae consist of two parts (one from L´evy walk phaseand another one from Brownian phase). The exponentsof evolution time t or time lag ∆ in these two parts mightbe different from the ones of the corresponding individ-ual L´evy walk and Brownian motion. This is becausethe PDF p ± ( t ) in (8) plays a weighted role on L´evy walkand Brownian motion. Besides, the sums of exponents ofthe time variables (including t, t a , T, ∆) in individual twoparts are ν and ν , respectively, whatever it is EAMSDor TAMSD, and weak or strong aging cases. (ii) The ex-ponents of time variables in weak and strong aging casesare closely related for TAMSD in (17). While keepingthe exponents of ∆ invariant and replacing measurementtime T by age t a , the result for strong aging case is ob-tained from the one of weak aging case. In other words,the TAMSD for weak aging case only depends on T and∆, while in the same way it counts on t a and ∆ for strongaging cases. (iii) The EAMSD and TAMSD in weak ag-ing case do not depend on the age t a , the results of whichare identical to the non-aging case t a = 0. In contrast,they explicitly depend on t a for strong aging case, whichimplies that the exponents β and ν − t a must be zeroif the equilibrium initial ensemble (i.e., t a → ∞ discussedin last section) of this system exists (see specific case 2 inTable I). And in this case, the TAMSD will be the samefor weak and strong aging cases, and only depends on ∆.(iiii) Comparing the strong aging EAMSD and the meanof TAMSD (17), it can be noted that h x t a (∆) i = h δ t a (∆) i for t a ≫ T, (18)which shows that the aging seemingly makes the weakergodicity breaking system to be ergodic. It is clear thatfor any α − Brownian motion is ergodic in its own phase.However, for TAMSD in L´evy walk phase, there are somedifferences between α + < < α + <
2. For
TABLE I. Values of several major parameters of EAMSD andTAMSD in (15) and (17) for six cases with different α ± .specific cases ν ν β α + = α − < < α ± < − α + α + < α − < α + − α − + 1 04. α + < < α − < α + α − < α + < α − − α + + 2 1 α − − α + α − < < α + < α − − α + + 2 1 α − − < α + <
2, the mean sojourn time in L´evy walk phaseis finite, individual trajectories become self-averaging atsufficiently long (infinite) times, such that there will be nodifference between δ t a (∆) obtained from different trajec-tories and ensemble-averaged quantity h δ t a (∆) i [23, 24].While for α + <
1, the characteristic time scale is infinite,then the individual TAMSD δ t a (∆) is irreproducible andinequivalent with the corresponding EAMSD. Specific cases . Since both α + and α − go through therange (0 , ν , ν , and β for these cases in (Supplemental Material).It seems tedious to discuss the EAMSDs and TAMSDsindividually for six different cases of α ± . In fact, they canbe organized into three categories to deepen understand-ings of the two-state process by considering the proper-ties of its ingredients — L´evy walk and Brownian motion.It is well-known that the standard L´evy walk performsballistic diffusion when the exponent of the distributionof running times α < < α <
2, which is faster than the normal diffusionof Brownian motion. Based on this understanding, theBrownian phase undoubtedly suppresses the diffusion be-havior of L´evy walk. This effect may be durable or tran-sient, which is completely determined by the fraction oftwo states p ± ( t ), or more essentially, the magnitude ofthe exponents α ± . From this point of view, the threecategories are: (i) α + and α − are comparable, includingthe first two cases in Table I; (ii) α + is smaller, includ-ing the middle two cases in Table I; (iii) α − is smaller,including the last two cases in Table I.As representatives of the above three situations, wechoose three sets of parameters: (i) α + = 1 . , α − = 1 . α + = 0 . , α − = 1 .
5, and (iii) α + = 1 . , α − = 0 . h δ t a (∆) i ≃ ( D ∆ − α + + Dµ − µ + + µ − ∆ , t a ≪ T, D ∆ − α + + Dµ − µ + + µ − ∆ , T ≪ t a , (19) (ii) h δ t a (∆) i ≃ ( v ∆ + Dµ − a + Γ(1+ α + ) T α + − ∆ , t a ≪ T,v ∆ + Dµ − a + Γ( α + ) t α + − a ∆ , T ≪ t a , (20)(iii) h δ t a (∆) i ≃ ( D T α − − ∆ − α + + 2 D ∆ , t a ≪ T, D α − t α − − a ∆ − α + + 2 D ∆ , T ≪ t a . (21)For the first category (i), a stationary of the fractions oftwo states p ± ( t ) can be achieved for long times, that is, p ± ( t ) tends to a constant not equal to 0 or 1 (see Sup-plemental Material). Then the EAMSD and TAMSD arethe combination of the fraction of analogues of individ-ual L´evy walk and Brownian motion whether it is weakaging or strong aging. For the second category (ii) with α + < α − where p + ( t ) → t → ∞ , the L´evy walkphase in state ‘+’ tends to occupy the whole time. Thenthe results are naturally similar to an individual L´evywalk, except for the small asymptotic form ∆ resultingfrom Brownian phase. For the third category (iii) with α + > α − , by contrast, now p − ( t ) → t → ∞ and theL´evy walk phase in state ‘+’ gradually withdraws fromthe two states in a power-law way. This power-law waysuppresses the diffusion of L´evy walk phase and givesthe opportunity to Brownian motion to be the leadingterm when α + − α − >
1. In conclusion, compared tothe EAMSD and TAMSD of individual aging L´evy walk[12] and Brownian motion, it can be found that the frac-tion p ± ( t ) in a two-state process plays a crucial role. Itcontributes a power term of ∆ to weak aging EAMSD,a power term of T to weak aging TAMSD, and a powerterm of t a to strong aging EAMSD and TAMSD.The model L´evy walk interrupted by rest has attractedconsiderable attention in physics [25, 26] and biology [9].The EAMSD and TAMSD for this model can be obtainedby taking the diffusivity D in Brownian phase to be zero.It has been pointed that all the results above consist twoparts corresponding to L´evy walk and Brownian motion,respectively. Taking D = 0 just eliminates the latterpart and brings no effect on the former part of L´evy walkphase. For L´evy walk interrupted by rest, the asymptoticbehavior of small ∆ in TAMSD disappears and subdiffu-sion behavior might exist if α + − α − > Initial ensemble . In general, the standard L´evy walkmodel is a non-Markovian process and so is the two-stateprocess alternating between L´evy walk and Brownianmotion with power-law distributed sojourn time. It isnatural to consider the effects of the initial ensembles ofthe particles. It is called a nonequilibrium initial ensem-ble [27, 28] if all particles are introduced to the systemat t = 0 without any prehistories. In contrast, if theparticles have been evolving for time t before we startto measure this system, we call this system with equi-librium initial ensemble when t → ∞ [27, 28]. TheEAMSD of standard L´evy walk has been shown to bedifferent for different initial ensemble [16, 29]. Note thatthe equilibrium initial ensemble exists only if the sojourntimes in two states ‘ ± ’ both have finite first moments,i.e., 1 < α ± < -2 -3 -2 -1 t a = 0t a = 2792 ~ + ~ (a): + = 1.5 - = 1.8 -2 -2 t a = 0t a = 388094 ~ t a = 0t a ~ (b): + = 0.6 - = 1.5 10 -2 -4 -2 t a = 0t a = 186760 ~ + ~ ~ (c): + = 1.5 - = 0.8 FIG. 1. TAMSD of the two-state process for different sets of α ± . Black circles and squares represent the simulation results ofthe mean value of TAMSD averaging over 200 realizations, and the solid lines are the theoretical ones (with small and largeasymptotic forms in (17)). (a): case (i) with α + = 1 . , α − = 1 . T = 200. The simulation resultsagree with the theoretical ones for small time ( ∼ ∆) and large time ( ∼ ∆ − α + ). (b): case (ii) with α + = 0 . , α − = 1 . T = 20426. The asymptotic behavior ∼ ∆ for large time is observed. The simulation and theoreticalresults do not coincide for small ∆ in strong aging case, since the coefficient t α + − a in front of ∆ in (20) is too small and anotherterm ∆ dominates. (c): case (iii) with α + = 1 . , α − = 0 . T = 9830. The asymptotic behavior ∼ ∆for short time can be observed. It does not coincide for large ∆ in strong aging case, since the coefficient t α − − a in front of∆ − α + in (21) is too small and another term ∆ dominates. Therefore, there is not much difference between the strong agingsimulations and the whole solid (yellow) line with slope 1 for large ∆. EAMSD h x (∆) i and TAMSD δ (∆) can be obtained bytaking t a = 0 in previous section, i.e., h x (∆) i = h x t a (∆) i| t a =0 , δ (∆) = δ t a (∆) | t a =0 . (22)Since the results of the weak aging case (i.e., t a ≪ ∆)with different sojourn time pairs ψ ± ( t ) in Eqs. (15) and(17) are independent of t a , they are indeed the results fornonequilibrium initial ensemble. When 1 < α ± <
2, theresults of the strong aging case (i.e., t a ≫ T ) in (19) areindependent of t a . Therefore, the EAMSD h x (∆) i andTAMSD h δ (∆) i for equilibrium initial ensemble ( t a →∞ ) are h x (∆) i = h δ (∆) i ≃ D ∆ − α + + 2 Dµ − µ + + µ − ∆ . (23)If the sojourn times are so long that the mean sojourntime diverges, there is no sense in talking about the equi-librated initial ensemble. However, the asymptotic be-haviors of strong aging case t a ≫ ∆ can still be inves-tigated (see Supplemental Material). There is a specialcase 0 < α + = α − <
1, where the particles reach a bal-ance that each half of them are located in each of the twostates and the EAMSD and TAMSD are both indepen-dent on the age t a , that is, x t a (∆) i = h δ t a (∆) i ≃ v + D ∆ (24)for sufficiently large t a . If α + = α − and at least oneof them less than 1, then neither an equilibrium initialensemble nor a balance for long time exists. The statewith small exponent α ± of sojourn time distribution willdominate the MSD for long times. One can see this phe-nomenon in the last four cases in Table I. In these cases, the EAMSD and TAMSD for strong aging cases all con-sist of two parts corresponding to L´evy walk and Brown-ian motion. One of the parts is independent on t a whileanother part contains a power term of t a with a negativeexponent. The latter part tends to zero as t a → ∞ andthe former one dominates, which corresponds to the statewith smaller exponent α ± of sojourn time distributions. Conclusion . It often happens that a single-state pro-cess cannot sufficiently describe the observed physicaland biological phenomena. Two-state process is a kindof simple but important model to characterize some ofthese phenomena. A Langevin equation with two diffu-sion modes (fast and slow diffusion modes) has been in-vestigated in [30], where a transient subdiffusion and thenon-Gaussian propagator for short time are observed fora nonequilibrium ensemble. In this report, we consider atwo-state process with fast phase (L´evy walk) and slowphase (Brownian motion), which is also the intermittentsearch process for finding rare hidden targets. It is noteasy to model the process with two completely differentphases by a Langevin equation. By contrast, we resortto the velocity process v ( t ), which also consists of twostates. Based on the velocity correlation function, we ob-tain the generic expressions of the EAMSD and TAMSDfor different sojourn time distributions.One of the key contributions of this report is to explic-itly discuss the relation between EAMSD and TAMSD.In particular, the weak and strong aging cases are alsoconsidered for these MSDs since the measurement in ex-periments might not begin at the start of the processconcerned. It is found that the occupation fraction playsa weighed role in L´evy walk phase and Brownian phase,and the MSDs are just a combination of these two parts.The meticulous discussions on the aging MSDs are help-ful to understand the two-state process and to analyzethe experimental data.If taking the diffusivity D to be zero in Brownianphase, we obtain another important process — L´evy walkinterrupted by rest. Taking D = 0 just eliminates thecontributions from Brownian phase. From another as-pect of the two-state process, we find the fact that theslow phase, whether it is rest or Brownian motion, sup-presses the diffusion behavior of L´evy walk if its sojourntime is longer than that of L´evy walk phase. The mech-anism is similar to the trap event [31] in CTRW mod- els. Compared to them, there exist some other modelsdescribing the suppression of the diffusion of L´evy walkwith different mechanism, such as the L´evy walk withmemory in running time [32] and the walker moving in aheterogeneous medium [33]. Acknowledgments . This work was supported by theNational Natural Science Foundation of China undergrant no. 11671182, and the Fundamental ResearchFunds for the Central Universities under grant no.lzujbky-2018-ot03. [1] J. W. Bell,
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