PHYSICAL REVIEW E 97, 052132 (2018)
Continuous time random walk with local particle-particle interaction Jianping Xu* Department of Petroleum Engineering, China University of Petroleum, Changping, Beijing 102249, China
Guancheng Jiang† State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Changping, Beijing 102249, China (Received 22 October 2017; revised manuscript received 9 May 2018; published 23 May 2018) The continuous time random walk (CTRW) is often applied to the study of particle motion in disordered media. Yet most such applications do not allow for particle-particle (walker-walker) interaction. In this paper, we consider a CTRW with particle-particle interaction; however, for simplicity, we restrain the interaction to be local. The generalized Chapman-Kolmogorov equation is modified by introducing a perturbation function that fluctuates around 1, which models the effect of interaction. Subsequently, a time-fractional nonlinear advection-diffusion equation is derived from this walking system. Under the initial condition of condensed particles at the origin and the free-boundary condition, we numerically solve this equation with both attractive and repulsive particle-particle interactions. Moreover, a Monte Carlo simulation is devised to verify the results of the above numerical work. The equation and the simulation unanimously predict that this walking system converges to the conventional one in the long-time limit. However, for systems where the free-boundary condition and long-time limit are not simultaneously satisfied, this convergence does not hold. DOI: 10.1103/PhysRevE.97.052132 I. INTRODUCTION
Ever since the pioneering work [1,2] on the continuous time random walk (CTRW) [3,4], it has witnessed extensive applications in various areas of research, e.g., matter transport in disordered solids [5], statistical mechanics of polymers subject to a force [6], transport in geological porous media [7,8], and reactive transport [9,10]. The CTRW is suitable for depicting the particle-medium interactions, which can stem from either geometrical confinement or physicochemical interaction. A famous example is the particle motion in a comblike structure [11], where a waiting time [12] must be introduced in the particulate dynamics. The particle-medium interaction can be long range and nonlocal, involving memory [13] (a long rest [12]) or a long jump [12,14,15]. Recently, increasing attention has been paid to the particleparticle interactions, especially colloidlike active particles [16] at the Brownian scale which interact with each other by hydrodynamic interaction [16], capillary repulsion or attraction [17], volume exclusion [18], etc. Such a many-body system [19,20] exhibits interesting behaviors such as jamming [21,22], swarming [23], and phasing [19]. In most scenarios, it might suffice to use the conventional CTRW to describe particle motion in disordered media; however, it may not suffice when the particle density grows high and the many-body problem becomes salient. If such a situation takes over, the particle’s jumping and lodging behaviors in the conventional CTRW must be perturbed or even substantially modified. Therefore, in this work we consider incorporating the particle-particle inter-
* †
action into the CTRW framework. For the sake of simplicity, this interaction is set to be local. The paper is organized as follows. In Sec. II we review some of the key elements of the conventional CTRW. In Sec. III we conduct the modification to incorporate the local particle-particle interaction. In Sec. IV we devise a Monte Carlo simulation to justify the main results obtained in Sec. III. We summarize in Sec. V. II. STANDARD CONTINUOUS TIME RANDOM WALK
In terms of a CTRW, particle motion can be viewed as a series of stochastic jumps in space. The jumps are characterized by the probability density function (PDF) (x,t) [10,12], the probability that a particle steps a distance x after waiting a duration of t [10]. Normally the decoupling of space and time is assumed, i.e., (x,t) = λ(x)w(t) [7,12], where λ(x) and w(t) are the jump length and waiting time PDFs, respectively, which read ∞ λ(x) = dt (x,t), w(t) =
∞
dx (x,t). −∞
Considering the conditional probability q(x,t|x0 ,0) that a particle starts from x0 at t = 0 and arrives at x at t, one can write down the celebrated generalized Chapman-Kolmogorov equation (GCKE) [10,24,25] ∞ t q(x,t|x0 ,0) = dx dt q(x ,t |x0 ,0) −∞
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2470-0045/2018/97(5)/052132(11)
0
0
×(x − x ,t − t ) + δ(t)δx,x0 , 052132-1
(1)
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PHYSICAL REVIEW E 97, 052132 (2018)
where δ stands for the Dirac delta function, x − x = x, and t − t = t. Another important PDF is the probability that a particle does not leave a position during the interval (0,t), which is t dt w(t ). (2) (t) = 1 −
For simplicity, we only investigate the Riemann-Liouville fractional operator here. Our previous work [13] discussed applications of more complicated fractional operators such as the Hadamard and Prabhakar types in the particle motion problem.
0
The conditional probability p(x,t|x0 ,0) that a particle starts from (x0 ,0) and is now at (x,t) is related to q(x,t|x0 ,0) by the equation [10] t dt (t − t )q(x,t |x0 ,0). (3) p(x,t|x0 ,0) = 0
Combining Eqs. (1) and (3) and using the Laplace transform, one gets the equation that p(x,t|x0 ,0) observes t ∞ p(x,t|x0 ,0) = dx dt p(x ,t |x0 ,0) −∞
0
×(x − x ,t − t ) + (t)δx,x0 .
(4)
Then the propagator p(x,t) of a particle for ∞being at position x at time t, which is expressed as p(x,t) = −∞ dx0 p(x,t|x0 ,0), satisfies the equation [10] ∞ t p(x,t) = dx dt p(x ,t )(x − x ,t − t ) −∞
0
+ (t)p(x,0).
(5)
ˆ A Fourier-Laplace transform of Eq. (5) gives [4,7,12] [(s) = 1/s − w(s)/s ˆ is used] pˆ0 (k) pˆ0 (k) 1 − w(s) ˆ 1 − w(s) ˆ ˆˆ , = p(k,s) = ˆ ˆ s s 1 − w(s) ˆ λ(k) ˆ 1 − (k,s) (6) where k and s are, respectively, the Fourier and Laplace variables. One circumflex denotes one Fourier or Laplace transform and two circumflexes denote both transforms. Equation (6) is a fundamental equation in the CTRW. Here we consider a special case: the subdiffusion, where the mean waiting time diverges and the mean jump length remains finite. ˆ In such a case, one can asymptotically expand w(s) ˆ and λ(k) in the limit (k,s) → (0,0) as [7,12] ˆ = 1 − ilk − σ 2 k 2 +O(k 4 ). w(s) ˆ = 1 − (τ s)α +O(s 2 ), λ(k) (7) An inverse Fourier-Laplace transform of Eq. (6) [using Eq. (7)] leads to the time-fractional advection-diffusion equation (8) ∂t p(x,t) = 0 Dt1−α D0 ∂x2 p(x,t) − u0 ∂x p(x,t) , 2
∂ ∂ where ∂t ≡ ∂t∂ , ∂x ≡ ∂x , ∂x2 ≡ ∂x 2 , and D0 and u0 are, respectively, the diffusion constant and the drift constant. One also , where ε0 is the notices that u0 can be expressed as −ε0 ∂U ∂x mobility and U is the external potential. Then Eq. (8) becomes the fractional linear Fokker-Planck equation [3]. Here 0 Dt1−α (0 < α < 1) is the Riemann-Liouville fractional derivative that is defined as t d 1 1−α f (t) = dt f (t )(t − t )α−1 . 0 Dt dt (α) 0
III. INCORPORATION OF LOCAL PARTICLE-PARTICLE INTERACTION A. Formalism
The GCKE describes a rule for state transition where those possible intermediate states x are traversed. It also applies to a single particle (walker) by the function q. When dealing with a many-particle system, usually one resorts to nonequilibrium statistical mechanics, which can sometimes be convoluted. An alternative is that we remain in the narrative of the GCKE, but with the consequence that the single-particle behavior is perturbed by the particle-particle interaction. This logic is analogous to the classical mean-field theory [26], where particle-particle interaction is translated into a so-called meanfield potential exerted on the individual particle. Here we also introduce a function to denote the perturbation that particleparticle interaction may introduce to the GCKE system. To determine the specific form of the function, let us return to Eq. (1). In Eq. (1), the probability (x,t) plays a central role, as it prescribes the probability of a given transition with jump distance x and waiting time t. However, does not discriminate between two different jumps that have identical jump length and waiting time. For example, if a transition is from (x ,t ) to (x,t) and another is from (x ,t ) to (x ,t ) (x = x = x and t > t > t ), one can actually write (x − x ,t − t ) = (x − x ,t − t ) as long as x − x = x − x = x and t − t = t − t = t. So whenever and wherever the transition occurs, its probability stays the same if x and t do not change. In such a way, the local event at (x ,t ) or (x ,t ) does not contribute to the transition probability. Nevertheless, when considering local particle-particle interaction, the local event matters. A particle’s transition behavior can be modified by its neighbors. For example, a planned hop may be interrupted or a resting particle may be excited to jump again. In these processes, the particle density field matters because it determines the intensity of the local interaction. Higher particle density leads to higher interaction intensity; consequently, the particle’s transition behaviors are more likely to be perturbed and modified. Therefore, the transition probability should be dependent on the particle density. In addition, under a nonuniform particle density field, the transition behavior at a densely populated location and that at a sparsely populated location will surely differ, which makes the transition space dependent. Then, if the density field evolves in time, the transition probability must also respond to that evolvement. As a result, the probability for transition varies spatiotemporally. So the introduced function must be able to account for the above properties of the transition probability. If we call this function, say, , then should be a function of the particle density function p(x,t), i.e., = [p(x,t)]. Further,
by itself must also be an explicitly spatiotemporal function
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to allow for the gradation and evolvement of the transition behavior. Finally, takes the form of = [x,t,p(x,t)]. This , combined with the previous , gives rise to the transition probability [x ,t ,p(x ,t )](x − x ,t − t ), which is able to allow for all the required properties discussed above. As the perturbation oscillates around 1, the transition from (x ,t ) to (x,t) is either more favored or less favored compared to the noninteractive case. The more deviates from 1, the higher the intensity of the interaction is. If instead
[x ,t ,p(x ,t )] = 1, we return to the classical CTRW. With this transition probability, one could modify the GCKE (1) as t ∞ q(x,t|x0 ,0) = dx dt q(x ,t |x0 ,0) −∞
0
B. Physical picture that reveals
× [x ,t ,p(x ,t )](x − x ,t − t )
+ δ(t)δx,x0 .
(9)
When fluctuates around 1, it models the influence of collective dynamics. Equation (9), after some procedures, yields an equation in parallel to Eq. (5), t ∞ dx dt p(x ,t ) [x ,t ,p(x ,t )] p(x,t) = −∞
the jump probability function ∗ becomes nonstationary. Similar work, such as Ref. [29], which reports a CTRW approach for pattern formation on networks with reaction, adopts a nonstationary ∗ (x,x ; t,t ) that is site dependent. Hence the essential difference of our work from previous works is the way we introduce nonstationarity into the classical CTRW. Of course it is straightforward to directly generalize the stationary to be nonstationary. However, it is also acceptable to introduce an extra function to characterize this nonstationarity if the physical picture revealed by this function could be elucidated. It might provide an alternative view to understand the problem.
0
×(x − x ,t − t ) + (t)p(x,0).
(10)
Nonetheless, Eq. (10) is an approximate result under the assumption that ≈ 1 [the detailed process of obtaining Eq. (10) is shown in Appendix A]. Fortunately, this assumption is not violated in the subsequent analysis. The Fourier-Laplace (F {·}-L {·}) transform of Eq. (10) results in ˆˆ ˆ p(k,0) ˆ ˆ p(k,s) = (s) + w(s) ˆ λ(k)F L { [x,t,p]p}(k,s), (11) ˆ where p ≡ p(x,t). By Eq. (2) one gets (s) = 1/s − w(s)/s. ˆ Moreover, to facilitate the deduction, we employ a memory ˆ = s w(s)/[1 ˆ − w(s)] ˆ kernel function ψ(t), which satisfies ψ(s) ˆ ˆ [7,27]. This gives w(s) ˆ = ψ(s)/[s + ψ(s)]. Substituting this ˆ relation and (s) into Eq. (11), one obtains ˆˆ ˆˆ ˆ ˆ p(k,s) + λ(k)F L { p}(k,s)], s p(k,s) − pˆ0 (k) = ψ(s)[− (12) ˆ and ≡ [x,t,p]. On adopting the where pˆ0 (k) = p(k,0) asymptotic relation (7) and Fourier-Laplace inverting Eq. (12), one obtains a time-fractional nonlinear advection-diffusion equation for the propagator p: ∂t p = 0 Dt1−α {[( − 1)/τ α ]p + D0 ∂x2 ( p) − u0 ∂x ( p)}, (13) where τ is inherited from Eq. (7) and is a characteristic time scale for particle motion [12]. The detailed process of deriving Eq. (13) is shown in Appendix B. The formulation above introduces nonstationarity and nonlinearity into the system by an extra function while preserving the stationary . This could be one way of thinking. There are some other works that directly consider a nonstationary ∗ (x − x ,t − t ; x ,t ). For example, Hansen and Berkowitz [28] considered an integro-differential formulation of the CTRW for solute transport subject to the bimolecular reaction A + B → 0. The annihilation of walkers is incorporated and
In Eq. (1), the transition probability from (x ,t ) to (x,t) is (x − x ,t − t ), while in our nonlinear CTRW the transition probability is [x ,t ,p(x ,t )](x − x ,t − t ). This could be understood in two ways. On the one hand, for the same transition from (x ,t ) to (x,t), the probability under the interactive regime and that under the noninteractive regime are unequal. On the other hand, if two transitions under the two regimes are equally probable, then their jump length and waiting time should be different. That is, provided that
= 1, there exist some appropriate location x ∗ and time t ∗ that fulfill the equation [x ,t ,p(x ,t )](x − x ,t − t ) = (x − x ∗ ,t − t ∗ ). This fact indicates that, equivalently, has led to the change of jump length and waiting time of otherwise noninteractive particles; the changes in jump length and waiting time are |x − x ∗ | and |t − t ∗ |. Now, before doing the integrals in Eq. (1), let us consider a procedure where we can sort all the (x ,t ) phase points in the R × (0,t) band in terms of their relative contribution to (x,t). When finishing this sorting, we will have traversed every point in R × (0,t) such that each of them occupies a place on the transition probability spectrum (from low probability to high probability). In the context of Eq. (1), this sorting is done according to the magnitude of (x − x ,t − t ). Regarding Eq. (9), the criterion for the sorting becomes [x ,t ,p(x ,t )](x − x ,t − t ). So every time one encounters an [x ,t ,p(x ,t )] = 1, this (x ,t ) must shift along the spectrum to a new place which once accommodates (x ∗ ,t ∗ ). When the sorting is completed, there can be some points in the previous spectrum that are visited more than once or some points that are not visited. It is similar to the concept of nonergodicity. In addition, when we add up the weights of these points together, the value of q(x,t|x0 ,0) will differ from the noninteractive one. Hence the physical picture that a nonuniform landscape of
reveals may lie in two aspects. First, if we are able to track the motion of an individual particle, we would find that, due to the particle-particle interaction, its jump length or waiting time is perturbed compared to the noninteractive one. Second, if we can monitor the movements of a large group of particles simultaneously, we would find that not every possibility in between (x0 ,0) and (x,t) can be visited by particles, while in the noninteractive regime there is no constraint on what particular path a particle will choose. For an individual particle, on the other hand, the perturbation on the waiting time and jump length appreciably changes
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the particle’s tendency to reside at a particular location, i.e., its residing probability. Let us revisit Eq. (2), which defines (t). Equation t (2) also exhibits itself in the form of (t) = ∞ 1 − −∞ dx 0 dt (x − x ,t − t ). However, in the nonlinear CTRW, is replaced by . Thus one could define a generalized residing probability g = g (x,t) as ∞ t g (x,t) = 1 − dx dt [x ,t ,p(x ,t )] −∞
0
×(x − x ,t − t ).
(14)
Therefore, due to the interaction, the particle’s residing probability changes from a temporal function (t) to a spatiotemporal function g (x,t). In addition, its physical meaning is the probability that a particle at location x does not leave x during (0,t). Equation (14) shows that the individual particle’s residing probability has been modified by . In addition, (t) and g (x,t) have some common properties, for example, (0) = g (x,0) = 1, and they both are decreasing functions with respect to time, which is evident from their definitions (2) and (14). They also give that limt→∞ (t) = 0 and limt→∞ g (x,t) = 0. In the computation part, these properties reemerge. At this point, there are several remarks about the physical picture of this interacting CTRW. (a) The nonuniform landscape of results in more active particle motion, where the residing behaviors of particles are dependent on space and the current particle configuration. This heterogeneity also hints at a bias in the direction of a jump even in the absence of an external field. The particles jump left or right, or fail the jump, according to their neighbors’ behaviors, provided the interaction is assumed local. (b) The particle configuration does not change between the jumps of the particles, which can be seen from Eq. (13), where the memory operator precedes the structure of p and ; otherwise a different subordination scheme would follow. (c) The bias in jumping direction is not exclusive here. For example, this bias is also observed in a CTRW subject to alternating external force [30]. A generalized master equation was employed to establish the model. C. Solution to Eq. (13)
In principle, Eq. (13) itself does not infer the specific form of the perturbation. To find the constraints on , we need to refer to other mechanistic model, e.g., the mean-field theory [20,26]. Obviously, Eq. (13) contains both memory and particle-particle interaction. In the limit of α → 1, the memory fades and the resultant equation is ∂t p = [( − 1)/τ ]p + D0 ∂x2 ( p) − u0 ∂x ( p), which merely models local particle-particle interaction. The local particle density change rate given by this equation is [( − 1)/τ ]p + D0 ∂x2 ( p) − u0 ∂x ( p). If using meanfield theory for the collective advection-diffusion, the den2 ∂ ε(p) ∂U p + D(p) ∂∂xp2 sity change rate is given by ∂t p = ∂x ∂x [20], where limp→0 ε(p) = ε0 and limp→0 D(p) = D0 when particle-particle interaction vanishes. Then, by equating the two local density change rates, after some algebraic manipulations we arrive at the constraint that needs to comply
with LFP = f (p),
(15)
where the operator LFP and function f (p) read (already use u = −ε ∂U ) ∂x LFP = p/τ + D0 ∂x2 p + ∂x (ε0 ∂x U )p, f (p) = p/τ + D(p)∂x2 p + ∂x [ε(p)∂x Up]. Equations (15) and (13) form a coupling system of p and
. The specific form of is implicitly given by Eq. (15), corresponding to a particular choice of ε(p) and D(p). In fact, the constraint that needs to conform to is not limited to Eq. (15); instead, it is an open question, depending on what mechanistic model for particle-particle interaction is selected. Here the mean-field theory is chosen to validate the model. To study particle-particle interaction, there are two elementary ways: attraction and repulsion. In these two scenarios, the dependence of ε and D on p is simplistic [20], ε(p) = ε0 (1 ± p)2γ −1 , D(p) = D0 (1 ± p)2γ −2 ,
(16)
where + and − denote the attractive (A) case and repulsive (R) case, respectively, and γ is a dimensionless variable ranging from 0 to 1 which characterizes the mean-field potential energy landscape [20]. Combined with the above two correlations, Eqs. (13) and (15) can be numerically solved to get p (pA and pR ) and ( A and R ). The classical equation (8) bears an analytical solution that is often expressed by the Fox H function [12,31]. To compare the results of Eq. (13) with those of Eq. (8), we use a simple external potential U = −κx, where κ is constant. The initial condition is p(x,0) = δ(x), indicating a particle spreading problem. Also, a free-boundary condition limx→±∞ p(x,t) = 0 is kept. For , the initial condition is set as |t=0 = 1. The boundary condition for is limx→±∞ = 1. For the purpose of exemplificative, we only simulate the α = 0.5 and γ = 0.8 case. The comparison is in Fig. 1. Figure 1(a) displays the solution of Eq. (8), where there is no particle-particle interaction. We denote this solution by pN , meaning noninteractive. Here pA denotes the solution under the attractive interaction and pR denotes the solution under the repulsive interaction. In addition, A and R are solutions for
under the attractive and repulsive interactions. Since p and
couple with each other, they must be solved simultaneously, that is, pA and A are solved at the same time and pR and R are solved at the same time. Further, pA and A are solved by Eq. (13) and pR and R are solved by Eq. (15) [combined with Eq. (16)]. In the numerical solution, a finite-difference scheme is employed. At each time step, p and are iterated until the error approaches zero. A key procedure in the finite-difference scheme is the discretization of the Riemann-Liouville fractional derivative. For this part, we follow the discretization rule in Refs. [13,33,34]. To compare pA and pR with pN , one could write pA = pA − pN and pR = pR − pN . Here pA and pR measure how close the interactive solution is to the noninteractive solution. Figure 1(b) shows the profile of pA , Fig. 1(c) shows the profile of pR , Fig. 1(d) exhibits the profile of A , and Fig. 1(e) exhibits the profile of R . Figure 1 suggests that in the long-time limit, pA and pR converge to 0, meaning that pA and pR converge to
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FIG. 2. (a) Spatiotemporal profile of A with no drift (U = 0 or u0 = 0), α = 0.5, γ = 0.8, and the same initial and boundary conditions as the previous computation. (b) Spatiotemporal profile of R . The unbiased walk witnesses a symmetric shape of . As expected, the perturbation in damps out in the long-time limit.
FIG. 1. (a) Solutions of Eq. (8) under fractional index α = 0.5 at t = 102 , 103 , 104 , and 105 s. This is a typical spatiotemporal profile of the propagator in such a Galilei-variant case [12,32], where the skewness of the curve keeps growing in the drift direction while the maximum of the curve remains at the origin. (b) Profile of pA = pA − pN , where pN denotes the noninteractive propagator and pA the attractive one, showing that in the early and middle period the particles are more likely to stay near the origin while in the long run this trend vanishes. (c) Repulsive case pR = pR − pN , showing that in the short period particles are less likely to stay near the origin. In the long run this trend damps out too. Also shown is the perturbation function of the (d) attractive and (e) repulsive cases. They exhibit excursion from 1 at early stages but ultimately converge to 1.
the noninteractive pN . In that limit, the propagator is flat enough in space, indicating the particles’ alienation from each other. When particles are separated from each other the local interaction no longer works, as if they are noninteractive. The figure also exhibits convergence of the perturbation to 1 in the long run. In addition, if = 1, one recovers the conventional CTRW. More interestingly, the external potential (or the drift) has rendered an asymmetrical shape of ’s spatial distribution. If there is no external field, the drift term in Eq. (13) disappears and Eq. (13) reduces to ∂t p = 0 Dt1−α [( − 1)/τ α ]p + D0 ∂x2 ( p) . (17)
method of solving the equations is the same as that used to solve Eqs. (15) and (13). The results of A and R are displayed in Fig. 2. As expected, the curves are symmetrical. In addition, the amplitude by which fluctuates is within a short range from 1, which corroborates the assumption behind Eq. (10) that
≈ 1. According to Eq. (14), contributes to the residing probability of a particle through a convolution structure with . Actually, for a particular regime of diffusion, one can calculate the spatiotemporal profile of g given that is known. Considering the subdiffusion regime of Figs. 1 and 2, (x,t) is a combined distribution of a Gaussian λ(x) and inverse power law w(t). For an exemplificative purpose, we present the g profile in Fig. 3, corresponding to Fig. 2. Figure 3 shows that the shape of g nearly reproduces that of
, yet the peaks in are largely smoothed by the Gaussian in . On the other hand, the overall magnitude of g drops as time elapses, which is the property in common with . However, the decrease of g is very slow over a long time. This is a result of the slowly decaying inverse power law w(t). Finally, as converges to 1 in the long-time limit, g converges to , becoming a purely temporal function, or straight lines in the figure. Similarly, for the subdiffusion with drift as shown in Fig. 1, the shape of g will reproduce that of
and exhibit an asymmetry in space instead. Hence we see that the shape of actually stands for the spatial distribution of residing probability of particles, at least in the subdiffusion regime. If > 1, it indicates a higher probability to stay local
s Accordingly, LFP simplifies to LFP = {p/τ + D0 ∂x2 p} and s f (p) reduces to f (p) = p/τ + D(p)∂x2 p, where the superscript s indicates the simplified operator and function. In this case, the constraint that observes is s LFP
= f s (p).
(18)
Solving Eqs. (17) and (18) [combined with D(p) in Eq. (16)] gives the profile under the no-external-field condition. The
FIG. 3. (a) Residing probability profile g (x,t) for attractive particles, corresponding to A (x,t) in Fig. 2(a). (b) Residing probability profile g (x,t) for repulsive particles, corresponding to R (x,t) in Fig. 2(b).
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FIG. 4. From (a) to (b), the most active particles first escape the source, approximately forming a ring. This ring is bigger in the repulsive case than in the attractive case, because repulsion facilitates the escape of these “forerunner” particles, while attraction hinders that. Thus, in (b), two minima of R around x = 0 and two maxima of A around x = 0 are observed (Fig. 2). In the repulsive case, the remaining particles in the red region are to some extent blocked by the first ring, contributing to a maximum of R at x = 0, while for the attractive particles in the red region, they experience the pull from the first ring, making them more inclined to leave, contributing to a minimum of A at x = 0. From (b) to (c), the first ring goes farther and gradually the second ring forms. Nevertheless, the overall particle density in this vicinity has gone down, thus the fluctuation in is attenuated. There can be a hierarchy of such rings. Finally, the particles are sparse enough and local interaction withdraws its influence, making = 1.
compared to the noninteractive case; if < 1, it implies a lower probability to stay local. Now the remaining question is how to understand the morphology of these curves of based on its association with particles’ residing probability. A possible physical context is sketched in Fig. 4. For better visualization, the particles are shown in two dimensions and no external field is present. According to the figure, ’s behavior could be explained by a hierarchy of particle rings and their interaction. The formation of these rings results from the fact that the kinetic energy is not evenly distributed among these particles. Some particles are more active. From Fig. 4(a) to Fig. 4(b), only the most active particles escaped, forming the first ring of particles and then the second, third, and so on. Thus this hierarchy of particle rings to some degree implies the energy levels of these particles. However, the presence of particle-particle interaction modifies the expanding behavior of these rings, which should have been independent of each other. As a result, it causes the nonuniform landscape of . For Figs. 1(d) and 1(e) with drift (rightward), the corresponding rings in Fig. 4 will deform into ellipses and extend to the right, with their long axis in parallel to the direction of the external force. Consequently, the central maximum or minimum zone of will extend its range to the right. Hence we see in Figs. 1(d) and 1(e) the disappearance of the right-hand-side maximum of A and the right-hand-side minimum of R .
IV. MONTE CARLO SIMULATION
Among the previous results, the convergence of to 1 in the long-time limit, or equivalently the convergence of pA and pR to pN , was not fully verified. The p- coupling equation system predicts this convergence, yet it is better to have more direct proof. Therefore, we design a Monte Carlo simulation (MCS) to simulate the mean squared displacement (MSD) of an interacting particle walking system and see whether this convergence ∞ is really there. Since x 2 (t) = −∞ dx x 2 p(x,t), if MSDA (t)
and MSDR (t) converge to MSDN (t) in the long-time limit, it is safe to claim that pA (x,t) and pR (x,t) converge to pN (x,t). For simplicity, this section focuses on the unbiased walk, where the drift is 0. In this case, the ε term in Eq. (15) and u0 term in Eq. (13) disappear. Theoretical lines in Fig. 5 are thus computed by Eqs. (17) and (18).
FIG. 5. Mean squared displacement of particles versus time from the α = 0.5, N = 400, tm = 107 MCS data in log-log scale. Squares represent the pure fractional case where no particle-particle interaction occurs, circles denote the attractive case, and pluses designate the repulsive case. Apart from the apparent convergence of pluses and circles to squares in the long run, which corroborates the previous calculation on the behaviors of pR , pA , and , the way in which they conduct this convergence is worth scrutiny. Compared to the noninteractive case, the repulsive MSD experiences a sequence of acceleration → deceleration → convergence, while the attractive MSD experiences a sequence of retardation → acceleration → convergence. Theoretical lines are also drawn to compare with the simulated data.
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The x axis is discretized into equispaced bins, each with width a. Particles (walkers) can jump into these bins, jump out of these bins, or lodge in these bins with a waiting time span. Due to the limitation of local interaction, particles only interact with those in the same or adjacent bin. The ith bin’s range is (i − 0.5)a x (i + 0.5)a. Allowing for the physical picture described in Sec. III B, we have the following rules. (a) Define s(j )|1j N as the displacement of the j th particle, where N is the total number of particles and the MSD 2 x 2 (t) = N1 N j =1 s(j ) . (b) Assign each particle a clock ξ (j )|1j N to record the remaining time to wait, with ξ (j ) sampled from a power-law kernel function w(t) ∼ Aα (τ/t)1+α [12]. (c) Here m0 (j ) is the number of particles in the bin where particle j resides; m− (j ) is the population in the left adjacent bin, while m+ (j ) is the population in the right adjacency. (d) For a particle that is ready to jump, we assign a probability P+ for jumping right, P− for jumping left, and P0 for a failed jump, i.e., a zero jump length, thus P+ + P− + P0 = 1. After the direction is chosen, we sample a positive jump length from the Gaussian PDF λ(x) ∼ norm(0,σ ). If the particle is still waiting, we assign a probability Pe to end its waiting process faster and 1 − Pe to keep normal waiting. We take σ = 2a. The maximum iteration time is tm . The specific formulas of P+ , P0 , P− , and Pe should vary according to the type of interaction. Attractive particles are more inclined to jump to bins that hold more particles, while the repulsive particles are more likely to jump to less populated bins. Based on this qualitative fact, the MCS adopts simple formulas for those probabilities: For the attractive case, P+ = m+ (j )/[m− (j ) + m0 (j ) + m+ (j )], and P− = P0 = m0 (j )/[m− (j ) + m0 (j ) + m+ (j )], for the repulsive m− (j )/[m− (j ) + m0 (j ) + m+ (j )]; case, P+ = [1/m+ (j )]/[1/m− (j ) + 1/m0 (j ) + 1/m+ (j )], and P0 = [1/m0 (j )]/[1/m− (j ) + 1/m0 (j ) + 1/m+ (j )], In P− = [1/m− (j )]/[1/m− (j ) + 1/m0 (j ) + 1/m+ (j )]. addition, Pe = P+ + P− = [m+ (j ) + m− (j )]/[m− (j ) + m0 (j ) + m+ (j )] for the attractive case and Pe = P+ + P− = [1/m+ (j ) + 1/m− (j )]/[1/m− (j ) + 1/m0 (j ) + 1/m+ (j )] for the repulsive case. If the condition m− (j ) 1 ∧ m0 (j ) 1 ∧ m+ (j ) 1 is not satisfied, the jump is considered free. Notwithstanding the simplicity of these formulas for P+ , P0 , P− , and Pe , they qualitatively represent the particles’ tendency in movement on the microscopic level and they suffice to generate the macroscopic MSD. Furthermore, it is necessary to point out how these rules for manipulating particles relate to the previous sections, especially and , otherwise the comparison between the MCS and the numerical solution would be meaningless. First, all through (a)–(d), the basic procedure is the sampling of jump length and waiting time, which corresponds to . [For the subdiffusive case, λ(x) is Gaussian and w(t) is an inverse power law, as mentioned in the rule description.] Then, as shown in Sec. III B, one of the essential results of is the modification on the jump length and waiting time of an equally probable transition in an otherwise noninteractive case. In the rules above, we use several local probabilities to fulfill this job: P+ , P0 , and P− govern the jump direction and chances of a failed jump and statistically they lead to a perturbed jump length compared to the noninteractive walk; Pe states that the waiting time can be ended faster due to the local
particle-particle interaction and this probability corresponds to ’s perturbation on particle’s waiting time. Therefore, this set of rules simply reiterates the transition probability
[x ,t ,p(x ,t )](x − x ,t − t ). In principle, these rules also refer to the coefficients D(p) and ε(p). [When drift is present, λ(x) ∼ norm(μ,σ ), where μ accounts for the drift.] The function has been calculated through Eq. (15) and apparently it is the deviation of D(p) and ε(p) from the constants D0 and ε0 that contributes to ’s deviation from 1. Since the above rules simulate a nonuniform landscape of , they also correspond to inconstant transport coefficients D(p) and ε(p). More information about the relation between the local probabilities (P+ , P0 , P− , and Pe ) and the transport coefficients (ε0 and D0 ) is in Appendix D. The result of an α = 0.5, N = 400, and tm = 107 MCS is presented in Fig. 5, along with the comparison to theoretical lines given by Eq. (19) and a nondrift version of Eq. (13) [i.e., Eq. (17)]. For a noninteracting fractional advection-diffusion 2D0 t α process governed by Eq. (8), the MSD is x 2 (t) = (1+α) + 2u20 t 2α (1+2α)
[12]. For unbiased diffusion, this reads x 2 (t) |u0 =0 =
2D0 t α . (1 + α)
(19)
The trace of squares in Fig. 5 matches well with Eq. (19). The major information given by Fig. 5 is in two aspects. The first is the verification of the convergence of interactive MSD to the noninteractive. The second is the way in which the convergence is accomplished, as mentioned in the caption of the figure. It is also explained by Fig. 4. For the attractive diffusion, the first few rings expand a relatively short range due to the attraction from the central region, yielding impeded diffusion at early stages. However, when more rings form, the outer rings’ pull on the inner rings actually assists the latter’s expansion, as if they are cooperating to make overall progress. From that point the MSD accelerates to catch up with the noninteractive MSD. For the repulsive diffusion, the first few rings expand to a relatively longer range due to the repulsion from the central region, resulting in an accelerated MSD at the inception. Nevertheless, when more rings form, those earlier rings barricade the inner rings’ expansion, contributing to decelerated MSD. That is, the results of this MCS are consistent with ’s behavior. It might seem that N = 400 is a rather low number for the MCS. Nevertheless, in our practice of the simulation, this number suffices to damp the noise in the MSD data. In fact, previous work [20] also uses particle population of several hundred to implement the Monte Carlo simulation for collective diffusion. Of course, larger N could be better, yet the improvement of accuracy is not tantamount to the soaring computation cost. In addition to the N = 400, tm = 107 MCS, we conducted more simulations with different particle population. It is shown that the convergence time rises drastically due to the increase of population N . Unfortunately, real systems often feature large particle population, leading to practically unreachable convergence time. Furthermore, particles can be confined in closed boundaries. If so, they cannot propagate far enough from the source and the local particle-particle interaction could by no means be neglected. Now we need to solve Eq. (13) with a closed boundary condition. To this end, one could expect that
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the perturbation in Figs. 2, 1(d), or 1(e) will not die out. Alternatively, if this physical boundary is very distant from the particle source, the perturbation behaves like that in the free-boundary condition. Therefore, the convergence of the interactive diffusion to the noninteractive in the long-time limit is only true under the free-boundary condition, where particles can travel infinitely far. The previous solution for p and and the Monte Carlo simulation are all based on this free-boundary condition. Nonetheless, these results reveal the connections between the classical CTRW and that with local particle-particle interaction and give the conditions for their equivalence: the long-time limit and free-boundary condition. V. CONCLUSION
We have modified the generalized Chapman-Kolmogorov equation by introducing a perturbation function which models the influence of local particle-particle interaction on an individual particle’s motion and the physical picture it reveals was presented. Consequently, we proposed a continuous time random walk model with local particle-particle interaction. We examined its properties in greater detail in a time-fractional regime. Under a free-boundary condition, the derived equation (13) solves the transient behaviors of the propagator p and perturbation . Both attractive and repulsive interactions were investigated, including direct numerical computation and Monte Carlo simulation. The results of the equation and the simulation have consistently shown that at early stages repulsion expedites the growth of mean squared displacement and attraction retards it, while for long enough time these two cases converge to the noninteracting one. Hence the conventional CTRW may be regarded as a special case of the interacting CTRW proposed here. The conditions for their equivalence are a long-time limit and free-boundary condition. In any other system where the two conditions are not simultaneously met, this nonlinear CTRW model works differently from the classical noninteractive one. ACKNOWLEDGMENTS
This work was supported by the National Science and Technology Major Project of China under Grant No. 2017ZX05009003 and the National Natural Science Foundation of China under Grants No. 51521063 and No. 51474231.
An inverse Laplace transform of Eq. (A2) leads to t ∞ dx dt pg (x ,t |x0 ,0) p(x,t|x0 ,0) = −∞
0
×(x − x ,t − t ) + (t)δx,x0 ,
(A3)
where ˆ pg (x ,t |x0 ,0) = L −1 {(s)L { [x ,t ,p]q(x ,t |x0 ,0)}(s)}(t ) t dt [x ,t ,p]q(x ,t |x0 ,0)(t − t ). = 0
(A4) Provided is a fluctuation around 1, if we assume ≈ 1, the above integral can be evaluated as t pg (x ,t |x0 ,0) ≈ [x ,t ,p] dt q(x ,t |x0 ,0)(t − t ) 0
= [x ,t ,p]p(x ,t |x0 ,0).
(A5)
Eq. (A5) into Eq. (A3) and considering p(x,t) = Substituting ∞ −∞ dx0 p(x,t|x0 ,0), one obtains Eq. (10). APPENDIX B: DERIVATION OF EQ. (13)
Starting from Eq. (12) and using Eq. (7), one has ˆˆ ˆˆ ˆ s p(k,s) − pˆ0 (k) = ψ(s)[− p(k,s) +(1 − ilk − σ 2 k 2 )F L { p}(k,s)]. (B1) ˆ Since w(s) ˆ = 1 − (τ s)α , ψ(s) = s w(s)/[1 ˆ − w(s)] ˆ scales as s 1−α /τ α , where τ is a scaling factor in Eq. (7) which is also a characteristic time scale for particle motion [12]. On writing ˆ ψ(s) = s 1−α /τ α and inserting it into Eq. (B1), one obtains ˆˆ F L { p}(k,s) − p(k,s) ˆˆ s p(k,s) − pˆ0 (k) = s 1−α α τ
2 − (iu0 k + D0 k )F L { p}(k,s) , (B2) where u0 = l/τ α and D0 = σ 2 /τ α are, respectively, the drift constant and diffusion constant. Since L −1 {s 1−α fˆ(s)} = 1−α f (t), an inverse Fourier-Laplace transform of Eq. (B2) 0 Dt gives ∂t p = 0 Dt1−α [( − 1)/τ α ]p + D0 ∂x2 ( p) − u0 ∂x ( p) , which is exactly Eq. (13).
APPENDIX A: DERIVATION OF EQ. (10) APPENDIX C: MORE DETAILS ON THE LOCALIZATION IN EQ. (A5)
Performing a Laplace transform over Eq. (9) yields ∞ ˆ q(x,s|x dx L { [x ,t ,p]q(x ,t |x0 ,0)}(s) 0 ,0) = −∞
ˆ − x ,s) + δx,x0 . ×(x
(A1)
ˆ ˆ ˆ Since Eq. (3) gives p(x,s|x 0 ,0) = q(x,s|x 0 ,0)(s), Eq. (A1) engenders ∞ ˆ p(x,s|x dx L { [x ,t ,p]q(x ,t |x0 ,0)}(s) 0 ,0) = −∞
ˆ ˆ (x ˆ − x ,s) + (s)δ ×(s) x,x0 .
(A2)
In Eq. (A5) the localization of in time is adopted as an approximation such that we can write pg (x,t|x0 ,0) =
[x,t,p]p(x,t|x0 ,0), which is a crucial step by which Eq. (13) is subsequently derived. From a purely computational perspective, following Eq. (A5) will not lead to significant error if the range by which fluctuates around 1 is small. Yet to provide more details as to why this localization is necessary, let us discard this approximation at first. If we start from Eq. (A3) and do not employ the localization, after the same procedure as in Appendix B, we will arrive at
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the governing equation for p as ∂t p = 0 Dt1−α (pg − p)/τ α + D0 ∂x2 ( pg ) − u0 ∂x ( pg ) ,
latter with the former. As shown in the main text, one must require that does not stray far from 1 to ensure that the transition probability is within 1.
(C1)
∞
where pg ≡ pg (x,t) = −∞ dx0 pg (x,t|x0 ,0) and pg (x,t|x0 ,0) is exactly Eq. (A4): t pg (x,t|x0 ,0) = dt [x,t ,p]q(x,t |x0 ,0)(t − t ). (C2) 0
Equation (C1) is the rigorous result without any approximation. When = 1, pg = p and it reduces to Eq. (8). To evaluate Eq. (C2), one can expand [x,t ,p] at t in a Taylor series. In doing so, pg (x,t|x0 ,0) is then ( ’s differentiability over time is assumed) t ∞ (−(t − t ))n ∂ n [x,t,p] pg (x,t|x0 ,0) = dt n! ∂t n 0 n=0 × q(x,t |x0 ,0)(t − t ) = [x,t,p]p(x,t|x0 ,0) +
∞ (−1)n ∂ n [x,t,p] n=1
where
t
R (n) (x,t) =
n!
∂t n
APPENDIX D: RELATION BETWEEN LOCAL PROBABILITIES AND TRANSPORT COEFFICIENTS
In the Monte Carlo simulation, the local probabilities (P+ , P0 , P− , and Pe ) play an important role. As mentioned, P+ is the probability for a particle to jump right, P− is the probability to jump left, P0 is the probability to abort a jump, and Pe = P+ + P− is defined as the probability to activate a residing particle. In this appendix we present more details about how they relate to the transport coefficients (ε0 and D0 ). First, let us consider the simplest situation: a noninteractive convection-diffusion process without waiting time. In this case, any intended jump does not fail and P+ + P− = 1, P0 = 0, and Pe = 1, which means that the particle jumps either left or right and never settles. Following Weiss [35], the probability of a particle to stay in the ith bin at t = t + t is determined by the discrete master equation
R (n) (x,t), (C3)
dt (t − t )n q(x,t |x0 ,0)(t − t ).
(C4)
0
Clearly, R (n) (x,t) is the average value of (t − t )n at location x until t, i.e., the nth moment of the particle’s residing time (waiting time) at location x until t. If we rely on Eq. (C1) to solve for p and , we would need preknowledge of R (n) (x,t). Unfortunately, R (n) (x,t) is dependent on q(x,t|x0 .0), which, according to Eq. (9), is dependent on . However, if we can discard the series items in Eq. (C3), the R (n) (x,t) term could be thrown away and we get a simple p- coupling. To achieve this, we mustnrequire that the change rate of at t is appreciably small, i.e., ∂∂t n ≈ 0. Then Eq. (C3) reduces to Eq. (A5) and Eq. (C1) reduces to Eq. (13). We notice that a slowly varying indeed emerges from the calculation results, at least in the subdiffusion case shown in Figs. 1 and 2. The oscillation range of from 102 to 105 s is appreciably small and the change of within (104 ,105 ) is much smaller than that within (102 ,103 ), which means it would be safer in the longer time to make the assumption of slowly varying . Another fact that compels us to keep the assumption of a slowly varying is from the asymptotic expansion in Eq. (7), through which the derivation of these equations becomes possible. The (k,s) → (0,0) condition already states a long-time condition. Practically speaking, this long-time condition can be valid in the time scale of our computation: 101 s. It is regarded as a long time relative to the characteristic time scale of those tiny particles’ motion (see Ref. [12], p. 18), which could possibly scale as less than 1 ms. However, when we mention that converges to 1 in the long-time limit, it means that this limit is really large even compared to our range of computation or the routine laboratory measure of time, e.g., a scale on the order of 106 or 107 s or even higher. Finally, although a slowly varying
is a weaker assumption than ≈ 1, we cannot replace the
pi (t + t) = P+ pi−1 (t) + P− pi+1 (t).
(D1)
In the continuous limit t → 0 and a → 0 (as mentioned in Sec. IV, a is the bin width), one could expand pi (t + t) and pi±1 (t) in a Taylor series pi (t + t) = pi (t) + t∂t pi (t) + O([t]2 ), pi±1 (t) = pi (t) ± a∂x pi (t) +
a2 2 ∂ pi (t) + O(a 3 ). (D2) 2 x
Substituting Eq. (D2) into Eq. (D1), after some algebraic manipulations one obtains the classical convection-diffusion equation ∂t p = −u0 ∂x p + D0 ∂x2 p,
(D3)
where u0 = D0 =
lim
(P+ − P− )
a , t
lim
(P+ + P− )
a2 . 2t
t→0,a→0
t→0,a→0
(D4)
The above procedure is available in, e.g., Ref. [36]. Considering u0 = −ε0 ∂x U , the relation (D4) also gives −ε0 ∂x U = a limt→0,a→0 (P+ − P− ) t . Hence we see that the difference of P+ and P− determines the direction of the drift, while the diffusion constant is not affected (because P+ + P− = 1 in this case). Practically, at the individual particle level, t and a do not approach absolute zero. Instead, they approach the characteristic length and time scales of the particle’s movement, which are very small. Based on Eq. (D1), it is convenient to consider a powerlaw waiting time PDF w(t). We follow the generalized master equation in Ref. [15]: t pi (t) = dt w(t − t )[P+ pi−1 (t )+P− pi+1 (t )]+δx,x0 (t).
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Repeating the procedures in Ref. [15], Eq. (D5) leads to u0 = lim (P+ − P− ) a→0
a a2 , D0 = lim (P+ + P− ) α , (D6) α a→0 τ 2τ
where τ is exactly the τ in Eq. (7). Thus −ε0 ∂x U = lima→0 (P+ − P− ) τaα if a long waiting time is involved. Since 2 P+ + P− = 1, we can also write D0 = lima→0 2τa α . Now let us consider both local particle-particle interaction and long-time waiting. As analyzed in the main text, in the interactive case, these local probabilities depend on the particle density field and evolve with respect to space and time. Furthermore, in the MCS we add a probability P0 to abort a jump. Now we have P+ + P− + P0 = 1. Similar to Ref. [15], considering expansion of P+ pi−1 (t) and P− pi+1 (t) at x, it reads P± pi∓1 (t) ∼ P± p(x,t) ∓ a∂x [P± p(x,t)] +
a2 2 ∂ [P± p(x,t)]. 2 x (D7)
Inserting Eq. (D7) into Eq. (D5) and then conducting a Laplace transform, after some algebraic manipulations one obtains 1 − w(s) ˆ ˆ p(x,s) − δx,x0 s = w(s) ˆ L {P+ p(x,t)}(s) + L {P− p(x,t)}(s) − a∂x L {(P+ − P− )p(x,t)}(s) +
a2 2 ∂x L {Pe p(x,t)}(s) , 2 (D8)
where Pe = P+ + P− is used. Note that L {P+ p(x,t)}(s) + L {P− p(x,t)}(s) + L {P0 p(x,t)}(s) = L {(P+ + P− + P0 )p ˆ (x,t)} = L {p(x,t)} = p(x,s). Using this relation, Eq. (D8) could be recast into ˆ s p(x,s) − δx,x0 s w(s) ˆ −L {P0 p(x,t)}(s) = 1 − w(s) ˆ
a2 2 − a∂x L {(P+ − P− )p(x,t)}(s) + ∂x L {Pe p(x,t)}(s) . 2 (D9)
[1] [2] [3] [4] [5] [6]
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s w(s) ˆ ˆ ˆ is exactly ψ(s) and ψ(s) = s 1−α /τ α . Taking Recall that 1− w(s) ˆ this into account, we can conduct an inverse Laplace transform of Eq. (D9), which yields ∂t p = 0 Dt1−α −(P0 /τ α )p + D0 ∂x2 (Pe p) − u0 ∂x [(P+ − P− )p] , (D10) 2
where u0 = lima→0 τaα and D0 = lima→0 2τa α . Equation (D10) has the same structure as Eq. (13), which means that the local probabilities in the MCS have led to the same mathematical structure that results from . This is a hidden reason why the data generated by the MCS can be fitted by the theoretical curve as shown in Fig. 5. The above analysis also reveals the relation between the local probabilities and the transport coefficients ε0 and D0 . When there is no particle-particle interaction, only P− and P+ work and they constitute the transport coefficients, as shown by Eqs. (D4) and (D6). The difference of P+ and P− governs the direction of the drift. When the local particle-particle interaction is present, local probabilities become dependent on space and time. Now they do not directly constitute the transport coefficients, but they collaborate with the coefficients to determine the mathematical structure of the local density change rate. Again, the difference of P+ and P− still governs the direction of the drift. Note that this drift is not caused by an external field but by local particleparticle interaction. Hence this drift is local and its direction can vary in space, according to the difference of P+ and P− . If P+ − P− at a particular location happens to be zero, there would be no drift locally. In contrast, the effect of an external field is global such that every particle experiences a drift. In Sec. IV we adopted simple expressions for P+ and P− , which depend on the local particle population. These expressions are qualitatively true in the repulsive and attractive cases that we simulated. More generally, P+ and P− are assumed to be P+ = P e−[(γ −1)Vi +γ Vi+1 +U/2]/kB T and P− = P e−[(γ −1)Vi+1 +γ Vi −U/2]/kB T [20], where Vi is the mean-field potential at the ith bin, U = Ui+1 − Ui , P is a constant within (0,1), kB is the Boltzmann constant, T is the temperature, and γ is the same γ in Eq. (16). In the special cases of repulsion and attraction, this formulation not only gives the relation (16), but also leads to the fact that attractive particles tend to jump to more populated bins and repulsive particles tend to jump to less populated bins (more details are in Ref. [20]). The expressions we adopted for P+ and P− in Sec. IV just approximate this tendency at the particle level.
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